Appendix A. Measure and integration

Transcription

1 Appendix A. Measure and integration We suppose the reader is familiar with the basic facts concerning set theory and integration as they are presented in the introductory course of analysis. In this appendix, we review them briefly, and add some more which we shall need in the text. Basic references for proofs and a detailed exposition are, e.g., [[ Hal 1 ]], [[ Jar 1,2 ]], [[ KF 1,2 ]], [[ LiL ]], [[ Ru 1 ]], or any other textbook on analysis you might prefer. A.1 Sets, mappings, relations A set is a collection of objects called elements. The symbol cardx denotes the cardinality of the set X. The subset M consisting of the elements of X which satisfy the conditions P 1 (x),...,p n (x) is usually written as M = { x X : P 1 (x),...,p n (x) }. A set whose elements are certain sets is called a system or family of these sets; the family of all subsystems of a given X is denoted as 2 X. The operations of union, intersection, and set difference are introduced in the standard way; the first two of these are commutative, associative, and mutually distributive. In a system {M α } of any cardinality, the de Morgan relations, X \ M α = (X \ M α ) and X \ M α = (X \ M α ), α α α α are valid. Another elementary property is the following: for any family {M n },whichis at most countable, there is a disjoint family {N n } of the same cardinality such that N n M n and n N n = n M n.theset (M \ N) (N \ M) is called the symmetric difference of the sets M,N and denoted as M N. It is commutative, M N = N M, and furthermore, we have M N =(M N)\(M N) andm N =(X\M) (X\N) for any X M N. The symmetric difference is also associative, M (N P )=(M N) P, and distributive with respect to the intersection, (M N) P =(M P ) (N P ). A family R is called a set ring if M N R and M N R holds for any pair M,N R. The relation M \ N =(M N) M also gives M \ N R, and this in turn implies R and M N R. If the symmetric difference and intersection are understood as a sum and product, respectively, then a set ring is a ring in the sense of the general algebraic definition of Appendix B.1. A.1.1 Example: Let J d be the family of all bounded intervals in R d,d 1. The family R d, which consists of all finite unions of intervals J J d together with the empty set, is a set ring, and moreover, it is the smallest set ring containing J d. As mentioned above, any R R d can be expressed as a finite union of disjoint bounded intervals. 595

2 596 Appendix A Measure and integration A set ring R 2 X is called a set field if it contains the set X (notice that in the terminology of Appendix B.1 it is a ring with a unit element but not an algebra). A set field A 2 X is called a σ field if n=1 M n A holds for any countable system {M n } n=1 A. De Morgan relations show that also n=1 M n A, and furthermore that a family A 2 X containing the set X is a σ field iff X \ M A for all M A and n M n A for any at most countable subsystem {M n } A. Given a family S 2 X we consider all σ fields A 2 X containing S (there is at least one, A =2 X ). Their intersection is again a σ field containing S; wecallittheσ field generated by S and denote it as A(S). A.1.2 Example: The elements of B d := A(J d ) are called Borel sets in R d. In particular, all the open and closed sets, and thus also the compact sets, are Borel. The σ field B d is also generated by other systems, e.g., by the system of all open sets in R d. In general, Borel sets in a topological space (X, τ) are defined as the elements of the σ field A(τ). A sequence {M n } n=1 is nondecreasing or nonincreasing if M n M n+1 or M n M n+1, respectively, holds for n =1, 2,... A set family M is monotonic if it contains the set n M n together with any nondecreasing sequence {M n },and n M n together with any nonincreasing sequence {M n }.Anyσ field represents an example of a monotonic system. To any S there is the smallest monotonic system M(S) containing S and we have M(S) A(S). If R is a ring, the same is true for M(R); in addition, if M R M M(R), then M(R) isaσ field and M(R) = A(R). A mapping (or map) f from a set X to Y is a rule, which associates with any x X a unique element y f(x) of the set Y ;wewrite f : X Y and also x f(x). If Y = R or Y = C the map f is usually called a real or a complex function, respectively. It is also often useful to consider maps which are defined on a subset D f X only. The symbol f : X Y must then be completed by specifying the set D f which is called the domain of f; wedenoteitalsoas D(f). If D f is not specified, it is supposed to coincide with X. The sets Ran f := { y Y : y = f(x), x D f } and Ker f := { x D f : f(x) =0} are the range and kernel of the map f, respectively. A map f : X Y is injective if f(x) =f(x ) holds for any x, x X only if x = x ; it is surjective if Ran f = Y. A map which is simultaneously injective and surjective is called bijective or a bijection. The sets X and Y have the same cardinality if there is a bijection f : X Y with D f = X. The relation f = g between f : X Y and g : X Y means by definition D f = D g and f(x) =g(x) for all x D f.if D f D g and f(x) =g(x) holds for all x D g we say that f is an extension of g while g is a restriction of f to the set D g ;wewrite f g and g = f \ D g. A.1.3 Example: For any X M we define a real function χ M : χ M (x) =1ifx M, χ M (x) =0ifx X \ M; it is called the characteristic (or indicator) function of the set M. The map M χ M is a bijection of the system 2 X to the set of all functions f : X R such that Ran f = {0, 1}. The set f ( 1) (N) :={ x D f : f(x) N } for given f : X Y and N Y is called the pull back of the set N by the map f. One has f ( 1) ( α I N ) α = α I f ( 1) (N α )for any family {N α } Y, and the analogous relation is valid for intersections. Furthermore, f ( 1) (N 1 \ N 2 )=f ( 1) (N 1 ) \ f ( 1) (N 2 )andf(f ( 1) (N) =Ranf N. On the other hand, f ( 1) (f(m)) M; the inclusion turns to identity if f is injective. A.1.4 Example: Let f : X Y with D f = X. Toaσ field B 2 Y we can construct the family f ( 1) := { f ( 1) (N) : N B}, which is obviously again a σ field. Similarly, if

3 A.1 Sets, mappings, relations 597 A 2 X is a σ field, then the same is true for { N Y : f ( 1) (N) A}. Hence for any family S 2 Y we can construct the σ field f ( 1) (A(S)) 2 X and the latter coincides with the σ field generated by f ( 1) (S), i.e., f ( 1) (A(S)) = A(f ( 1) (S)). Given f : X Y and g : Y Z we can define the composite map g f : X Z with the domain D(g f) :=f ( 1) (D g )=f ( 1) (D g Ran f) by (g f)(x) :=g(f(x)). We have (g f)(p ) ( 1) = f ( 1) (g ( 1) (P )) for any P Z. If f : X Y is injective, then for any y Ran f there is just one x y D f such that y = f(x y ); the prescription y g(y) :=x y defines a map g : Y X, whichis called the inverse of f and denoted as f 1.Wehave D(f 1 )=Ranf, Ran f 1 = D f and f 1 (f(x)) = x, f(f 1 (y)) = y for any x D f and y Ran f, respectively. These relations further imply f ( 1) (N) =f 1 (N) for any N Ran f. Oftenwehaveapairof mappings f : X Y and g : Y X and we want to know whether f is invertible and f 1 = g; this is true if one of the following conditions is valid: (i) D g =Ranf and g(f(x)) = x for all x D f (ii) Ran f D g, g(f(x)) = x for all x D f and Ran g D f, f(g(y)) = y for all y D g If f : X Y is injective, then f 1 is also injective and (f 1 ) 1 = f. If g : Y Z is also injective, then the composite map g f is invertible and (g f) 1 = f 1 g 1. The Cartesian product M N is the set of ordered pairs [x, y] with x M and y N; the Cartesian product of the families S and S is defined by S S := { M N : M S,N S }. For instance, the systems of bounded intervals of Example 1 satisfy J m+n = J m J n.if M N is empty, then either M = or N =. On the other hand, if M N is nonempty, then the inclusion M N P R implies M P and N R. Wehave (M P ) N =(M N) (P N) and similar simple relations for the intersection and set difference. Notice, however, that (M N) (P R) can be expressed in the form S T only if M = P or N = R. The definition of the Cartesian product extends easily to any finite family of sets. Alternatively, we can interpret M 1 M n as the set of maps f : {1,...,n} n j=1 M j such that f(j) M j. This allows us to define X α I M α for a system { M α : α I } of any cardinality as the set of maps f : I α I M α which fulfil f(α) M α for any α I. The existence of such maps is related to the axiom of choice (see below). Given f : X C and g : Y C, we define the function f g on X Y by (f g)(x, y) :=f(x)g(y). Let M X Y ; then to any x X we define the x cut of the set M by M x := { y Y :[x, y] M}; we define the y cuts analogously. Let A 2 X, B 2 Y be σ fields; then the σ field A(A B) is called the direct product of the fields A and B and is denoted as A B. A.1.5 Example: The Borel sets in R m and R n in this way generate all Borel sets in R m+n, i.e., wehave B m B n = B m+n. On the other hand, the cuts of a set M A B belong to the original fields: we have M x B and M y A for any x X and y Y, respectively. A subset R ϕ X X defines a relation ϕ on X: if [x, y] R ϕ we say the element x is in relation with y and write xϕy. A common example is an equivalence, which is a relation on X that is reflexive ( x x for any x X ), symmetric ( x y implies y x ), and transitive ( x y and y z imply x z ). For any x X we define the

4 598 Appendix A Measure and integration equivalence class of x as the set T x := { y X : y x }. Wehave T x = T y iff x y, so the set X decomposes into a disjoint union of the equivalence classes. Another important example is a partial ordering on X, which means any relation that is reflexive, transitive, and antisymmetric, i.e., such that the conditions x y and y x imply x = y. If X is partially ordered, then a subset M X is said to be (fully) ordered if any elements x, y M satisfy either x y or x y. An element x X is an upper bound of a set M X if y x holds for all y M; itisamaximal element of M if for any y M the condition y x implies y = x. A.1.6 Theorem (Zorn s lemma): Let any ordered subset of a partially ordered set X have an upper bound; then X contains a maximal element. Zorn s lemma is equivalent to the so called axiom of choice, which postulates for a system { M α : α I } of any cardinality the existence of a map α x α such that x α M α for all α I see, e.g., [[ DS 1 ]], Sec.I.2, [[ Ku ]], Sec.I.6. Notice that the maximal element in a partially ordered set is generally far from unique. A.2 Measures and measurable functions Let us have a pair (X, A), where X is a set and A 2 X a σ field. A function f : X R is called measurable (with respect to A )if f ( 1) (J) A holds for any bounded interval J R, i.e., f ( 1) (J ) A. This is equivalent to any of the following statements: (i) f ( 1) ((c, )) A for all c R, (ii) f ( 1) (G) A for any open G R, (iii) f ( 1) (B) A. If X is a topological space, a function f : X R is called Borel if it is measurable w.r.t. the σ field B of Borel sets in X. A.2.1 Example: Any continuous function f : R d R is Borel. Furthermore, let f : X R be measurable (w.r.t. some A )and g : R R be Borel; then the composite function g f is measurable w.r.t. A. If functions f,g : X R are measurable, then the same is true for their linear combinations af + bg and product fg as well as for the function x (f(x)) 1 provided f(x) 0 forall x X. Even if the last condition is not valid, the function h, defined by h(x) :=(f(x)) 1 if f(x) 0 and h(x) := 0 otherwise, is measurable. Furthermore, if a sequence {f n } converges pointwise, then the function x lim n f n (x) is again measurable. The notion of measurability extends to complex functions: a function ϕ : X C is measurable (w.r.t. A ) if the functions Re ϕ( ) and Im ϕ( ) are measurable; this is true iff ϕ ( 1) (G) A holds for any open set G C. A complex linear combination of measurable functions is again measurable. Furthermore, if ϕ is measurable, then ϕ( ) is also measurable. In particular, the modulus of a measurable f : X R is measurable, as are the functions f ± := 1 2 ( f ±f). A function ϕ : X C is simple (σ simple) if ϕ = n y nχ Mn, where y n C and the sets M n A form a finite (respectively, at most countable) disjoint system with n M n = X. By definition, any such function is measurable; the sets of (σ )simple functions are closed with respect to the pointwise defined operations of summation, multiplication, and scalar multiplication. The expression ϕ = n y nχ Mn is not unique, however, unless the numbers y n are mutually different.

5 A.2 Measures and measurable functions 599 A.2.2 Proposition: A function f : X R is measurable iff there is a sequence {f n } of σ simple functions, which converges to f uniformly on X. If f is bounded, there is a sequence of simple functions with the stated property. In fact, the approximating sequence {f n } canbechoseneventobenondecreasing. If f is not bounded it can still be approximated pointwise by a sequence of simple functions, but not uniformly. Given (X, A) and(y,b) we can construct the pair (X Y,A B). Let ϕ : M C be a function on M A B; then its x cut is the function ϕ x defined on M x by ϕ x (y) :=ϕ(x, y); we define the y cut similarly. Cuts of a measurable functions may not be measurable in general, however, it is usually important to ensure measurability a.e. cf. Theorem A.3.13 below. A mapping λ defined on a set family S and such that λ(m) is either non negative or λ(m) =+ for any M S is called (a non negative) set function. Itismonotonic if M N implies λ(m) λ(n), additive if λ(m N) =λ(m)+λ(n) for any pair of sets such that M N S and M N =, andσ additive if the last property generalizes, λ ( n M n)= n λ(m n), to any disjoint at most countable system {M n } S such that n M n S. A set function µ, which is defined on a certain A 2 X,isσ additive, and satisfies µ( ) = 0 is called a (non negative) measure on X. If at least one M A has µ(m) <, then µ( ) = 0 is a consequence of the σ additivity. The triplet (X, A,µ) is called a measure space; the sets and functions measurable w.r.t. A are in this case often specified as µ measurable. A set M A is said to be µ zero if µ(m) = 0, a proposition valued function defined on M A is valid µ almost everywhere if the set N M, onwhichit is not valid, is µ zero. A measure µ is complete if N M implies N A for any µ zero set M; below we shall show that any measure can be extended in a standard way to a complete one. Additivity implies that any measure is monotonic, and µ(m N) = µ(m) + µ(n) µ(m N) µ(m) + µ(n) for any sets M,N A which satisfy µ(m N) <. Using the σ additivity, one can check that lim k µ(m k )=µ ( n=1 M n) holds for any nondecreasing sequence {M n } A, and a similar relation with the union replaced by intersection is valid for nonincreasing sequences. A measure µ is said be finite if µ(x) < and σ finite if X = n=1 M n, where M n A and µ(m n ) < for n =1, 2,... Let (X, τ) be a topological space, in which any open set can be expressed as a countable union of compact sets (as, for instance, the space R d ; recall that an open ball there is a countable union of closed balls). Suppose that µ is a measure on X with the domain A τ; then the following is true: if any point of an open set G has a µ zero neighborhood, then µ(g) = 0. Given a measure µ we can define the function ϱ µ : A A [0, ) byϱ µ (M N) := µ(m N). The condition µ(m N) = 0 defines an equivalence relation on A and ϱ µ is a metric on the corresponding set of equivalence classes. A point x X such that the one point set {x} belongs to A and µ({x}) 0 is called a discrete point of µ; the set of all such points is denoted as P µ. If µ is σ finite the set P µ is at most countable. A measure µ is discrete if P µ A and µ(m) =µ(m P µ ) for any M A.

6 600 Appendix A Measure and integration A measure µ is said to be concentrated on a set S A if µ(m) =µ(m S) for any M A. For instance, a discrete measure is concentrated on the set of its discrete points. If (X, τ) is a topological space and τ A, then the support of µ denoted as supp µ is the smallest closed set on which µ is concentrated. Next we are going to discuss some ways in which measures can be constructed. First we shall describe a construction, which starts from a given non negative σ additive set function µ defined on a ring R 2 X ; we assume that there exists an at most countable disjoint system {B n } R such that n B n = X and µ(b n ) < for n =1, 2,... Let S be the system of all at most countable unions of the elements of R; it is closed w.r.t. countable unions and finite intersections, and M \R Sholds for all M Sand R R. Any M S can be expressed as M = j R j, where {R j } R is an at most countable disjoint system; using it we can define the set function µ on S by µ(m) := j µ(r j). It is monotonic and σ additive. Furthermore, we have µ ( n M n) n µ(m n); this property is called countable semiadditivity. Together with the monotonicity, it is equivalent to the fact that M k=1 M k implies µ(m) k=1 µ(m k). The next step is to extend the function µ to the whole system 2 X by defining the outer measure by µ (A) :=inf{ µ(m) : M S,M A }. The outer measure is again monotonic and countably semiadditive; however, it is not additive so it is not a measure. Its importance lies in the fact that the system A µ := {A X : inf M S µ (A M) =0} is a σ field. This finally allows us to define µ := µ \ A µ ;itisacompleteσ additive measure on the σ field A µ A(R), which is determined uniquely by the set function µ in the sense that any measure ν on A(R), which is an extension of µ, satisfies ν = µ \ A(R). The measure µ is called the Lebesgue extension of µ. A measure µ on a topological space (X, τ) is called Borel if it is defined on B A(τ) and µ(c) < holds for any compact set C. We are particularly interested in Borel measures on R d, where the last condition is equivalent to the requirement µ(k) < for any compact interval K R d. Any Borel measure on R d is therefore σ additive and corresponds to a unique σ additive set function µ on R d. The space R d, however, has the special property that for any bounded interval J J d we can find a nonincreasing sequence of open intervals I n J and a nondecreasing sequence of compact intervals K n J such that n I n = n K n = J. This allows us to replace the requirement of σ additivity by the condition µ(j) =inf{ µ(i) : I G J } =sup{ µ(k) : K F J } for any J J d, where G J J d is the system of all open intervals containing J, and F J J d is the system of all compact intervals contained in J. A set function µ on J d which is finite, additive, and fulfils the last condition is called regular. A.2.3 Theorem: There is a one to one correspondence between regular set functions µ and µ := µ \ B d on R d. In particular, Borel measures µ and ν coincide if µ(j) =ν(j) holds for any J J d. A.2.4 Example: Let f : R R be a nondecreasing right continuous function. For any a, b R, a<b,weset µ f (a, b) :=f(b 0) f(a), µ f (a, b] :=f(b) f(a), and analogous expressions for the intervals [a, b] and [a, b]. This defines the regular set function µ f on

7 A.3 Integration 601 J 1 ; the corresponding Borel measure µ f is called the Lebesgue Stieltjes measure generated by the function f. In particular, if f is the identical function, f(x) = x, wespeak about the Lebesgue measure on R. Let us remark that the Lebesgue Stieltjes measure is sometimes understood as a Lebesgue extension with a domain which is generally dependent on f; however, it contains B in any case. A.2.5 Example: Let µ and ν be regular set functions on J m R m and J n R n, respectively; then the function ϱ on J m+n defined by ϱ(j L) := µ(j) ν(l) is again regular; the corresponding Borel measure is called the direct product of the measures µ and ν which correspond to µ and ν, respectively, and is denoted as µ ν. In particular, repeating the procedure d times, we can in this way construct the Lebesgue measure on R d which associates its volume with every parallelepiped. A.2.6 Proposition: Any Borel measure on R d is regular, i.e., µ(b) =inf{µ(g) : G B, G open} =sup{µ(c) : C B, C compact}. As a consequence of this result, we can find to any B B d a nonincreasing sequence of open sets G n B and a nondecreasing sequence of compact sets C n B (both dependent generally on the measure µ ) such that µ(b) = lim n µ(g n )=µ ( n G n)= lim n µ(c n )=µ ( n C n). Proposition A.2.6 generalizes to Borel measures on a locally compact Hausdorff space, in which any open set is a countable union of compact sets see [[ Ru 1 ]], Sec Let us finally remark that there are alternative ways to construct Borel measures. One can use, e.g., the Riesz representation theorem, according to which Borel measures correspond bijectively to positive linear functionals on the vector space of continuous functions with a compact support cf. [[ Ru 1 ]], Sec.2.14; [[ RS 1 ]], Sec.IV.4. A.3 Integration Now we shall briefly review the Lebesgue integral theory on a measure space (X, A,µ). It is useful from the beginning to consider functions which may assume infinite values; this requires to define the algebraic operations a + :=, a := for a > 0 and a := 0 for a =0,etc., to add the requirement f ( 1) ( ) A to the definition of measurability, and several other simple modifications. Given a simple non negative function s := n y nχ Mn on X, we define its integral by X sdµ:= n y nµ(m n ); correctness of the definition follows from the additivity of µ. In the next step, we extend it to all measurable functions f : X [0, ] putting X fdµ X { } f(x) dµ(x) :=sup sdµ : s S f, X where S f is the set of all simple functions s : X [0, ) such that s f. Wealso define M fdµ:= X fχ M dµ for any M A; in this way we associate with the function f and the set M anumberfrom [0, ], which is called the (Lebesgue) integral of f over M w.r.t. the measure µ. A.3.1 Proposition: Let f, g be measurable functions X [0, ] andm A; then M (kf) dµ = k M fdµ holds for any k [0, ), and moreover, the inequality f g implies M fdµ M gdµ.

8 602 Appendix A Measure and integration Notice that the integral of f = 0 is zero even if µ(x) =. On the other hand, the relation X ϕ dµ = 0 for any measurable function ϕ : X C implies that µ({ x X : ϕ(x) 0}) =0,i.e., that the function ϕ is zero µ a.e. Let us turn to limits which play the central role in the theory of integration. A.3.2 Theorem (monotone convergence): Let {f n } be a nondecreasing sequence of nonnegative measurable functions; then lim n X f n dµ = X (lim n f n ) dµ. The right side of the last relation makes sense since the limit function is measurable. However, we often need some conditions under which both sides are finite. The corresponding modification is also called the monotone convergence (or Levi s) theorem: if {f n } is a nondecreasing sequence of non negative measurable functions and there is a k>0such that X f n dµ k for n =1, 2,..., then the function x f(x) := lim n f n (x) is µ a.e. finite and lim n X f n dµ = X fdµ k. The monotone convergence theorem implies, in particular, that the integral of a measurable function can be approximated by a nondecreasing sequence of integrals of simple functions. A.3.3 Corollary (Fatou s lemma): X (liminf n f n ) dµ liminf n X f n dµ holds for any sequence of measurable functions f n : X [0, ]. This result has the following easy consequence: let a sequence {f n } of non negative measurable functions have a limit everywhere, lim n f n (x) =f(x), and X f n dµ k for n =1, 2,...; then X fdµ k. Applying the monotone convergence theorem to a sequence {f n } of non negative measurable functions we get the relation X ( n=1 f n) dµ = n=1 X f n dµ. In particular, if f : X [0, ] is measurable and {M n } n=1 A is a disjoint family with n=1 M n = M, then putting f n := fχ Mn we get M fdµ = n=1 M n fdµ. This relation is called σ additivity of the integral; it expresses the fact that the function f together with the measure µ generates another measure. A.3.4 Proposition: Let f : X [0, ] be a measurable function; then the map M ν(m) := M fdµ is a measure with the domain A, and X gdν = X gf dµ holds for any measurable g : X [0, ]. Let us pass to integration of complex functions. A measurable function ϕ : X C is integrable (over X w.r.t. µ )if X ϕ dµ < (recall that if ϕ is measurable so is ϕ ). The set of all integrable functions is denoted as L(X, dµ); in the same way we define L(M,dµ) for any M A. Given ϕ L(X, dµ) wedenotef := Re ϕ and g := Im ϕ; then f ± and g ± are non negative measurable functions belonging to L(X, dµ). This allows us to define the integral of complex functions through the positive and negative parts of the functions f,g as the mapping ϕ ϕdµ := f + dµ f dµ + i g + dµ i g dµ X X X X X of L(X, dµ) to C. If, in particular, µ is the Lebesgue measure on R d we often use the symbol L(R d )orl(r d,dx) instead of L(R d,dµ), and the integral is written as ϕ(x) dx, or occasionally as ϕ( x) d x. The above definition has the following easy consequence: if M ϕdµ =0 holdsfor all M A, then ϕ(x) =0 µ a.e. in X. Similarly M ϕdµ 0 for all M A implies ϕ(x) 0 µ a.e. in X; further generalizations can be found in [[ Ru 1 ]], Sec The integral has the following basic properties:

9 A.3 Integration 603 (a) linearity: L(X, dµ) is a complex vector space and X (αϕ + ψ) dµ = α X ϕdµ+ X ψdµ for all ϕ, ψ L(X, dµ) and α C, (b) X ϕdµ X ϕ dµ holds for any ϕ L(X, dµ). A.3.5 Examples: A simple complex function σ = n η nχ Mn on X is integrable iff n η n µ(m n ) <, and in this case X σdµ = n η nµ(m n ). The same is true for σ simple functions. Further, let functions f : X [0, ] and ϕ : X C be measurable and dν := fdµ; then ϕ L(X, dν) iff ϕf L(X, dµ) and Proposition A.3.4 holds again with g replaced by ϕ. For a finite measure, we have an equivalent definition based on approximation of integrable functions by sequences of σ simple functions. A.3.6 Proposition: If µ(x) <, then a measurable function ϕ : X C belongs to L(X, dµ) iff there is a sequence {τ n } of σ simple integrable functions such that lim n (sup x X ϕ(x) τ n (x) ) = 0; if this is the case, then X ϕdµ = lim n X τ n dµ. Moreover, if ϕ is bounded the assertion is valid with simple functions τ n. One of the most useful tools in the theory of integral is the following theorem. A.3.7 Theorem (dominated convergence, or Lebesgue): Let M A and {ϕ n } be a sequence of complex measurable functions with the following properties: ϕ(x) := lim n ϕ n (x) exists for µ almost all x M and there is a function ψ L(X, dµ) such that ϕ n (x) ψ(x) holds µ a.e. in M for n =1, 2,...Then ϕ L(X, dµ) and lim ϕ ϕ n dµ =0, lim ϕ n dµ = ϕdµ. n M n M M Suppose that we have non negative measures µ and ν on X (without loss of generality, we may assume that they have the same domain) and k>0; then we can define the non negative measure λ := kµ + ν. We obviously have L(X, dλ) = L(X, dµ) L(X, dν) and X ψdλ= k X ψdµ+ X ψdν for any ψ L(X, dλ). In particular, if non negative measures µ and λ with the same domain A satisfy µ(m) λ(m) for any M A, then L(X, dλ) L(X, dµ) and X fdµ X fdλ holds for each non negative f L(X, dλ). Let µ, ν again be measures on X with the same domain A. We say that ν is absolutely continuous w.r.t. µ and write ν µ if µ(m) = 0 implies ν(m) = 0for any M A. On the other hand, if there are disjoint sets S µ,s ν A such that µ is concentrated on S µ and ν on S ν we say that the measures are mutually singular and write µ ν. A.3.8 Theorem: Let λ and µ be non negative measures on A, the former being finite and the latter σ finite; then there is a unique decomposition λ = λ ac + λ s into the sum of non negative mutually singular measures such that λ ac µ and λ s µ. Moreover, there is a non negative function f L(X, dµ), unique up to a µ zero measure subset of X, such that dλ ac = fdµ, i.e., λ ac (M) = M fdµ for any M A. The relation λ = λ ac + λ s is called the Lebesgue decomposition of the measure λ. The second assertion implies the Radon Nikodým theorem: let µ be σ finite and λ finite; then λ µ holds iff there is f L(X, dµ) such that dλ = fdµ.

10 604 Appendix A Measure and integration A.3.9 Remark: There is a close connection between these results (and their extensions to complex measures mentioned in the next section) and the theory of the indefinite Lebesgue integral, properties of absolutely continuous functions, etc. We refer to the literature mentioned at the beginning; in this book, in fact, we need only the following facts: a function ϕ : R C is absolutely continuous on a compact interval [a, b] if for any ε>0 there is δ>0 such that j ϕ(β j) ϕ(α j ) <ε holds for a finite disjoint system of intervals (α j,β j ) [a, b] fulfilling j (β j α j ) <δ. The function ϕ is absolutely continuous in a (noncompact) interval J if it is absolutely continuous in any compact [a, b] J. A function ϕ : R C is absolutely continuous in R iff its derivative ϕ exists almost everywhere w.r.t. the Lebesgue measure and belongs to L(J, dx) for any bounded interval J R with the endpoints a b; in such a case we have ϕ(b) ϕ(a) = b a ϕ (x) dx. Next we shall mention integration of composite functions. Let w : X R d be a map such that w ( 1) (B d ) A; this requirement is equivalent to measurability of the component functions w j : X R, 1 j d. Suppose that µ(w ( 1) (J)) < holds for any J J d ; then the relation B µ (w) (B) :=µ(w ( 1) (B)) defines a Borel measure µ (w) on R d. A.3.10 Theorem: Adopt the above assumptions, and let a Borel function ϕ : R d C belong to L(R d,dµ (w) ); then ϕ w L(X, dµ) and ϕdµ (w) = (ϕ w) dµ B w ( 1) (B) holds for all b B d. In particular, if X = R d, A = B d,and µ (w) is the Lebesgue measure on R d, the latter formula can under additional assumptions be brought into a convenient form. Suppose that w : R d R d is injective, its domain is an open set D R d, the component functions w j : D R have continuous partial derivatives, ( k w j )( ) forj, k =1,...,d, and finally, the Jacobian determinant, D w := det( k w j ), is nonzero a.e. in D. Such a map is called regular; its range R := Ran w is an open set in R d, and the inverse w 1 is again regular. A.3.11 Theorem (change of variables): Let w be a regular map on R d with the domain D and range R; then a Borel function ϕ : R C belongs to L(R,dx) iff (ϕ w)d w L(D,dx); in that case we have ϕ(x) dx = ((ϕ w) D w )(x) dx B w ( 1) (B) for any Borel B R. In Example A.2.5 we have mentioned how the measure µ ν can be associated with a pair of Borel µ, ν on R d. An analogous result is valid under much more general circumstances. A.3.12 Theorem: Let (X, A,µ)and(Y,B,ν) be measure spaces with σ finite measures; then there is just one measure λ on X Y with the domain A B such that λ(a B) = µ(a)ν(b) holds for all A A,B B; this measure is σ finite and satisfies λ(m) = X ν(m x) dµ(x) = Y µ(m y ) dν(y) for any M A B.

11 A.4 Complex measures 605 The measure λ is again called the product measure of µ and ν and is denoted as µ ν. Using it we can formulate the following important result. A.3.13 Theorem (Fubini): Suppose the assumptions of the previous theorem are valid and ϕ : X Y C belongs to L(X Y,d(µ ν)). Then the cut ϕ x L(Y,dν) for µ a.a. x M and the function Φ : Φ(x) = Y ϕ x dν belongs to L(X, dµ); similarly ϕ y L(X, dµ) ν a.e. in N and Ψ : Ψ(y) = X ϕy dµ belongs to L(Y,dν). Finally, we have X Φ dµ = Y Ψ dν = X Y ϕd(µ ν), or in a more explicit form, ( ) ( ) ϕ(x, y) dν(y) dµ(x) = ϕ(x, y) dµ(x) dν(y) = ϕ(x, y) d(µ ν)(x, y). X Y Y X X Y We should keep in mind that the latter identity may not be valid if at least one of the measures µ and ν is not σ finite. It is also not sufficient that both double integrals exists counterexamples can be found, e.g., in [[ Ru 1 ]], Sec.7.9 or [[ KF ]], Sec.V.6.3. However, if at least one double integral of the modulus ϕ is finite, then all the conclusions of the theorem are valid. A.4 Complex measures A σ additive map ν : A C corresponding to a given (X, A) is called complex measure on X; if ν(m) R for all m A we speak about a real (or signed) measure. Any pair of non negative measures µ 1,µ 2 with a common domain A determines a signed measure by ϱ := µ 1 µ 2 ; similarly a pair of real measures ϱ 1,ϱ 2 defines a complex measure by ν := ϱ 1 + iϱ 2. Any at most countable system {M j }, which is disjoint and satisfies M = j M j, will be called decomposition of the set M; the family of all decompositions of M will be denoted { by S M. To a given complex measure ν and M A, we define ν (M) := } sup j ν(m j) : {M j } S M. One has ν (M) ν(m) ; the set function ν ( ) is called the (total) variation of the measure ν. A.4.1 Proposition: The variation of a complex measure is a non negative measure; it is the smallest non negative measure such that µ(m) ν(m) holds for all M A. Using the total variation, we can decompose in particular any signed measure ϱ in the form ϱ = µ + µ, where µ ± : µ ± (M) = 1 2 [ ϱ (M) ± ϱ(m)]. Since in general the decomposition of a signed measure into a difference of non negative measures is not unique, one is interested in the minimal decomposition ϱ = µ + ϱ µ ϱ such that any pair of non negative measures µ 1,µ 2 on A with the property ϱ = µ 1 µ 2 satisfies µ 1 (M) µ + ϱ (M) and µ 2 (M) µ ϱ (M) for each M A. The minimality is ensured if there is a disjoint decomposition Q + Q = X such that µ ± ϱ := ±ϱ(m Q ± ) 0; the pair {Q +,Q } is called Hahn decomposition of X w.r.t. the measure ϱ. The Hahn decomposition always exists but it is not unique. Nevertheless, if { Q +, Q } is another Hahn decomposition, one has ϱ(m Q ± )=ϱ(m Q ± ) for any M A, so the measures µ ± ϱ depend on ϱ only; we call them the positive and negative variation of the measure ϱ. The formula ϱ = µ + ϱ µ ϱ is named the Jordan decomposition of the measure ϱ.

12 606 Appendix A Measure and integration One has µ + ϱ (M)+µ ϱ (M) = ϱ (M) andµ ± ϱ (M) =sup{±ϱ(a) : A M, A A} for any M A. As a consequence, the positive and negative variations of a signed measure as well as the total variation of a complex measure are finite. One can introduce also infinite signed measures; however, we shall not need them in this book. Complex Borel measures on R d have A = B d for the domain. Variation of a complex Borel measure is a non negative Borel measure. As in the non negative case, a complex Borel measure can be approximated using monotonic sequences of compact sets from inside and open sets from outside of a given M B d. Also the second part of Theorem A.2.3 can be generalized. A.4.2 Proposition: Let complex Borel measure ν and ν on R d satisfy ν(j) = ν(j) for all J J d ; then ν = ν. Before proceeding further, let us mention how the notion of absolute continuity extends to complex measures. The definition is the same: a complex measure ν is absolutely continuous w.r.t. a non negative µ if µ(m) = 0 implies ν(m) = 0 for all M A. There is an alternative definition. A.4.3 Proposition: A complex measure ν satisfies ν µ iff for any ε > 0 there is a δ>0 such that µ(m) <δ implies ν(m) <ε. In particular, if ϕ L(X, dµ) andν is the measure generated by this function, ν(m) := M ϕdµ, then ν µ, so for any ε>0 there is a δ>0 such that µ(m) <δ implies M ϕdµ <ε; this property is called absolute continuity of the integral. Theorem A.3.8 holds for a complex measure λ as well. The measure can be even σ finite; however, then the function f belongsnolongerto L(X, dµ), it is only measurable and integrable over any set M A with λ(m) <. The Radon Nikodým theorem yields the polar decomposition of a complex measure: A.4.4 Proposition: For any complex measure ν there is a measurable function h such that h(x) =1 forall x X and dν = hd ν. Let us pass now to integration with respect to complex measures. We start with a signed measure ϱ = µ + ϱ µ ϱ : a function ϕ : X C is integrable w.r.t. ϱ if it belongs to L(X, dµ + ϱ ) L(X, dµ ϱ )=:L(X, dϱ). Its integral is then defined by ϕdϱ := ϕdµ + ϱ ϕdµ ϱ ; X X X the correctness follows from the uniqueness of the Jordan decomposition. The integral w.r.t. a complex measure ν represents then a natural extension of the present definition: for any function ϕ : X C belonging to L(X, dν) :=L(X, d Re ν) L(X, d Im ν) weset ϕdν := ϕdre ν + ϕdim ν. X X The set of integrable functions can be expressed alternatively as L(X, dν) = L(X, d ν ); it is a complex vector space and the map ϕ X ϕdν is again linear. Also other properties of the integral discussed in the previous section extend to the complex measure case. For instance, the inequality X ϕdν X ϕ d ν holds for any ϕ L(X, dµ). We shall not continue the list, restricting ourselves by quoting the appropriate generalization of Proposition A.3.4. X

13 A.5 The Bochner integral 607 A.4.5 Proposition: Let ϕ L(X, dν) for a complex measure ν; then the map M γ(m) := M ϕdν defines a complex measure γ on A. The conditions ψ L(X, dγ) and ψϕ L(X, dν) are equivalent for any measurable γ : X C; if they are satisfied one has X ψdγ = X ψϕ dν. A.5 The Bochner integral The theory of integration recalled above can be extended to vector valued functions F : Z X, where X is a Banach space; they form a vector space denoted as V(Z, X ) when equipped with pointwise defined algebraic operations. Let (Z, A,µ) be a measure space with a positive measure µ. A function S V(Z, X ) is simple if there is a disjoint decomposition {M j } n j=1 A of the set Z and vectors y 1,...,y n X such that S = n j=1 y jχ Mj. The integral of such a function is defined by Z S(t) dµ(t) := n j=1 y jµ(m j ); as above, it does not depend on the used representation of the function S. To any F V(Z, dµ) we define the non negative function F := F ( ). A vector valued function F is integrable w.r.t. µ if there is a sequence {S n } of simple functions such that S n (t) F (t) holds for µ a.a. t Z and Z F S n dµ 0. The set of all integrable functions F : Z X is denoted by B(Z, dµ; X ). If F is integrable, the limit Z F (t) dµ(t) := lim n Z S n (t) dµ(t) exists and it is independent of the choice of the approximating sequence; we call it the Bochner integral of the function F. The function χ M F is integrable for any set M A and F B(Z, dµ; X ), so we can also define M F (t) dµ(t) := Z χ M(t)F (t) dµ(t). If {M k } is a finite disjoint decomposition of M, wehave F (t) dµ(t) = F (t) dµ(t), M k M k which means that the Bochner integral is additive. A.5.1 Proposition: The map F Z F (t) dµ(t) from the subspace B(Z, dµ; X ) V(Z, X ) tox is linear. Suppose that for a vector valued function F there is a sequence {S n } of simple functions that converges to F µ a.e.; then F belongs to B(Z, dµ; X ) iff F L(Z, dµ), and in that case F (t) dµ(t) F (t) dµ(t). Z Z The existence of an approximating sequence of simple functions has to be checked for each particular case; it is easy in some situations, e.g., ifz is a compact subinterval in R and F is continuous, or if Z is any interval, F is continuous and its one sided limits at the endpoints exist. The continuity of F : R X in an interval [a, b] also implies the relation d t dt a F (u) du = F (t) for any t (a, b). Proposition A.5.1 shows that the Bochner integral is absolutely continuous: for any ε>0 there is a δ>0 such that N F (t) dµ(t) <ε holds for any N A with µ(n) <δ. Another useful result is the following.

14 608 Appendix A Measure and integration A.5.2 Proposition: If B : X Y is a bounded linear map to a Banach space Y; then the condition F B(Z, dµ; X ) implies BF B(Z, dµ; Y), and ( ) (BF)(t) dµ(t) =B F (t) dµ(t). Z Z Many properties of the Lebesgue integral can be extended to the Bochner integral. Probably the most important among them is the dominated convergence theorem. A.5.3 Theorem: Let {F n } B(Z, dµ; X ) be a sequence such that {F n (t)} converges for µ a.a. t Z and F n (t) g(t), n=1, 2,..., for some g L(Z, dµ). Assume further that there is a sequence {S n } of simple functions, which converges to the limiting function F : F (t) = lim n F n (t) µ a.e. in Z; then F B(Z, dµ; X ) and lim n Z F n (t) dµ(t) = Z F (t) dµ(t). An analogue to Theorem A.3.10 can be proven for some classes of functions, e.g., for a monotonic w : R R. Since B(X ) is a Banach space, the Bochner integral is also used for operator valued functions. For instance, suppose that a map B : R B(X ) is such that the vector valued function t B(t)x is continuous for any x X. Further, let K R be a compact interval and µ a Borel measure on R; then lim n K B(t)x n dµ(t) = K B(t)xdµ(t) holds for any sequence {x n } X converging to a point x. Moreover, if an operator T C commutes with B(t) for all t K, then K B(t)ydµ(t) belongs to D(T ) for any y D(T ) and T K B(t)ydµ(t) = K B(t)Tydµ(t).

15 Appendix B. notions Some algebraic In this appendix we collect some algebraic definitions and results needed in the text. There are again many textbooks and monographs in which this material is set out extensively; let us name, e.g., [[ BR 1 ]], [[ Nai 1 ]], [[ Ru 2 ]], or [[ Ti ]] for associative algebras, and [[ BaR ]], [[ Kir ]], [[ Pon ]], or [[ Žel ]] for Lie groups and algebras. B.1 Involutive algebras A binary operation in a set M is a map ϕ : M M M; itisassociative or commutative if ϕ(ϕ(a, b),c)=ϕ(a, ϕ(b, c)) or ϕ(a, b) =ϕ(b, a), respectively, holds for all a, b, c M. Aset G equipped with an associative binary operation is called a group if there exist the unit element e G, ϕ(g, e) =ϕ(e, g) =g for any g G, and the inverse element g 1 G to any g G, ϕ(g, g 1 )=ϕ(g 1,g)=e. Consider next a set R equipped with two binary operations, which we call summation, ϕ a (a, b) :=a+b,andmultiplication, ϕ m (a, b) :=ab. The triplet (R, ϕ a,ϕ m ) is a ring if (R, ϕ a ) is a commutative group and the two operations are distributive, a(b+c) =ab + ac and (a+b)c = ac + bc for all a, b, c R. If there is an e R such that ae = ea = a holds for all a R,wecallittheunit element of R. Let A be a vector space over a field F. The vector summation gives it the structure of a commutative group; if we define a multiplication which is distributive with the summation and satisfies α(ab) =(αa)b = a(αb) for any a, b A,α C, then A becomes a ring, which we call a linear algebra over the field F, in particular, a real or complex algebra if F = R or F = C, respectively. An algebra is said to be associative if its multiplication is associative. The term algebra without a further specification always means a complex associative algebra in what follows; we should stress, however, that many important algebras are nonassociative, e.g., the Lie algebras discussed in Sec.B.3 below. An algebra is Abelian or commutative if its multiplication is commutative. A subalgebra of an algebra A is a subset B, which is itself an algebra with respect to the same operations. If A has the unit element, which is not contained in B, then we can extend the subalgebra to B := {αe + b : α C, b B};inasimilarway, any algebra can be completed with the unit element by extending it to the set of pairs [α, a], α C, a A, with the appropriately defined operations. A proper subalgebra B A is called a (two sided) ideal in A if the products ab and ba belong to B for all a A,b B; we define the left and right ideal analogously. A trivial example of an ideal is the zero subalgebra {0} A. The algebra A itself is not regarded as an ideal; 609

16 610 Appendix B Some algebraic notions thus no ideal can contain the unit element. A maximal ideal in A is such that it is not a proper subalgebra of another ideal in A; any ideal in an algebra with the unit element is a subalgebra of some maximal ideal. An algebra is called simple if it contains no nontrivial two sided ideal. The intersection of any family of subalgebras (ideals, one sided ideals) in A is respectively a subalgebra (ideal, one sided ideal), while the analogous assertion for the unions is not valid. Let A be an algebra with the unit element. We say that an element a A is invertible if there exists an inverse element a 1 A such that a 1 a = aa 1 = e; we define the left and right inverse in the same way. For any a A there is at most one inverse; an element is invertible iff it belongs to no one sided ideal of the algebra A, which means, in particular, that in an algebra without one sided ideals any nonzero element is invertible. Recall that a field is a ring with the unit element which has the last named property; the examples are R, C or the noncommutative field Q of quaternions. We define the spectrum of a A as the set σ A (a) :={λ :(a λe) 1 does not exist}. The complement ρ A (a) :=C \ σ A (a) is called the resolvent set; its elements are regular values for which the the resolvent r a (λ) :=(a λe) 1 exists. B.1.1 Proposition: Let A be an algebra with the unit element; then (a) If a, ab are invertible, b is also invertible. If ab, ba are invertible, so are a and b. (b) If ab = e, the element ba is idempotent but it need not be equal to the unit element unless dim A <. (c) If e ab is invertible, the same is true for e ba. (d) σ A (ab) \{0} = σ A (ba) \{0}, and moreover, σ A (ab) =σ A (ba) provided one of the elements a, b is invertible. (e) σ A (a 1 )={λ 1 : λ σ A (a)} holds for any invertible a A. For any set S A we define the algebra A 0 (S) generated by S as the smallest subalgebra in A containing S; it is easy to see that it consists just of all polynomials composed of the elements of S without an absolute term. We say that S is commutative if ab = ba holds for any a, b S; the algebra A 0 (S) is then Abelian. A maximal Abelian algebra is such that it is not a proper subalgebra of an Abelian subalgebra; any Abelian subalgebra in A can be extended to a maximal Abelian subalgebra. We also define the commutant of a set S Aas S := { a A: ab = ba, b S}; in particular, the center is the set Z := A. We define the bicommutant S := (S ) and higher order commutants in the same way. B.1.2 Proposition: Let S, T be subsets in an algebra S; then (a) S and S are subalgebras containing the center Z, and also the unit element if A has one. Moreover, S = S = and S = S IV =. (b) The inclusion S T implies S T. (c) S S,and S is commutative iff S S, which is further equivalent to the condition that S is Abelian. (d) A 0 (S) = S and A 0 (S) = S. (e) A subalgebra B A is maximal Abelian iff B = B ; in that case also B = B. Let us turn to algebras with an additional unary operation. Recall that an involution a a on a vector space A is an antilinear map A A such that (a ) = a holds for all a A;aninvolution on an algebra is also required to satisfy the condition (ab) = b a for any a, b A. An algebra equipped with an involution is called an involutive algebra

17 B.1 Involutive algebras 611 or briefly a algebra. A subalgebra in A, which is itself a algebra w.r.t. the same involution, is called a subalgebra; we define the ideal in the same way. The element a is said to be adjoint to a. Given a subset S Awe denote S := {a : a S}; the set S is symmetric if S = S; in particular, an element a fulfilling a = a is called Hermitean. By A 0 (S) we denote the smallest subalgebra in A containing the set S. B.1.3 Proposition: Let A be a algebra; then (a) Any element is a linear combination of two Hermitean elements, and e = e provided A has the unit element. (b) a is invertible iff a is invertible, and (a ) 1 =(a 1 ). (c) σ A (a )=σ A (a) holds for any a A. (d) A subalgebra B A is a subalgebra iff it is symmetric; the intersection of any family of subalgebras ( ideals) is a subalgebra ( ideal). (e) Any ideal in A is two sided. (f) A 0 (S) =A 0(S S ) holds for any subset S A;if S is symmetric, then S and S are subalgebras in A. B.1.4 Example (bounded operator algebras): The set B(H) with the natural algebraic operations and the involution B B is a algebra whose unit element is the operator I. Let us mention a few of its subalgebras: (a) If E is a nontrivial projection, then {EB : B B(H)} is a right ideal but not a subalgebra; on the other hand, {EBE : B B(H)} is a subalgebra but not an ideal. (b) If dim H =, the sets K(H) J 2 (H) J 1 (H) of compact, Hilbert Schmidt, and trace class operators, respectively, are ideals in B(H); similarly J p (H) is an ideal in any J q (H), q>p, etc. (c) The algebra A 0 (B) generated by an operator B B(H) consists of all polynomials in B without an absolute term. It is a algebra if B is Hermitean, while the opposite implication is not valid; for instance, the Fourier Plancherel operator F is non Hermitean but A 0 (F ) is a algebra because F 3 = F 1 = F. The algebras of bounded operators, which represent our main topic of interest, inspire some definitions. We have already introduced the notions of spectrum and hermiticity; similarly an element a A is said to be normal if aa = a a,aprojection if a = a = a 2, and unitary if a = a 1, etc. Of course, we also employ other algebras than B(H) and its subalgebras, e.g., the Abelian algebra C(M) of continuous complex functions on a compact space M with natural summation and multiplication, and the involution given by complex conjugation, (f )(x) :=f(x). An ideal J in an algebra A is a subspace, so we can construct the factor space A/J. It becomes an algebra if we define on it a multiplication by ã b := ãb, where a, b are any elements representing the equivalence classes ã and b; it is called the factor algebra (of A w.r.t. the ideal J ). If A has the unit element, then the class ẽ := {e+c : c J} is the unit element of A/J. A morphism of algebras A, B is a map ϕ : A B which preserves the algebraic structure, ϕ(αa + b) = αϕ(a) + ϕ(b) and ϕ(ab) = ϕ(a)ϕ(b) for all a, b A,α C. In particular, if ϕ is surjective, then the image of the unit element (an ideal, maximal ideal, maximal Abelian subalgebra) in A is respectively the unit element (an ideal,... ) in B.If ϕ is bijective, we call it an isomorphism; in the case A = B one uses the terms

18 612 Appendix B Some algebraic notions endomorphism and automorphism of A, respectively. The null space of a morphism ϕ is the pull back ϕ 1 (0 B ) of the zero element of the algebra B; itisanidealin A.If A, B are algebras and ϕ preserves the involution, ϕ(a )=ϕ(a), it is called morphism. B.1.5 Example: Let J be a ( )ideal in a ( )algebra A; then the map ϕ c : ϕ c (a) =ã isa( )morphism of A to A/J. It is called a canonical morphism; its null space is just the ideal J. The factor algebra A/J is simple iff the ideal J is maximal. Moreover, any ( )morphism ϕ : A B can be expressed as a composite mapping, ϕ = π ϕ c, where ϕ c is the canonical morphism corresponding to the ( )ideal J := ϕ 1 (0 B ) and π : A/J ϕ(a) is the ( )isomorphism defined by π(ã) :=ϕ(a) for any a A. Let us finally recall a few notions concerning representations. This term usually means the mapping of an algebraic object onto a suitable set of operators, which preserves the algebraic structure. We shall most often (but not exclusively) use representations by bounded operators: by a representation of a ( )algebra A we understand in this case a ( )morphism π : A B(H), the space H is called the representation space and dim H the dimension of the representation π. If the morphism π is injective, the representation is said to be faithful. Representations π j : A B(H j ),j=1, 2, are equivalent if there is a unitary operator U : H 1 H 2 such that π 2 (a)u = Uπ 1 (a) holds for any a A. A representation π : A B(H) is called irreducible if the operator family π(a) has no nontrivial closed invariant subspace. A vector x H is cyclic for the representation π if the set π(a)x := { π(a)x : a A} is dense in H. Representations of groups, Lie algebras, etc., are defined in the same way. B.2 Banach algebras Algebras can be equipped with a topological structure. Suppose that an algebra A is at the same time a locally convex topological space with a topology τ; thenwecallita topological algebra if the multiplication is separately continuous, i.e., the maps a ab and a ba are continuous w.r.t. the topology τ for any fixed b A. A subalgebra B A is closed if it is closed as a subset in A. The closed subalgebra A(S) generated by a set S Ais the smallest closed subalgebra in A containing S. Isomorphisms ϕ : A B of topological algebras are classified by their continuity: the algebras A, B are topologically isomorphic if there is a continuous isomorphism ϕ such that ϕ 1 is also continuous. B.2.1 Proposition: Let A be a topological algebra and S, B its subset and subalgebra, respectively; then (a) B is a closed subalgebra in A.If B is Abelian, the same is true for B,andany maximal Abelian subalgebra is closed. (b) A(S) =A 0 (S). (c) The subalgebras S, S are closed and (S) = S. (d) If B is an ideal, B A, then B is also an ideal in A. Any maximal ideal is closed. (e) the null space of a continuous morphism ϕ : A C is a closed ideal in A. There are various ways how of defining a topology on an algebra. B.2.2 Example: The strong and weak operator topologies on B(H) are both locally convex, and the operator multiplication is separately continuous with respect to them (compare with Theorem and Problem 3.9); thus B s (H) andb w (H) are topological algebras.

19 B.2 Banach algebras 613 One of the most natural ways is to introduce a topology by means of a norm. An algebra A is called a normed algebra provided (i) A is a normed space with a norm. (ii) ab a b for any a, b A. (iii) If A has the unit element, then e =1. The last condition may be replaced by e 1 because the opposite inequality follows from (ii). The multiplication in a normed algebra is jointly continuous. If A is complete w.r.t. the norm, it is called a Banach algebra. We can again assume without loss of generality that a normed algebra A has the unit element; otherwise we extend it in the above described way, defining the norm by [α, a] := α + a A.If J is a closed ideal in a Banach algebra A, then A/J is a Banach algebra w.r.t. the norm a := inf b J a b A. A complete envelope of a normed algebra A is a Banach algebra B such that it contains A as a dense subalgebra and a A = a B holds for any a A. B.2.3 Theorem: Any normed algebra A has a complete envelope, which is unique up to an isometric isomorphism preserving the elements of A. The space B(H) equipped with the operator norm provides an example of a Banach algebra; the complete envelope of any subalgebra B B(H) is its closure B. By a direct generalization of the methods of Section 1.7, we can prove the following claims. B.2.4 Theorem: Let A be a Banach algebra with the unit element; then (a) Any element a A fulfilling a e < 1 is invertible. The set R of all invertible elements in A is open and the map a a 1 is continuous in it. (b) The resolvent set ρ A (a) of any element a A is open in C and the resolvent r a : ρ A A is analytic. (c) The spectrum σ A (a) of any element a A is a nonempty compact set. (d) The spectral radius r(a) :=sup{ λ : λ σ A (a) } is independent of A and equals r(a) = lim n an 1/n =inf n an 1/n ; it does not exceed the norm, r(a) a. The independence feature of part (d) is not apparent in the case of bounded operators, where the spectral quantities are related to a single algebra B(H). To appreciate this result, notice that the spectrum, and in particular its radius, is a purely algebraic property, while the right side of the formula depends on the metric properties of the algebra A. B.2.5 Proposition: Let A, B be Banach algebras; then (a) A morphism ϕ : A B is continuous iff there is C such that ϕ(a) B C a A for any a A; ifϕ is a continuous isomorphism, then the algebras A, B are topologically isomorphic. (b) If A, B are complete envelopes of normed algebras A 0, B 0, then any continuous morphism ϕ 0 : A 0 B 0 has just one continuous extension ϕ : A B. (c) If J is the null space of a continuous surjective morphism ϕ : A B, then A/J and B are topologically isomorphic. An isomorphism ϕ : A B is called isometric if ϕ(a) B = a A holds for all a A. B.2.6 Theorem (Gel fand Mazur): A Banach algebra with the unit element, in which any nonzero element is invertible, is isometrically isomorphic to the field C of complex numbers.

20 614 Appendix B Some algebraic notions B.3 Lie algebras and Lie groups A Lie algebra (real, complex, or more generally, over a field F ) is a finite dimensional (nonassociative) linear algebra L with the multiplication which we conventionally denote as (a, b) [a, b]. The latter is antisymmetric, [a, b] = [b, a], and satisfies the Jacobi identity [a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0 for any a, b, c L.Thedimension of L is its vector space dimension; if {e j } n j=1 isabasisin L, the product is fully determined by the relations [e j,e k ]=c i jk e i, where the coefficients c i jk are called the structure constants (one uses the summation convention, according to which the sum is taken over any repeated index). A complex extension L C is a complex extension of L as a vector space with the product [a 1 + ib 1,a 2 + ib 2 ]:=[a 1,a 2 ] [b 1,b 2 ]+i[a 1,b 2 ]+i[b 1,a 2 ]. A Lie algebra is commutative if [a, b] = 0 for all a, b L, i.e., all the structure constants are zero; this definition differs from the associative case. In contrast, other definitions like those of a subalgebra, ideal, and also morphisms, representations, etc., modify easily for Lie algebras. B.3.1 Examples: Consider the following matrix algebras: (a) The set of all n n real matrices forms an n 2 dimensional real Lie algebra called the general linear algebra and denoted as gl(n, R); its subalgebra sl(n, R) consists of all traceless g gl(n, R). Its complex extension is the algebra gl(n, C) ofn n complex matrices; similarly traceless complex matrices form the algebra sl(n, C), which is often also denoted as A n 1. (b) The algebra gl(m, C) has other Lie subalgebras. A matrix g is said to be skew symmetric if g t = g, where g t denotes the transposed matrix of g (this property should not be confused with antihermiticity). The set of skew symmetric m m matrices forms the orthogonal Lie algebra, which is denoted as o(m, C); we alternatively speak about the algebras B n and D n for o(2n +1, C) ando(2n, C), respectively. On the other hand, consider the subset in gl(2n, C) consisting of matrices such that g t ) j 2n + j 2n g = 0, where j 0 en 2n :=( and e e n 0 n is the n n unit matrix. The corresponding Lie algebra is called symplectic and denoted as sp(n, C) orc n. Given a Lie algebra L, we define the subalgebras L (n) by the recursive relations L (0) := L and L (n+1) := [L (n),l (n) ]; similarly we define L (n) by L (0) := L and L (n+1) := [L (n),l]. The algebra is solvable if L (n) = {0} for some n; itisnilpotent if L (n) for some n. Any nilpotent algebra is solvable. A commutative Lie algebra is, of course, nilpotent; a less trivial example is the Heisenberg Weyl algebra, which is nilpotent of order two. On the other hand, a Lie algebra L is semisimple if it has no commutative ideal; it is simple if it has no (nontrivial) ideal at all. An equivalent characterization leans on the notion of the Cartan tensor g : g rs = c i rk ck si, through which one defines the Killing form L L F by (a, b) :=g rs a r b s. The algebra L is semisimple iff its Killing form is nondegenerate, i.e., detg 0. Forreal Lie algebras, the Killing form may be used to introduce the following notion: L is compact if the form is positive, and noncompact otherwise. A compact L can be expressed as L = Z S = Z S 1 S n, where Z is its center, S is semisimple, and S 1,...,S n are simple algebras. Simple Lie algebras allow a full classification. It appears that, up to an isomorphism, complex simple algebras are almost exhausted by the types A n,b n,c n,and D n listed in Example 1; there are just five more simple Lie algebras called exceptional. For small

21 B.3 Lie algebras and Lie groups 615 values of n, some of these algebras are isomorphic, namely A 1 B 1 C 1, B 2 D 2, and A 3 D 3, while D 2 is semisimple and isomorphic to A 1 A 1. One can classify real forms of simple complex algebras in a similar way. The notion of a group was introduced above; for simplicity we shall here denote the group operation as a multiplication. A subgroup of a group G is a subset H G which is itself a group w.r.t. the same operation. H is a left invariant subgroup if hg H holds for any h H and g G. We define the right invariant subgroup in a similar way; a subgroup is invariant provided it is left and right invariant at the same time (these notions play a role analogous to ideals in algebras). The notions of the direct product of groups and a factor group are easy modifications of the above discussed algebraic definitions. A topological group is a group G, which is simultaneously a T 1 topological space such that the map g g 1 is continuous and the group multiplication is jointly continuous. An isomorphism of topological groups is a map which is a group isomorphism and, at the same time, a homeomorphism of the corresponding topological spaces. One introduces various classes of topological groups according to the properties of G as a topological space, e.g., compact groups or locally compact groups. In a similar way, one defines a connected group. If G is not connected, it can be decomposed into connected components; the component containing the unit element is a closed invariant subgroup. B.3.2 Examples: (a) The group T n of translations of the Euclidean space R n is a commutative topological group, which is locally but not globally compact. (b) The orthogonal group O(n) consists of real orthogonal n n matrices, i.e., such that g t g = e. It is compact and has two connected components specified by the conditions det g = ±1; the connected component of the unit element is the rotation group denoted as SO(n). (c) The group U(n) of unitary complex n n matrices is locally compact and connected; the same is true for its subgroup SU(n) of matrices with det g =1. (d) The substitution operators U ϕ of Example form a group. If we equip the set of mappings ϕ with the metric ϱ(ϕ, ϕ) := sup x R n ϕ(x) ϕ(x), it becomes a topological group which is not locally compact. A topological group can also be equipped with a measure. The easiest way to introduce it is through linear functionals cf. the concluding remark in Section A.2. Consider such a functional µ on the space C 0 (G) of continuous functions with compact supports which is positive, i.e., µ(f) 0 holds for all f 0. If it satisfies µ(f(g 1 )) = µ(f) for any f C 0 (G) andg G, it is called a left Haar measure on G. On a locally compact group G, there is always a left Haar measure and it is unique up to a multiplicative constant. We introduce the right Haar measure in a similar way; a measure on G is said to be invariant if it combines the two properties. An important class of topological groups consists of those which allow a locally Euclidean parametrization. To be more precise, the notion of an analytic manifold is needed. This is a Hausdorff space M together with a family of pairs (U α,ϕ α ),α I, where U α is an open set in M and ϕ α is a homeomorphism U α R n for a fixed n with the following properties: α I U α = M and for any α, β I the component functions of the map ϕ β ϕ 1 α are real analytic on ϕ α (U α U β ). The number n is called the dimension of the manifold; replacing R n by C n in the definition we introduce complex analytic manifolds in the same way. Given analytic manifolds M, N, one can associate with their topological product (M N,τ M N ) the family of pairs (U α V β,ϕ α ψ β ). The obtained structure again

22 616 Appendix B Some algebraic notions satisfies the above conditions; we call it the product manifold of M and N. The dimension of the product manifold is m = m M + m N. A map M N is analytic if the component functions of all the maps ψ β ϕ 1 α are analytic on their domains. Agroup G is called a (real or complex) Lie group if it is an analytic manifold (real or complex, respectively) and its multiplication and inversion as maps G G G and G G, respectively, are analytic. For instance, the groups of Examples B.3.2a c belong to this class; this is not true for the group of Example B.3.2d, where the dimension of the group manifold is infinite. Any Lie group is locally compact. A subgroup of G which is itself a Lie group with the same multiplication is called a Lie subgroup. As an analytic manifold, at the vicinity of any point a Lie group admits a description through local coordinates defined by the corresponding map ϕ α. This concerns, in particular, the unit element e: there is a neighborhood U of the point 0 R n where we can parametrize the group elements by g (g 1,...,g n ). The composition law of G is then locally expressed by real analytic functions f j,j=1,...,n,from U U to R n so that (gh) j = f j (g 1,...,g n,h 1,...,h n ); the consistency requires f j (g, 0) = g j,f j (0,h)=h j and ( f j / g k ) (0,0) =( f j / h k ) (0,0) = δ jk.thestructure constants of the group G are defined by c i jk := ( 2 f i / g j h k 2 f i / h j g k ) (0,0) ; they satisfy the same conditions as the structure constants of Lie algebras. This is not a coincidence; there is a close connection between Lie groups and Lie algebras. Let U be the neighborhood of the parameter space origin used above and consider the space C (U). The operators T j : (T j φ)(g) =( φ/ g j )(ggj 1 ) g=0 are then well defined and span a Lie algebra L which is said to be associated with G. The correspondence extends to subalgebras: if H is a Lie subgroup of G, then its Lie algebra M is a subalgebra of L. Moreover, if H is an invariant subgroup, then M is an ideal in L, etc.on the other hand, the association G L is not injective. B.3.3 Examples: (a) The group of rotations (translations on a circle) SO(2) can be expressed as SO(2) = T 1 /Z, where T 1 means translations on a line and Z is the additive group of integers. Both SO(2) and T 1 have the same (one dimensional) Lie algebra so(2). (b) Let Z 2 be the two point group {0, 1} with the addition modulo 2. The Lie groups SU(2) and SO(3) = SU(2)/Z 2 have the same Lie algebra so(3) as was discussed in Example e. Basic notions concerning the representation theory of Lie groups and algebras can be readily adapted from the preceding sections. The correspondence discussed above induces a natural relation between some representations of a Lie group and those of its Lie algebra; in the simplest case of a one dimensional G this is the content of Stone s theorem. In general, however, the representation theory of Lie groups and algebras is a complicated subject which goes beyond the scope of the present book; we refer the reader to the literature quoted at the beginning of the appendix.

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53 List of symbols A 2 (C) 48 a(f) annihilation operator 410 a (f) creation operator 410 A 0 (S) 610 A + positive cone 208 ( A) W standard deviation 293 A W mean value of an observable 256 ac(j) 126 AC(J) 97 AC 2 (J) 143 A E projections in an algebra 216 B d Borel sets 596 B ε (p 1,...,p n ) 14 B(V 1,V 2 ) bounded operators 17 B(H) bounded operators on a Hilbert space 63 B(Z, µ; X ) 607 bd boundary of a set 5 C complex numbers C n 1 C(H) scalar operators 217 C0 (Rn ) 15 C(X ) bounded continuous functions 2 C (X) 16 C(X ) closed linear operators 25 card cardinality 595 D(T ), D T domain of an operator 24 D (G) 473 dim dimension 2, 44 E A ( ) projection valued measure 165 {E t } spectral decomposition 152 F(H) Fock space 403 F s (H), F a (H) (anti)symmetric Fock space 404 HD G Dirichlet Laplacian 472 H α,y point interaction Hamiltonian 475,

54 648 List of symbols H n Hermite polynomials 46 id identical mapping 167 J d 595 J 1 (H) trace class operators 84 J 2 (H) Hilbert Schmidt operators 81 J (H), K(H) compact operators 77 Ker kernel of a mapping 596 L j angular momentum 365 L(X, dµ) integrable functions 602 L 2 (X, dµ; G) 47 L (α) k Laguerre polynomials 46 L loc (X), L p loc (X) 126, 446 l p 1 l bounded sequences 1 L p (M,dµ) 2 L p (M,dµ), L p (R n ) 2 L (M,dµ) 4 L p + L 444 L p + L ε 486 L (R d,de) 156 L(H) densely defined operators 93 L b,sa (H) bounded and self adjoint operators 228 L c (H) closed densely defined operators 97 L cs (H) closed symmetric operators 120 L n (H) normal operators 100 L s (H) symmetric operators 94 L sa (H) self adjoint operators 94 l.i.m. limes in medio 19 lin linear envelope 2 M b (H) bound states 493 M s (H) scattering states 493 N natural numbers N(V ) number of bound states 455 N (H) bounded normal operators 74 O, O b observables, bounded observables 270 P, P j momentum operators 97, 261 P l Legendre polynomials 46 P ψ ( ) decay law 341 P Poincaré group 371 Q, Q j position operators 94, 261 Q(A) form domain 115 R real numbers R n 1 R d 595

55 List of symbols 649 R + positive semiaxis [0, ) R T (λ) (T λ) 1 27 RH u (λ) reduced resolvent 343 R ess essential range 102 Ran range 596 S(R n ) Schwartz space 15 s lim strong operator limit 66 supp support 600 sup ess essential supremum 4 T ( ), T b ( ) functional calculus 156, 160 T Σ (T ), T Π (T ) second quantization 405 U ε (x) ε neighborhood 5 u lim operator norm limit V(X, G) vector valued functions 47 w(,a; ψ) measurement outcome probability 256 W j (W ) reduced states 384 w lim weak limit, weak operator limit 22, 67 Y lm spherical functions 395 Z integers Γ(T ) operator graph 24 G D Dirichlet Laplacian 472 Θ(T ), Θ(s) numerical range 68, 111 ρ(t ), ρ A (a) resolvent set 26, 610 σ(t ), σ A (a) spectrum 26, 610 σ ac absolutely continuous spectrum 175 σ c continuous spectrum 26 σ ess essential spectrum 99 σ p point spectrum 26 σ r residual spectrum 26 σ s singular spectrum 175 σ sc singularly continuous spectrum 175 τ s strong operator topology 66 τ u operator norm topology 66 τ w weak operator topology 67 τ σs σ strong operator topology 213 τ σw σ weak operator topology 214 Φ E (R d ) 160 Φ S (f) Segal field operator 414 χ M characteristic function of a set 596 ψ p,q canonical coherent states 297 Ω ± wave operators X subsets of set X algebraic sum 33

56 650 List of symbols form sum 116 direct, orthogonal sum direct, orthogonal sum direct integral 54, 140 tensor product, product measure algebraic tensor product 56, 108 Cartesian product 597 / factorization 2 o interior of a set 23 commutant, dual space bicommutant 214, 610 adjoint, involution, dual space ˆ ˇ Fourier transform 18 orthogonal complement 4 t transposition 614 closure composite mapping inverse 597 ( 1) pull-back 596 \ restriction 596 extension, inclusion intersection union \ set difference symmetric difference, Laplacian limit, map map (point to point) s strong operator limit 66 w weak limit, weak operator limit 22, 67 implication modulus, norm in R n 3 [ ] integer part norm 3 p norm 3, 88 norm 3, 4 (, ) inner product 4 [, ] commutator, boundary form

57 Index accidental degeneracy 374 adjoint, element of a algebra 611 to a bounded operator 63 to an unbounded operator 93 algebra, Abelian 609 associative 609 Banach 613 commutative 609 complex 609 discrete 218 generated by a set 610 Heisenberg Weyl 364 involutive 611 Jordan 288 Lie 614 linear 609 maximal Abelian 610 normed 613 quasilocal 441 real 609 Segal 289 exceptional 289 special 289 simple 610 topological 612 von Neumann 214 algebraic sum, of subspaces 50 of vector spaces 33 alternative, Fredholm 79 Weyl 132 analytic family, Kato type 480 type (A) 480 manifold 615 vector 183 angular momentum 365 orbital 366 total 366 anomalous magnetic moment 486 anticommutator 312, 411 approach, active 359 passive 359 approximation, adiabatic 351 pole 352 semiclassical 307 asymptotic conditions 492 autoionization of helium 353 automorphism of an algebra 612 axiom of choice 598 axioms, countability 9 Gårding Wightman 435 Haag Araki 441 Haag Kastler 437 JNW 288 Kuratowski 35 Nelson 441 of topology 7 Osterwalder Schrader 441 Segal 288 separability 9 Wightman 441 band spectrum 483 baryon number 368 basis, of a vector space 2 Hamel 29 local 9 measurability 53 occupation number 410 of a measure 57 of a topological space 9 orthonormal 44 trigonometric 45 Berry phase 351 bicommutant, extended 179 in an algebra 610 of a bounded-operator set 214 bijection

58 652 Index binary operation 609 Bose Einstein condensation 420 bosons 391 bound, Bargmann 455 Birman Schwinger 457 Calogero 480 Cwikel Lieb Roseblium 458 Ghirardi Rimini 480 GMGT 455 relative 104, 116 bound state 497 of subsystems 399 zero energy 488 bound states, number of 455 boundary condition, Dirichlet 137 Neumann 137 boundary conditions free 564 separated 136 bounds, of a Hermitean operator 68 bracketing 474 C algebra 206 C product of C algebras 241 canonical anticommutation relations 312, 413 commutation relations 300, 413 Weyl form of 302 form of a self adjoint operator 188 CAR 312, 413 Cartan tensor 614 cascading phenomenon 312 Cayley transformation 120 CCR 300, 412 center, of a lattice 427 of an algebra 217, 610 of mass, separation of 392 chain rule 498 classical limit 306 closure, in a metric space 5 of a form 113 of an operator 25 cluster 464 coherent set 269 maximal 269 coherent states 49, 310 canonical 310 spin 310 coherent subspace 269 coherent system 270 commutant, in an algebra 610 of a bounded operator set 214 of an operator set 179 commutativity, of operators 107 of self adjoint operators 170 with an antilinear operator 375 commutator 411 compatibility, of propositions 427 compatible observables 274 a complete set of 280 complete envelope, of a normed algebra 613 metric space 7 set of commuting operators 232 completion, of a metric space 7 of a pre Hilbert space 41 complex extension of a real Lie algebra 614 scaling 353 condition, Agmon 509 asymptotic 492 Enss 509 Friedrichs 345 semigroup 342 spectral 435 Tang 557 conductance 538 cone 208 closed future light 435 conjecture, Dirac 288 conservative system 318 conserved quantity 322 contact interaction 479 continuity, global 6 local 5 sequential 66 contraction, of Lie algebras 375 convergence, dominated 603 monotone 602 of a net 8 of a sequence 5 strong operator 66 weak operator 67

59 Index 653 convex set 29 extremal point of 29 convolution 31 coordinates, atomic 394 geodesic polar 539 Jacobi 394 coordinates, Jacobi cluster 485 spherical 394 generalized 397 core, for an operator 96 for a form 113 coupling constant 345, 512 renormalization 484 coupling, weak 514 covariance, w.r.t. transformations 374 covering 10 open 10 creation operator 410 criterion, Cook 499 self adjointness 103 essential self adjointness 104 cross norm 58 CSCO 232 curvature Gauss 539 mean 539 principal 540 signed 528 curve 30 Peano 244 cut, of a function 599 of a set 597 cyclic vector, of a representation 612 of an operator set 244 decay law 341 for a mixed state 352 decomposition, Bloch 483 cluster 482 Floquet 483 Hahn 605 Jordan 605 Lebesgue, of a measure 603 of a set 605 of unity 152 polar, of a complex measure 606 of a bounded operator 74 spectral 152, 165 to partial waves 396 deficiency indices 118 deficiency subspace 118 degree of freedom 299, 380 dimension, of a vector space 2 Hilbertian 44 of a manifold 615 of a representation 612 of an atomic lattice 429 relative 242 direct integral of a Hilbert space field 54 of an operator valued function 140 direct product, of groups 615 of measures 601 of projection valued measures 155 of σ fields 597 direct sum, of vector spaces 2 of Banach spaces 16 of C algebras 208 of Hilbert spaces 52 of operators 140 of W algebras 218 discrete point of a measure 599 dispersion, of a random variable 287 of a state 430 total 431 distribution 32 operator valued 419 tempered 31 distributive law, on a lattice 427 domain, of a form 111 of a mapping 596 of an operator 24 dual space 19 algebraic 29 of a topological vector space 21 second 21 Dyson expansion 334 effect, Aharonov Bohm 588 Stark 353 Zeeman 448 eigenfunction expansion method 287 eigenspace 26 eigenvalue 25 degenerate 480

60 654 Index eigenvalue, continued embedded 176 multiplicity of 26 simple 26 eigenvalue spacing 484 eigenvector 25 endomorphism of an algebra 612 energy, kinetic 260 threshold 511 total 262 equation, almost Mathieu 576 Schrödinger 319 equivalence 597 equivalence class 598 e.s.a. 96 essential range 198 Euler angles 365 evolution operator 318 example, Nelson 300 Wigner von Neumann 469 extension, Friedrichs 115 Lebesgue 600 of a map 596 of an operator 17 symmetric 94 F topology 32 factor 217 algebra 611 group 615 space 2 family, analytic 480 monotonic 596 separating points 29 Fermi golden rule 347 fermions 391 Feynman integral 332 product 333 field 610 measurable Hilbert space 54 field theory, constructive 440 filter 257 Fock representation, of the CCR 416 Fock space 403 antisymmetric 404 symmetric 404 form 3 below bounded 111 bilinear 3 closable 112 closed 112 densely defined 111 generated by an operator 111 Killing 614 positive 3, 111 strictly 3 quadratic 3, 111 real 3 relatively bounded 116 sectorial 140 sesquilinear 3 unbounded 111 symmetric 3 form domain, of an operator 115 form sum, of self adjoint operators 116 formal operator, regular 126 singular 126 formula, Eisenbud Wigner 523 Feynman Kac 333 Lagrange 148 Landauer 538 Landauer Büttiker 554 polarization 3,4 Stone 178 Trotter 194 Fourier, coefficient 43 expansion 44 transformation 19 function 596 absolutely continuous 604 almost periodic 57 Borel 598 characteristic 596 cylindrical 332 homogeneous of order β 470 integrable 602 locally 126 w.r.t. a signed measure 606 measurable 598 of a self adjoint operator 177 of commuting s a operators 181 of positive type 196 rapidly decreasing 15 simple 598

61 Index 655 function, continued spherical 395 vector valued 53 analytic 27 σ simple 598 functional 3 additive 3 antilinear 3 bounded linear 20 homogeneous 3 linear 3 positive on a algebra 208 normal 221 real 3 strongly positive 240 gap in the spectrum 483 generator, of a unitary group 191 of an operator semigroup 197 GNS representation 210 GNS triplet 210 Gram determinant 34 graph 24, 561 boundary 570 degree of a vertex 561 edge of 561 finitely periodic 588 generalized 579 interior 571 leaky quantum 581 metric 561 neighboring vertices 570 of an operator 24 star 593 vertex of 561 ground state 462 group 609 connected 615 compact 615 Euclidean 370 Galilei 370 Heisenberg Weyl 364 Lie 616 locally compact 615 Lorentz 372 Poincaré 371 of transformations 360 of unitary operators 191 symmetry 362 topological 615 velocity 327 Hamiltonian 262 atomic 446 in the axiomatic approach 435 interaction 388 time dependent 319 hedgehog manifold 589 Heisenberg relations 295 Hermitean, element of a algebra 611 operator 67 hidden parameters 432 Hilbert space 41 of analytic functions 49 of vector valued functions 48 rigged 287 Hilbertian sum of subspaces 57 homeomorphism, linear 6 of metric spaces 6 Hofstadter butterfly 588 Hölder, continuity 349 inequality 32 Hughes Eckart term 481 ideal 609 gas 420 maximal 610 one sided 545 identities, resolvent 40 identity, Hilbert 40 Jacobi 614 parallelogram 4 Parseval 44 of propositions 425 implication between propositions 425 inequalities between operators 69 inequality, Bessel 43 Faber Krahn 583 Herdy 311 Hölder 32 Kato 479 Minkowski 1, 33 Lieb Thirring 458, 481 Schwarz 4, 111

62 656 Index inequality, Bessel, continued triangle 5 infimum, of a proposition set 425 of an operator set 69 inner product 4 integrability 602 of vector valued functions 607 integral 601 absolute continuity of 606 Bochner 607 of a complex function 602 of motion 322 complete system of 323 w.r.t. a projection valued measure 161 interior, of a set 23 point 5 invariance w.r.t. transformations 361 inverse, decay problem 351 element 610 one sided 610 involution, on an algebra 610 involutive automorphism 429 irreducibility, algebraic 244 of an operator set 235 topological 244 isometry, linear 6 of metric spaces 6 partial 73 isomorphism, isometric 613 of algebras 611 of Hilbert spaces 41 of proposition systems 429 of topological groups 615 of vector spaces 2 spatial 217 topological 13 isospin 368 Jacobi matrix 572 Kato class 479 kernel, of a mapping 596 Kramers degeneracy 374 Laplacian, Dirichlet 472 Neumann 472 lattice, atomic 427 Boolean 427 complete orthocomplemented 426 countably complete 427 generated by a set 427 irreducible 427 weakly modular 427 layer, asymptotically planar 541, 543 curved 539 locally curved 542 lemma, Riesz 42 Riemann-Lebesgue 19 Schur 236, 244 uncertainty principle 311 Zorn 598 Lie algebra 614 symplectic 614 commutative 614 compact 614 dimension of 614 exceptional 614 general linear 614 nilpotent 614 of a Lie group 616 orthogonal 614 semisimple 614 solvable 614 Lie group 616 Lie subgroup 616 limit, thermodynamic 422 limit circle case 132 limit point case 132 linear, homeomorphism 13 hull 2 independence, of vectors 2 modulo L 135 linearly independent set 2 loop 538 Lorentz group 372 proper orthochronous 375 map 596 composite 597 injective 596 inverse 597 regular 604 surjective 596 uniformly continuous 30

63 Index 657 mapping 596 Maslov correction 349 mass operator 435 matrix representation of a bounded operator 66 matrix, skew symmetric 614 maximal common part of self adjoint operators 125 maximal element 598 mean value, of a random variable 287 mean square deviation 287 measurability of a complex function 598 of a vector valued function 47 measure absolutely continuous w.r.t. another measure 603 complex 606 Borel 600 complete 599 complex 605 Borel 606 concentrated on a set 600 discrete 599 finite 599 generated by a function and a measure 602 invariant 615 Lebesgue 601 Lebesgue Stieltjes 601 left Haar 615 non negative 599 outer 600 product 605 projection valued 151 real 605 regular 601 signed 605 space 599 spectral 151 Wiener 333 σ finite 599 measures, mutually singular 603 measurement 251 first kind 286 preparatory 280 second kind 286 method eigenfunction expansion 287 Howland 351 of induced representations 375 Rayleigh Ritz 452 WKB 312 metric 5 induced by a norm 5 metric space 5 completely bounded 11 complete 6 separable 5 microcausality 436 mixed state, on a C algebra 242 second kind 399 mixture, of states 287 mode matching technique 553 model, Friedrichs 345 Kronig Penney 490 Lee 352 standard 374 Wannier Stark 485 moment, of a probability measure 287 momentum, canonically conjugate 414 of a particle 259 monodromy operator 339 morphism, canonical 612 of algebras 611 multiplication 609 multiplicativity of operator integrals 157 multiplicity, of an eigenvalue 26 of the spectrum 514 multiplier of a projective representation 361 neighborhood 5 in a topological space 7 net 8 norm 3 dominated by another norm 35 Hilbert Schmidt 82 induced by an inner product 4 normal element, of a algebra 611 normal form of second quantized operator 421 normed space 3 reflexive 21

64 658 Index norms, equivalent 35 null space of a morphism 612 numerical range, of a form 111 of an operator 68 O algebra 240 observable, bounded 270 fundamental 283 in a broader sense 279 operational definition of 252 preserved by a transformation 361 quasilocal 437 observables, algebra of 270, 284 Op algebra 239 closed 239 self adjoint 239 standard 239 operator adjoint 63, 93 associated with a form 115 Banach adjoint 87 conjugation 148 dual 87 essentially self adjoint 96 Fourier Plancherel 19 Laplace Beltrami 400, 541 in a layer 540 left shift 64 Møller 522 norm 17 product 17 right shift 64 Schrödinger 443 splitting of 130 Weyl 304 operator set, commutative 228 irreducible 235 nondegenerate 215 reduced by a subspace 235 reducible 235 symmetric 227 operator set, invariant 243 operator sets, commuting 228 operators, accretive 139 antihermitean 87 antiunitary 87 below bounded 139 bounded 17 closable 25 closed 25 compact 77 canonical form of 81 densely defined 93 dissipative 139 essentially self adjoint 96 finite dimensional 77 Hermitean 67 Hilbert Schmidt 81 Hilbert Schmidt integral 89 irreducible 106 isometric 73 linear 17 local 370 maximal symmetric 94 mutually dominated 504 n particle 404 normal 74, 100 of multiplication 101 of the class J p 88 of the number of particles 407 positive 68, 139 pseudo differential 245 reducible 105 relatively bounded 104 relatively compact 173 scalar 211 sectorial 139 self adjoint 94 dominated by a s a operator 504 statistical 86 strictly positive 68 Sturm Liouville 141 superselection 272 symmetric 94 trace class 84 on a Banach space 88 unitary 72 with a compact resolvent 173 ordering, partial 598 orthocomplementation 426 orthogonal, complement 4 projection of a vector 42 set 5

65 Index 659 sum of operators 105 vectors 4 orthonormal, basis 44 set 5 parastatistics 400 parity 73, 323, 367 internal 367 violation 373 part of an operator 105 absolutely continuous 175 singular 175 singularly continuous 175 partial isometry 73 final subspace of 74 initial subspace of 74 partition 464 picture, Heisenberg 323 interaction (Dirac) 324 Schrödinger 323 Poincaré group 371 point, accumulation 5 in a lattice 427 interaction in one dimension 475, 477 in three dimensions 478 isolated 5 limit 5 Poisson bracket 308 polarization 266 formula 3, 4 pole, of a surface 539 polynomials, Hermite 46 Laguerre 46 Legendre 46 position, of a particle 259 positive element, of a C algebra 208 postulate (ao c ) 284 (ao s ) 284 (ao w ) 284 (pl) 426 (pl q ) 429 (Q1a) 253, 255 (Q1b) 253, 265 (Q1c) 255 (Q2a) 256 (Q2b) 256, 265 (Q3) 257 (Q4a) 318 (Q4b) 318 (Q4c) 319 (Q5a) 359, 360 (Q6a) 379 (Q6b) 392 (Q6c) 398 potential 443 centrally symmetric 394 periodic 483 purely attractive 456 repulsive 471 predual, of a W algebra 242 pre Hilbert space 4 principle, Birman Schwinger 456, 547 invariance of wave operators 506 minimax 80, 448 of uniform boundedness 23 Pauli 392 superposition 254 uncertainty 298 unique continuation 588 probability density 260 product, manifold 616 Cartesian 593 symmetrized 288 projection 70 Abelian 217 in a algebra 611 minimal 247 orthogonal 87 projections, complete system of 72 equivalent 217 projection valued measure 151 discrete 152 propagator, reduced 341 unitary 317 proposition 425 absurd 426 false 425 lattice 426 modular 438 not valid 425 system 429 trivial 426

66 660 Index proposition, continued true 425 valid 425 propositions, disjoint 428 simultaneously valid 425 pull back 532 quantization, Dirac 308 practical 309 second 406 quantum computing 399 quantum field, at an instant 422 free 413, 419 generalized 440 Hermitean 419, 440 quantum kinematic of decays 351 quantum logic 438 quantum theory, quaternionic 430 real 430 quantum waveguide 527 a local deformation 530 quantum wire 552 quarks 374 quasienergy 339 quaternions 438 qubit 399 range, essential 103 numerical 68, 111 of a mapping 596 ray 253 reduction formulas 386 reflection 73 regular endpoint 126 regular value 27 of element of an algebra 610 regularity domain 40, 75 relation 597 intertwining 498 of equivalence 597 of spin to statistics 390, 440 relations, de Morgan 595 uncertainty 294, 295 relative bound, of a form 116 of an operator 104 representation completely reducible 244 faithful 612 irreducible 612 isometric, of a C algebra 212 of a group 612 of an algebra 612 of a Lie algebra 612 projective 361 space 612 representations, equivalent 612 resolvent, first identity 40 of an operator 27 of element of an algebra 610 reduced 343 second identity 40 set 27 of element of an algebra 610 resonance, shape 353 zero energy 513 restriction, of a map 596 of an operator 17 ring 609 set 595 Rollnik class 479 rotation, in plane 365 in space 365 of spin 366 S matrix 493 scalar 1 product 4 scattering, direct problem 522 inverse problem 522 of subsystems 399 operator 493 Schrödinger equation 319 Schrödinger operator 443 Schrödinger representation of the CCR 302 Schwarz inequality 3, 4, 111 on a C algebra 209 Schwinger functions 441 second quantization 406 Segal field operator 414 semianalytic vector 184 seminorm 3 sequence, Cauchy 6 occupation number 409

67 Index 661 series, Neumann 32 Puiseaux 516 Rayleigh Schrödinger 454 set, arcwise connected 30 balanced 38 Borel 596 bounded 30 closed 5 compact 10 weakly 22 completely bounded 11 connected 30 dense, everywhere 5 in a set 5 nowhere 5 directed 8 field 596 function 599 regular 600 open 5, 7 precompact 11 system, centered 37 total 15 weakly compact 22 sets, monotonic family of 596 shape of the drum 484 simple algebra, type A n 614 type B n 614 type C n 614 type D n 614 function, vector valued 607 eigenvalue 26 singular, endpoint 126 value, of a compact operator 81 space, Banach 15 Bose Fock 421 Fermi Fock 421 Fréchet 15 Hilbert 41 metric 5 normed 3 pre Hilbert 4 reflection 367 Schwartz 15 Sobolev 287 topological 7 space, topological vector 13 vector 1 complex 1 real 1 space like separation 436 spectral, decomposition 152, 165 radius 28 representation 188 theorem 165 spectrum, absolutely continuous 175 continuous 26 of a self adjoint operator 176 discrete 173 essential 99 of an operator 26 of element of an algebra 610 point 26 pure point 76, 103 purely discrete 173 residual 26 simple 188 singular 175 singularly continuous 175 spin 276 relation to statistics 399, 440 splitting trick 130 square root, of an operator 69 stability of matter 311, 482 standard, completion procedure 7 deviation of a random variable 287 state antibound 513 asymptotic 492 bound 497 bounded energy 273 completely additive 243 component 384 conditions of preparation 252 dispersionless 431 finite energy 272 Gibbs 420 minimum uncertainty 297 mixed 265 normal 221 on a Banach algebra 208 on a proposition system 430 mixed, pure 430

68 662 Index state, continued on an Op algebra 240 operational definition of 252 preparation of 251 pure 253, 265 on a C algebra 213 reduced 384 scattering 492 space 253 Banach 289 n particle 404 one particle 404 stationary 322 Werner 399 statistics, Bose Einstein 391 Fermi Dirac 391 step function 16 strangeness 368 structure constants 614 of a Lie group 616 subalgebra 609 closed 612 subgroup 615 invariant 615 sublattice 427 subnet 30 subspace 2 invariant, of an operator 26 of an operator set 235 Lagrangean 562 of a topological space 9 singlet 384 triplet 384 substitution operators 72 subsystem 379 subsystems, noninteracting 388 summation 609 convention 276, 614 superposition, of states 254 principle 254 superselection rule 272 Bargmann 371 superselection rules commutativity of 272 support, of a measure 600 supremum, of a proposition set 426 of an operator set 69 surface, Cartan Hadamard 542 complete 543 end of 545 symmetric, operatorquad 94 subset of a algebra 611 symmetry group 362 commutative 362 continuous 362 discrete 362 finite 362 Lie 362 noncommutative 362 hidden 369 w.r.t. transformations 362 system, composed 379 of seminorms 14 separating points 14 tensor C product 218 tensor product, algebraic 56, 58 of bounded operators 108 of Hilbert spaces 55 of unbounded operators 109 of W algebras 218 realization of 54 tensor W product 218 theorem, adiabatic 351 Baire 35 bicommutant 242 Birman 504 Birman Kuroda 504 Bochner 197 Bolzano Weierstrass 30 Burnside 245 closed graph 25 dominated convergence 603 edge of the wedge 352 Ehrenfest 321 embedding 288 Gleason 434 Haag 422 Heine Borel 10 Hellinger Toeplitz 95 HVZ 466 inverse mapping 24 Jauch Piron 433

69 Index 663 theorem, adiabatic, continued Jordan Wigner 312 Kaplansky density 242 Kato, on atomic Hamiltonians 445 on helium spectrum 461 on positive eigenvalues 469 Kato Rellich 105 Kato Rosenblum 504 KLMN 117 Kupsch Sandhas 500 Levi 602 Molčanov 464 monotone convergence 602 Noether 369 open mapping 23 orthogonal decomposition 42 Pearson 500 Piron 429 Radon Nikodým 603 reconstruction of fields 441 representation for forms 114 Riesz representation 601 Riesz Markov 31 Sakai 242 SNAG 197 spectral 165 Uryson 30 virial 469 von Neumann, density 247 on a generating operator 243 on deficiency indices 148 on the generating operator 229 Weyl 201 Wüst 140 theory, Gel fand 241 Kato Birman 522 time delay 523 time ordered exponential 334 topological isomorphism of algebras 612 topological product 9 topological space 7 arcwise connected 30 compact 10 connected 30 first countable 9 Hausdorff 10 locally compact 11 locally convex 14 normal 10 regular 10 second countable 9 σ compact 11 topological vector space 13 complete 31 weakly complete 22 topology 7 discrete 7 F weak 8 relative 9 strong operator 66 trivial 7 ultrastrong 242 ultraweak 242 weak 14, 21 weak operator 67 σ strong 213 σ weak 214 trace class 84 trace, of an operator 85 on a W algebra 220 faithful 220 finite 220 normal 220 semifinite 220 translation, n dimensional 364 on line 363 tree graph 588 regular 588 rooted 588 triangle inequality 5 uncertainty principle 298 local 311 uncertainty relations 294 uniform topology on an Op algebra 240 unit element, of a group 609 of an algebra 609 unitary element, of a algebra 611 unitary equivalence 107 unitary group, one parameter 191 strongly continuous 191

70 664 Index unitary invariant 140 upper bound 598 vacuum 403 in the axiomatic approach 435 mean 436 variation 408 of a measure, total 605 negative 605 positive 605 vector 1 analytic 183 cyclic 244, 612 generating 188 semianalytic 184 vertex coupling free 564 Kirchhoff 587 permutation invariant 564 δ 563 δ 564 δ s 564 δ p 590 W algebra 214 continuous 219 finite 220 homogeneous 219 properly infinite 220 purely infinite 220 reduced 217 semifinite 220 type I 218 type I n, I 219 type II, II 1, II 220 type III 220 Wannier ladder 354 wave function 259 wave operators 492 asymptotic completeness 497 completeness 497 existence 497 generalized 500 Weyl relations 302 Wiener integral 333 Wightman functions 440 Wronskian 149 yes no experiment 256 yes no experiments, compatible 291 disjoint 291 δ interaction, in one dimension 475 in three dimensions 478 δ interaction 477 ε lattice 11 σ additivity, of the integral 602 σ field 596 algebra 611 Banach 205 normed 205 partial 245 topological 205 ideal 611 morphism 612 subalgebra 611