Holomorphic vector-valued functions Paul Garrett garrett/ 1. Definition, examples

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1 (February 9, 2005) Holomorphic vector-valued fuctios Paul Garrett garrett/ Abstract: Oe of the first goals of a presetatio of classical complex fuctio theory is to prove Goursat s refiemet of Cauchy s theorem: complex differetiability implies the coclusio of Cauchy s theorem, hece Cauchy s itegral formula, hece complex aalyticity (expadability i power series). Thereafter, oe is typically very casual about termiology, usig complex differetiable ad aalytic ad holomorphic iterchageably. Ideed, use of the term holomorphic ofte sigals the completio of this basic Cauchy theory. Our goal here is to achieve the same effect for vector-valued fuctios. This does require a bit of rethikig power series with coefficiets i topological vector spaces. Defiitio, examples Appedix: Vector-valued power series, Abel s theorem. Defiitio, examples Oe of the first goals of a presetatio of classical complex fuctio theory is to prove Goursat s refiemet of Cauchy s theorem: complex differetiability implies the coclusio of Cauchy s theorem, hece Cauchy s itegral formula, hece complex aalyticity (expadability i power series). Thereafter, oe is typically very casual about termiology, usig complex differetiable ad aalytic ad holomorphic iterchageably. Ideed, use of the term holomorphic ofte sigals the completio of this basic Cauchy theory. Our goal here is to achieve the same effect for vector-valued fuctios. This does require a bit of rethikig power series with coefficiets i topological vector spaces (see the Appedix). Let V be a topological vector space ad f : D V be a V -valued fuctio o a ope set D C. Defiitio: The fuctio f is (strogly) complex-differetiable if, for all D, exists (i V ). lim w (f(w) f()) w Defiitio: The fuctio f is (strogly) aalytic if it is locally expressible as a coverget power series (with coefficiets i V ). Defiitio: The fuctio f is weakly holomorphic if, for all λ i the cotiuous dual V, the C-valued fuctio λ f is holomorphic i the classical sese. Remark: It is a classical fact, Goursat s refiemet of Cauchy s results, that complex-differetiable scalarvalued fuctios are, i fact, complex aalytic (locally represetable by coverget power series), ad, thus, by Abel s theorem, certaily ifiitely differetiable. Further, oe has the Cauchy formulas, otios of pole versus essetial sigularity, Lauret expasios at poles, ad so o. (Ideed, the aalyticity is prove via the Cauchy theory.) Theorem: For locally covex quasi-complete topological vector space V a weakly holomorphic V -valued fuctio f is strogly holomorphic. Ad the usual Cauchy-theory itegral formulas apply, for example f() = ζ dζ

2 Paul Garrett: Holomorphic vector-valued fuctios (February 9, 2005) where is a closed path with havig widig umber +. Ad f() is ifiitely differetiable, i fact expressible as a coverget power series f() = 0 c ( o ) with c =! f ( o ) = dζ (ζ ) + Remark: The (strog) cotiuity follows without the quasi-completeess, but the formulatio of Cauchy theory makes best sese if V is assumed quasi-complete (ad locally covex). Proof: First we show that weak holomorphy of f implies that f : D V is (strogly) cotiuous (that is, i the origial topology o V ). Without loss of geerality, we prove cotiuity at 0. We may also suppose that f(0) = 0 V. Let λ V. Sice λ f is holomorphic ad vaishes at 0, the fuctio (λ f)()/ iitially defied oly o a puctured disk at 0 exteds to a holomorphic fuctio o a disk about 0. By Cauchy theory for scalar-valued holomorphic fuctios, (λ f)() = (λ f)(ζ) dζ ζ ζ where is a circle of radius 2r cetered at 0, ad < r. Let M λ be the sup of λ f o. The the elemetary estimate (λ f)() (2π 2r) 2π r Mλ 2r = M λ r Thus, the set of values f() : r is weakly bouded (meaig that it is a bouded set whe V is give the weak topology from V ). But we kow that weak boudedess implies (strog) boudedess, so this set is bouded. That is, give a balaced covex eighborhood N of 0 i V, there is t o > 0 such that for complex w with w t o that set of values lies iside wn. The f() wn ad for < w we have f() N. sufficietly ear 0 This is (strog) cotiuity. We had take f(0) = 0, so we have prove that, give N, for f() f(0) N Now that we have the (strog) cotiuity, the rest of the argumet is early obvious, keepig i mid properties of Gelfad-Pettis itegrals. We certaily use the quasi-completeess for this. First, sice f() is ow kow to be (strogly) cotiuous, the itegral I() = exists as a Gelfad-Pettis itegral, ad thus for ay λ V λ(i()) = (λ f)(ζ) ζ ζ dζ dζ = (λ f)() by the holomorphy of λ f. Sice liear fuctioals separate poits, ecessarily I() is oe other tha f(), so we have the Cauchy itegral formula f() = ζ dζ 2

3 Paul Garrett: Holomorphic vector-valued fuctios (February 9, 2005) That is, the basic Cauchy formula is correct. However, otice that complex differetiability is ot really immediate. To prove complex differetiability of f at o, take o = 0 ad use f(0) = 0, for coveiece. Thus, there is a disk < 3r such that for every λ V (λ f)()/ exteds to a holomorphic fuctio F () o < r. The cotiuity assures that the itegral exists, ad by Cauchy theory for scalar-valued fuctios so sice fuctioals separate poits Now Thus, (λ f)() f() ζ(ζ ) dζ = (λ F )() = f() = ζ(ζ ) dζ ζ(ζ ) = ζ 2 + ζ 2 (ζ ) = ζ 2 dζ + Give a covex balaced eighborhood U of 0 i V, the set K = { : ζ = 2r} is compact, so cotaied i some multiple t o U of U. Thus, for < r, f() (λ f)(ζ) ζ(ζ ) dζ ζ 2 (ζ ) dζ ζ 2 dζ (2r) 2 r t ou Thus, as 0 the limit of f()/ exists. Sice f(0) = 0, this proves the complex differetiability of f. We leave the derivatio of the power series as a exercise i similar techiques. /// 2. Appedix: Vector-valued power series, Abel s theorem I this sectio V is a locally covex topological vector space, ad we further assume that V is quasi-complete, so that (for example) Cauchy sequeces i V coverge. Lemma: Let c be a bouded sequece of vectors i the locally covex quasi-complete topological vector space V. Let be a sequece of complex umbers, let 0 r be real umbers such that r, ad suppose that r < +. The the series c coverges i V. Further, give a covex balaced eighborhood U of 0 i V let t be a positive real such that for all complex ζ with ζ t we have {c } tu. The ( ) ( ) c tu r tu 3

4 Paul Garrett: Holomorphic vector-valued fuctios (February 9, 2005) Proof: If N is a covex balaced eighborhood of 0 i the topological vector space ad ad w are complex umbers with w, the N wn, sice /w implies (/w)n N, or N wn. Further, for a absolutely coverget series α of complex umbers, for ay o (α V ) = ( α V ) ( ) α N α N o o o < For a balaced ope U cotaiig 0, let t be large eough such that for ay complex ζ with ζ t the sequece c is cotaied i ζu. The previous discussio shows that m l c l l ( m ) tu Give ε > 0, ivokig absolute covergece, take m sufficietly large such that for all m The m l m < t ε c l l t (ε/t) U = U Thus, the origial series is coverget. Sice X is quasi-complete the limit exists i V. The last cotaimet assertio follows from this discussio, as well. /// Corollary: Let c be a bouded sequece of vectors i a locally covex quasi-complete topological vector space V. The o < the series f() = c coverges ad gives a holomorphic (ifiitely-may times complex-differetiable) V -valued fuctio. Proof: The lemma shows that the series expressig f() ad its apparet k th derivative c ( k) k all coverge for <. The usual direct proof of Abel s theorem o the differetiability of (scalar-valued) power series ca be adapted to prove the ifiite differetiability of the X-valued fuctio give by this power series, as follows. Let g() = c 0 The f() f(w) w g(w) = c ( w w w ) For =, the expressio i the paretheses is. For >, it is ( + 2 w w 2 + w ) w = ( w ) + ( 2 w w ) ( 2 w 3 w ) + (w 2 w ) + (w w ) = ( w) [ ( w 2 ) + w( w 3 ) w 3 ( + w) + w ] 2 = ( w) (k + ) 2 k w k k=0 For r ad w r the latter expressio is domiated by 2 ( ) w r < w 2 r 2 2 4

5 Paul Garrett: Holomorphic vector-valued fuctios (February 9, 2005) Let U be a balaced eighborhood of 0 i X, ad t a sufficietly large real umber such that for all complex ζ with ζ t all c lie i ζu. For r < ad w r <, by the lemma, f() f(w) w g(w) = ( w) 2 c ( 2 ) ( ) (k + ) 2 k w k ( w) 2 r 2 tu Thus, for ay give covex balaced eighborhood U of 0 i X, as w k=0 f() f(w) w g(w) evetually lies i U. /// Corollary: Let c be a sequece of vectors i a Baach space X such that for some r > 0 the series c r coverges i X. The for < r the series f() = c coverges ad gives a holomorphic (ifiitely-may times complex-differetiable) X-valued fuctio. /// 5