B 2 L(H) with a bounded everywhere defined inverse such that the operators BS i B 1, i =1;:::;n, are all normal. We write ±(H) for the class of all sc

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1 Seminar LV, No. 9, 12 pp., Operator Exponentials on Hilbert Spaces Christoph Schmoeger Abstract Let H be a complex Hilbert space and let L(H) be the Banach algebra of all bounded linear operators on H. In this paper we consider the following class of operators: ^±(H) = fs 2L(H): S is a scalar type operator and ff(s) ff(s +2kßi) fkßig for k =1; 2;:::g. The main results of this paper read as follows: 1. If T;S 2 ^±(H) ande T e S = e S e T then T 2 S 2 = S 2 T If S 2 ^±(H), T 2L(H) ande T = e S then TS 2 = S 2 T. Math. Subject Classification: 47A10, 47A60 Key words and phrases: scalar type operators, exponentials 1 Terminology and results Throughout this paper let H denote a complex Hilbert space and L(H) the Banach algebra of all bounded linear operators on H. For A 2L(H) the spectrum and the spectral radius of A are denoted by ff(a) and r(a), respectively. The set of eigenvalues of A is denoted by ff p (A). For the resolvent setofa we write ρ(a). We usen(a) anda(h) to denote the kernel and the range of A, respectively. An operator S 2L(H) is called a scalar type operator if S admits a representation Z S = E(d ); ff(s) where E(d ) denotes integration with respect to a spectral measure E( ) on H. See [1], [2] and [14] for properties of spectral measures and scalar type operators. If A 2 L(H) is normal (AA Λ = A Λ A) then A is a scalar type operator and the values of the spectral measure of A are selfadjoint projections (see [1], Theorem 7.18). J. Wermer [14] hasshown that the scalar type operators on H are those operators which are similar to normal operators. More precisely, Wermer has shown that for every finite set S 1 ;:::;S n of commuting scalar type operators on H there is a selfadjoint operator 1

2 B 2 L(H) with a bounded everywhere defined inverse such that the operators BS i B 1, i =1;:::;n, are all normal. We write ±(H) for the class of all scalar type operators on H. In the present paper we consider the following class of operators: ^±(H) =fs 2 ±(H) :ff(s) ff(s +2kßi) fkßig for k =1; 2;:::g: Now we state the main results. Proofs will be given in Section 3, in Section 4 we present some corollaries. Theorem 1.1 If T 2 ^±(H), S 2L(H) and e T e S = e S e T then e S T 2 = T 2 e S. If in addition ff p (T ) fkßi : k =1; 2;:::g = ; then e S T = Te S. Theorem 1.2 If T;S 2 ^±(H) and e T e S = e S e T then T 2 S 2 = S 2 T 2. Theorem 1.3 Suppose that T;S 2 ^±(H) and that e T e S = e S e T. (a) If ff p (T ) fkßi : k =1; 2;:::g = ; then TS 2 = S 2 T. (b) If ff p (T ) fkßi : k =1; 2;:::g = ff p (S) fkßi : k =1; 2;:::g = ; then TS = ST. For related results concerning the equation e A e B = e B e A see [10], [11], [12] and [15]. Theorem 1.4 Suppose that T;S 2L(H), T + S 2 ^±(H) and that e T +S = e T e S = e S e T : If ff p (T + S) fkßi : k =1; 2;:::g = ; then TS = ST. Theorem 1.5 If S 2 ^±(H), T 2L(H) and e T = e S then TS 2 = S 2 T. If in addition ff p (S) fkßi : k =1; 2;:::g = ; then TS = ST. For related results concerning the equation e A = e B see [3], [9] and [11]. 2 Preparations In this section we collect some results which we need for the proofs of the theorems in Section 1. Proposition 2.1 Suppose that A 2L(H) is normal. (a) If μ 2 C then (A μ)(h) =(A Λ μ)(h). 2

3 (b) If B 2L(H) then E(ff(A) ff(b))(h) = 2ρ(B) (A )(H); where E( ) denotes the spectral measure of A. Proof. (a) Since A is normal, A μ is normal. Exercise in [8] gives the result. (b) is shown in [7, Theorem 1], see also [6]. Let A 2L(H). The map ffi A : L(H)!L(H), defined by ffi A (C) =CA AC (C 2L(H)) is called the inner derivation determined by A. It is clear that ffi A operator on L(H) withkffi A k»2kak. is a bounded linear Throughout this paper let f denote the entire function f : C! C given by f(z) = Let M A = f 2 ff(ffi A ):f( ) =0g. Proposition 2.2 Let A 2L(H). ( z 1 (e z 1); if z 6= 0; 1; if z =0: (a) If M A = ;, then f(ffi A ) is an invertible operator on L(H). (b) If 2 M A then is a simple zero of f and there is j 2 Z nf0g with =2jßi. (c) M A is a finite set, M A f±2ßi; ±4ßi;:::g. (d) If M A 6= ; and M A = f 1 ;:::; p g with j 6= k for j 6= k then (e) ff(ffi A )=f μ : ; μ 2 ff(a)g. (f) e ffi A(C) =e A Ce A for all C 2L(H). N(f(ffi A )) = N(ffi A 1 ) Φ :::Φ N(ffi A p ): (g) f(ffi A )(ffi A (C)) = e A Ce A C for all C 2L(H). Proof. (a) If M A = ;, then f( ) 6= 0for all 2 ff(ffi A ), thus f(ffi A ) is invertible. (b), (c) and (d) are shown in [11]. (e) follows from [4], and Proposition in [5] shows that (f) holds. (g) follows from (f) and zf(z) =f(z)z = e z 1. 3

4 Proposition 2.3 Let A be a normal operator in L(H) and let E( ) be its spectral measure. If 0 2 C, C 2 N(ffi A 0 ), D 2 N(ffi A + 0 ) then C(H) E(ff(A) ff(a 0 ))(H) and D Λ (H) E(ff(A) ff(a 0 ))(H): Proof. From CA AC = 0 C we get AC = C(A 0 ). Put B = A 0. Now take μ 2 ρ(b). Then (A μ)c(b μ) 1 = AC(B μ) 1 μc(b μ) 1 = CB(B μ) 1 μc(b μ) 1 = C(B μ)(b μ) 1 = C; thus C(H) (A μ)(h). Since μ 2 ρ(b) was arbitrary, we derive C(H) (A μ)(h): Proposition 2.1(b) implies now that μ2ρ(b) C(H) E(ff(A) ff(b))(h) =E(ff(A) ff(a 0 ))(H): Now suppose that D 2 N(ffi A + 0 ), hence DA = (A 0 )D = BD. Therefore D Λ B Λ = A Λ D Λ. A similar computation as above shows that for μ 2 ρ(b Λ )we have (A Λ μ)d Λ (B Λ μ) 1 = D Λ ; thus D Λ (H) μ2ρ(b Λ ) (A Λ μ)(h): Since ρ(b Λ )=f : 2 ρ(b)g, we get from Proposition 2.1 that D Λ (H) (A ) Λ (H) = (A )(H) 2ρ(B) 2ρ(B) = E(ff(A) ff(b))(h) =E(ff(A) ff(a 0 ))(H): The following propositions are of central importance for our investigations. 4

5 Proposition 2.4 Let A be a normal operator in L(H) and suppose that ff(a) ff(a +2kßi) fkßig for k =1; 2;:::: If k 2 N nf0g, C 2 N(ffi A +2kßi) and D 2 N(ffi A 2kßi) then AC = kßic = CA and DA = kßid = AD: Proof. Put 0 = 2kßi. From C 2 N(ffi A 0 ) we get from Proposition 2.3 that Since ff(a) ff(a +2kßi) fkßig, C(H) E(ff(A) ff(a +2kßi))(H): E(ff(A) ff(a +2kßi))(H) E(fkßig): From Theorem in [8] it follows that E(fkßig) =N(A kßi). Thus C(H) N(A kßi); hence AC = kßic. From CA AC = 2kßiC we conclude that CA = kßic = AC. For D 2 N(ffi A 2kßi) =N(ffi A + 0 ) we get from Proposition 2.3 that D Λ (H) E(ff(A) ff(a +2kßi))(H) N(A kßi): Thus AD Λ = kßid Λ. Therefore AD Λ x = kßid Λ x for each x 2H. The normalityofa gives A Λ D Λ x = kßid Λ x, hence A Λ D Λ = kßid Λ, thus DA = kßid. From DA AD =2kßi we derive AD = DA 2kßi = kßid = DA: Proposition 2.5 Suppose that S 2 ^±(H) and k 2 N nf0g. (a) If C 2 N(ffi S +2kßi) then SC = kßic = CS. (b) If D 2 N(ffi S 2kßi) then DS = kßid = SD. (c) If U 2 N(f(ffi S )) then SU + US =0. (d) If ff p (S) fnßi : n =1; 2;:::g = ; then N(f(ffi S )) = f0g. 5

6 Proof. We know that there are operators X and A in L(H) such that X is invertible in L(H), A is normal and S = X 1 AX: Therefore we have S = X 1 (A )X for each 2 C and ff(s) =ff(a) andff(s ) = ff(a ). Since S 2 ^±(H), we derive that (Λ) ff(a) ff(a +2nßi) fnßig for n =1; 2;:::. (a) From CS SC = 2kßiC, we get CX 1 AX X 1 AXC = 2kßiC; therefore (XCX 1 )A A(XCX 1 ) = 2kßi(XCX 1 ). This shows that XCX 1 2 N(ffi A +2kßi). From (Λ) and Proposition 2.4 we see that hence SC = kßic = CS. (b) Similar. AXCX 1 = kßixcx 1 = XCX 1 A; (c) Follows from (a), (b) and Proposition 2.2(d). (d) Let n 2 N nf0g. Since nßi 62 ff p (S), we see from (a) that N(ffi S +2nßi) =f0g. In view of Proposition 2.2(d) it remains to show that N(ffi S 2nßi) =f0g. Take D 2 N(ffi S 2nßi) and put ~ D = XDX 1. As in the proof of (a) we see that ~ D 2 N(ffiA 2nßi). From Proposition 2.3 it follows that ~D Λ (H) E(ff(A) ff(a +2nßi))(H): By (Λ) we get ~ DΛ (H) E(fnßig) =N(A nßi). Since ff p (A) =ff p (S) and nßi 62 ff p (S), it follows that N(A nßi) =f0g. Thus ~ DΛ =0,hence D =0. 3 Proofs Proof of Theorem 1.1. Use Proposition 2.2(g) to see that f(ffi T )(ffi T (e S )) = e T e S e T e S =0; hence V = ffi T (e S )=e S T Te S 2 N(f(ffi T )). Proposition 2.5(c) shows that 0=TV + VT = Te S T T 2 e S + e S T 2 Te S T = e S T 2 T 2 e S : If ff p (T ) fkßi : k =1; 2;:::g = ;, then by Proposition 2.5(d), V =0,thus e S T = Te S. 6

7 Proof of Theorem 1.2. It follows from Theorem 1.1 that T 2 e S = e S T 2. By Proposition 2.2(g) we derive f(ffi S )(ffi S (T 2 )) = e S T 2 e S T 2 =0; hence U = ffi S (T 2 )=T 2 S ST 2 2 N(f(ffi S )). Proposition 2.5(c) gives now 0=SU + US = ST 2 S 2 T 2 + T 2 S 2 ST 2 = T 2 S 2 S 2 T 2 : Proof of Theorem 1.3. (a) We know from Theorem 1.1 that e S T = Te S, thus f(ffi S )(ffi S (T )) = e S Te S T =0; therefore TS ST 2 N(f(ffi S )). Use again Proposition 2.5(c) to see that 0=S(TS ST)+(TS ST)S = TS 2 S 2 T (b) Proposition 2.5(d) gives N(f(ffi S )) = f0g. Hence TS = ST. Proof of Theorem 1.4. Proposition 2.2(g) shows that f(ffi T +S )(ffi T +S (e T )) = e (T +S) e T e T +S e T = e S e T e T e T +S e T = e S e S e T e T =0; therefore U = e T (T +S) (T +S)e T = e T S Se T 2 N(f(ffi T +S )). Since N(f(ffi T +S )) = f0g (Proposition 2.5(d)), it follows that U = 0, hence e T S = Se T, therefore f(ffi T +S )(ffi T +S (S)) = e (T +S) Se T +S S = e S e T Se T e S S = 0: Hence we see that S(T + S) (T + S)S = ST TS 2 N(f(ffi T +S )) = f0g. Proof of Theorem 1.5. Since f(ffi S )(ffi S (T )) = e S Te S T = e T Te T T =0; we have TS ST 2 N(f(ffi S )), thus, by Proposition 2.5(c) 0=S(TS ST)+(TS ST)S = TS 2 S 2 T; hence TS 2 = S 2 T. If ff p (S) fkßi : k =1; 2;:::g = ;, we see from Proposition 2.5(d) that N(f(ffi S )) = f0g, thus TS = ST. 7

8 " ( : Since A is normal, e A+A Λ = e A e AΛ = e A (e A ) Λ = I = e 0 : 4 Corollaries Corollary 4.1 If A 2L(H) then A is normal, e A e AΛ = e A+AΛ = e AΛ e A : Proof. The implication " ) is clear. " ( : Since A + AΛ is selfadjoint, ff(a + A Λ ) R. Thus A + A Λ 2 ^±(H) andff p (A + A Λ ) fkßi : k =1; 2;:::g = ;. Theorem 1.4 shows now that AA Λ = A Λ A. Corollary 4.2 If A; B 2L(H) are selfadjoint then A = B, e A = e B : Proof. The implication " ) is clear. " ( : Since A 2 ^±(H) and ff p (A) fkßi : k = 1; 2;:::g we see from Theorem 1.5 that AB = BA. Thus A B is selfadjoint ande A B = I. Take 2 ff(a B). Thus 2 R and e =1,hence =0.This gives ff(a B) =f0g. From ka Bk = r(a B) =0we get A = B. Corollary 4.3 Suppose that A and B are normal operators in L(H) and that e A = e B. Then A + A Λ = B + B Λ : Proof. Use Corollary 4.1 to see that e A+AΛ = e B+BΛ. By Corollary 4.2, A + A Λ = B + B Λ. Corollary 4.4 If A 2L(H) is normal then A = A Λ, e A is unitary. Proof. The implication " ) is clear. Now use Corollary 4.2 to derive A + A Λ =0. For our next result we need the following lemma (see also [8, Theorem 12.37]). 8

9 Lemma 4.1 If T 2 L(H) is invertible then there are selfadjoint operators A and B in L(H) such that T = e ia e B ; ff(a) [ ß; ß] and ß 62 ff p (A): Proof. If T is invertible, so are T Λ and T Λ T. Theorem in [8] shows that the positive square root (T Λ T ) 1=2 is also invertible. By [8, Theorem 12.35] there is a unitary U 2L(H) with T = U(T Λ T ) 1=2. Since ff((t Λ T ) 1=2 ) (0; 1), log is a continuous real function on ff((t Λ T ) 1=2 ). Thus the symbolic calculus for selfadjoint operators shows that there is a selfadjoint B 2L(H) such that (T Λ T ) 1=2 = e B. A. Wintner has shown in [16] that there is a selfadjoint A 2L(H) such that U = e ia, ff(a) [ ß; ß] and ß 62 ff p (A). Remarks. (1) It is shown in [13] that if U 2 L(H) is unitary then there is a unique selfadjoint operator A 2L(H) such that For related results see [9]. U = e ia ; ff(a) [ ß; ß] and ß 62 ff p (A): (2) Lemma 4.1 shows that an invertible operator in L(H) is the product of two exponentials. It is natural to ask whether every invertible operator is an exponential, rather than merely the product of two exponentials. The answer is affirmative ifdimh < 1, as a consequence of [8, Theorem 10.30]. But in general the answer is negative, as one can see from [8, Theorem 12.38]. For normal and invertible operators we have the following results. Corollary 4.5 Suppose that T 2L(H) is invertible. The following assertions are equivalent: (a) T is normal. (b) There is some normal S 2L(H) such that T = e S. Proof. (b) ) (a): Clear. (a) ) (b): By Lemma 4.1 there are selfadjoint operators A; B 2L(H) such that and T = e ia e B (1) ff(a) [ ß; ß] and ß 62 ff p (A): From T Λ = e B e ia and the normality oft we see that thus e 2B = T Λ T = TT Λ = e ia e 2B e ia ; 9

10 (2) e 2B e ia = e ia e 2B : Use (1) to get (3) ia 2 ^±(H) andff p (ia) fkßi : k =1; 2;:::g = ;. Since 2B is selfadjoint, we have (4) 2B 2 ^±(H) and ff p (2B) fkßi : k =1; 2;:::g = ;: Therefore it follows from (2), (3), (4) and Theorem 1.3(b) that AB = BA. Thus T = e ia+b. Put S = ia + B. ThenT = e S and S is normal. Corollary 4.6 Suppose that T 2L(H) is invertible and normal. Then there is a unique normal operator S 2L(H) such that T = e S ; r(s S Λ )» 2ß and 2ßi 62 ff p (S S Λ ): Proof. The proof of Corollary 4.5 shows that there is a normal S 2 L(H) with T = e S, S = ia + B, where A and B are selfadjoint, AB = BA, ff(a) [ ß; ß] and ß 62 ff p (A). Since S S Λ = 2iA, we get r(s S Λ )» 2ß and 2ßi 62 ff p (S S Λ ). Now suppose that R 2 L(H) is normal, T = e R, r(r R Λ )» 2ß and 2ßi 62 ff p (R R Λ ). Then there are selfadjoint operators C; D 2L(H) with From R R Λ =2iC we see that R = ic + D and CD = DC: ff(c) [ ß; ß] and ß 62 ff p (C): It follows from e S = e R that T Λ = e B e ia = e D e ic, thus e 2B = T Λ T = e 2D. Now use Corollary 4.2 to derive B = D. From e ia e B = e ic e D we see that e ia = e ic : It is shown in [13] that then A = C (see Remark (1)). Hence S = T. Our final result reads as follows: Corollary 4.7 For P 2L(H) the following assertions are equivalent: (a) e T +P = e T for all T 2L(H). (b) There is some k 2 Z such that P =2kßiI. 10

11 Proof. (b) ) (a): Clear. (a) ) (b): Take T 2 L(H) with r(t ) < ß. Proposition 2.2(e) shows that r(ffi T ) < 2ß. Thus, by Proposition 2.2(c), M T = ;, hence N(f(ffi T )) = f0g (Proposition 2.2(a)). From f(ffi T )(ffi T (T + P )) = e T (T + P )e T (T + P ) = e (T +P ) (T + P )e T +P (T + P ) = 0 we see that (T + P )T = T (T + P ), hence TP = PT. Therefore we have shown that (5) TP = PT for each T 2L(H) with r(t ) <ß. Now take T 2L(H) withr(t ) ß and put T 0 = ß T.Thenr(T 2r(T ) 0)= ß. (5) shows that 2 T 0 P = PT 0. Therefore we have that TP = PT for all T 2L(H). Thus P = ffi for some ff 2 C. Since e P = I, I = e ff I,hencee ff =1. References [1] H.R. Dowson: Spectral theory of linear operators. Academic Press (1978). [2] N. Dunford: Spectral operators. Pac. J. Math. 4 (1953), [3] E. Hille: On roots and logarithms of elements of a complex Banach algebra. Math. Ann. 136 (1958), [4] G. Lumer, M. Rosenblum: Linear operator equations. Proc. Amer. Math. Soc. 10 (1959), [5] T.W. Palmer: Banach algebras and the general theory of *-algebras. Volume I: Algebras and Banach algebras. Cambridge University Press (1994). [6] V. Pták, P. Vrbová: On the spectral function of a normal operator. Csechoslovak Math. J. 23 (1973), [7] C.R. Putnam: Ranges of normal and subnormal operators. Michigan Math. J. 18 (1971), [8] W. Rudin: Functional analysis. McGraw-Hill (1991). [9] Ch. Schmoeger: Über die Eindeutigkeit des Logarithmus eines unitären Operators. Nieuw Arch. Wiskd., IV, Ser. 15 (1997), [10] Ch. Schmoeger: Remarks on commuting exponentials in Banach algebras. Proc. Amer. Math. Soc. 127 (1999), [11] Ch. Schmoeger: Remarks on commuting exponentials in Banach algebras, II. Proc. Amer. Math. Soc. 128 (2000),

12 [12] Ch. Schmoeger: On normal operator exponentials. Proc. Amer. Math. Soc. (to appear). [13] M.H. Stone: Linear transformations in Hilbert space. American Mathematical Society (1932). [14] J. Wermer: Commuting spectral measures on Hilbert space. Pac. J. Math. 4 (1953), [15] E.M.E. Wermuth: A remark on commuting operator exponentials. Proc. Amer. Math. Soc. 125 (1997), [16] A. Wintner: Zur Theorie beschränkter Bilinearformen. Math. Z. 30 (1929), Christoph Schmoeger Mathematisches Institut I Universität Karlsruhe D Karlsruhe Germany christoph.schmoeger@math.uni-karlsruhe.de 12