Angular Derivatives of Holomorphic Maps in Infinite Dimensions

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1 JOURNAL OF MATHEMATCAL ANALYSS AND APPLCATONS 0, ARTCLE NO. 01 Angular Derivatives of Holomorphic Maps in nfinite Dimensions Kazimierz Włodarczyk Alina Szałowska nstitute of Mathematics, Uniersity of Łodz, Banacha, Łodz, Pol Submitted by John Horath Received February 3, 1995 Let H be a complex Hilbert space let L Ž H, H. be the complex Banach space of all bounded linear operators from H to H with the operator norm. n the generalized right half-planes the unit balls contained in H L Ž H, H., infinite-dimensional angular sets are defined, various new generalizations of the classical PickJulia theorem to infinite dimensions are proved, conditions of CaratheodoryFan type, guaranteeing the existence of angular limits angular derivatives of holomorphic maps of these generalized right half-planes unit balls, are established Academic Press, nc. 1. NTRODUCTON Extensions applications of Caratheodory s conditions which guarantee the existence of angular limits angular derivatives of holomorphic maps have proved to be very useful in studying many problems in complex analysis have recently been studied extensively. However, they are mainly considered in the finite-dimensional setting Ža survey appears in R. B. Burckel, C. Caratheodory 35, B. G. Eke 6, J. L. Goldberg 8, E. Lau G. Valiron 10, B. D. MacCluer J. H. Shapiro 11, K. Nevanlinna 13, Ch. Pommerenke 1, W. Rudin 15, D. Sarason 16, J. H. Shapiro 17, J. H. Shapiro, W. Smith, D. A. Stegenga 18, R. K. Singh J. S. Manhas 19, G. Valiron 0, E. Warschawski 1, others.. n this paper, using some methods of nonlinear functional analysis, infinite dimensional holomorphy, spectral theory, operator theory, the theory of infinite-dimensional bounded symmetric homogeneous domains, we study problems concerning the existence of angular limits angular X96 $18.00 Copyright 1996 by Academic Press, nc. All rights of reproduction in any form reserved.

2 WŁODARCZYK AND SZAŁOWSKA derivatives by eliminating the assumption that the spaces considered are of finite dimensions. The writing of the present paper was inspired by the results of T. Ando Ky Fan 1 Ky Fan 7. Before summarizing our work, we give a brief review of some known results. Let x : Rex0, for a positive number k, let x:mxkre x. Ž 1.1. k Let x : x 1 let ABC be the triangle whose interior lies entirely within, where A is the point x 1 BC is perpendicular to the real axis. There is a positive number M such that, for all values of x within the triangle ABC, 1xMŽ 1x.. Ž 1.. Caratheodory proved: THEOREM 1.1. Let f:, be a holomorphic map. f then, for any k 0, we hae Re fž x. L inf, x Re x fž x. Re fž x. lim lim lim Df Ž x. L. x Re x x, xk x, xk x, xk THEOREM 1.. Let F: be a holomorphic map. f x n is any sequence of numbers lying within the triangle ABC tending to x 1, then lim1 FŽ x. n 1xn 1 exists as n. This limit is either or a number L 0. n the second case, we also hae lim DFŽ x. Lasn. n these two cases, the numbers L are referred to as angular limits angular derivatives. The sets defined by Ž 1.1. Ž 1.. are called angular sets. Let H be a complex Hilbert space, let ², : denote the inner product on H, let denote the norm on H defined by the formula xž² x, x :. 1, xh. Let L Ž H, H. be the complex Banach space of all bounded linear operators from H to H with the operator norm, let denote the identity map on H, let X L Ž H, H.:ReX0. Ky Fan 7, using the results of T. Ando Ky Fan 1 concerning some generalization of the PickJulia theorem, attempted to extend Theorem n

3 ANGULAR DERVATVES to operator-valued holomorphic maps. However, his formulation of the result is different from that in Caratheodory. He proved the following result: THEOREM 1.3. Let f: be a holomorphic map. Suppose there is a Hermitian operator A L Ž H, H. satisfying Re fž x. A for all x Re x, for any 0, there is z such that Then, for any k 0, we hae Re fž z. A. Re z fž x. Re fž x. lim A lim A x x, x Re x x, xk k lim Df Ž x. A 0. x, x k From a number of theoretical points of view, it is desirable to possess generalizations of Caratheodory s results in higher dimensions. The case of n n holomorphic maps f: Bn Bn of the unit ball Bn in, Bn z : z1, zž² z,z :. 1, was studied by W. Rudin 15 in angular sets D B n, 1, n D z z,..., z :1z 1z. 1 n 1 n infinite-dimensional angular sets, the situation is much more complicated it is not easy to find formulations which are analogous to the finite-variable results likely to hold true. The object of this note is to study the above mentioned problems considered by Caratheodory, Pick, Julia, Rudin, Fan to infinite dimensions. Let H xh:re² x,: x² x,: 0, where H, 1, H xh: x1, X LŽ H, H.:ReX0, 0 X LŽ H, H.: X1. 0

4 WŁODARCZYK AND SZAŁOWSKA The biholomorphic map f of H onto H is defined by the formula 0 Ž. see, Theorem 5, p. 99; 9, Theorem 1, p. 35, moreover, 1 Ž ² :. 0 f x x 1 x,, xh, f y Q y 1² y, : V,QVŽ H E V., yh, Ž 1.. where E denotes a linear projection of H onto the subspace V u:u.the biholomorphic map f of 0 onto is defined by the formula Žsee, Theorem 5, p. 99; 9, Theorem 1, p. 35., moreover, 1 0 f X X X, X, f Y Y Y, Y. 1.6 Let T y: H0 H 0, y H 0, denote the Mobius biholomorphic map of the form Žsee 9, Theorem, p E 1y y Ey Ž xy. TyŽ x., xh 0, Ž ² x, y: where E denotes a linear projection of H onto the subspace uy: u y. For Y 0, let A Y *Y B YY *, Ž 1.8. Y let T Y : 0 0 denote the Mobius biholomorphic map of the form Žsee 9, Theorem, p Y Y Y 0 T X B XY Y*X A, X. 1.9 The unit balls H0 0 are bounded symmetric homogeneous domains. n this paper, using the maps defined by formulae Ž 1.3. Ž 1.9., we first define characterize infinite-dimensional angular sets in H, H 0,, Ž see Definitions.1, 3.1,.1, 5.1 Propositions We next generalize the classical PickJulia theorem to holomorphic maps F: H, F: H, F:, F: Ž 0 0 see Theorems.1, 3.1,.1, 5.1., then, as an application, we show the conditions of CaratheodoryFan type, guaranteeing the existence of angular limits angular derivatives of holomorphic maps F: H H, F: H0 H 0, Y

5 ANGULAR DERVATVES 5 F:, F: in such angular sets Ž 0 0 see Theorems., 3.,., 5... Also, included are some examples. This paper is a continuation of the studies in 3, 5.. PCKJULA THEOREMS, CHARACTERZATON OF ANGULAR SETS, AND THE EXSTENCE OF ANGULAR LMTS AND DERVATVES OF HOLOMORPHC MAPS F: HV HV For H, 1, set where For x, z H, let H xh:rex,: 0, Ž.1. RE² x, : Re² x, : x ² x, :. P ² x,: ², z: ² x, z: ², z:² x, :. Ž.., x, z t is evident that P, x, z P, z, x P Re² x,: x² x,: Re² x, :., x, x Our first result of PickJulia type is THEOREM.1. Let H be a complex Hilbert space let H where 1. f F: H is a holomorphic map such that FŽ z. 1 for some z H, then 1, z, x Ž, x, x, z, z. F x P P P.3 for all x H. Here H P are defined by.1., respectiely., x, z Proof. Let f:, x : x 1, be a Cayley biholomorphic map of the form 1 fž x. Ž 1x.Ž 1x., x. 1. Ž 1 Let r f z here f is defined by Ž Applying the Schwarz lemma to the holomorphic map g: H, gž , defined by the formula gž y. f 1 Ff T 1 Ž y., yh r 0

6 6 WŁODARCZYK AND SZAŁOWSKA Žhere T is defined by Ž we obtain gž y.y r, yh 0. n particular, Ž 1 for y T f.ž x., xh, we get Thus r 1 1 f F x Trf x, xh f F Ž x. f F Ž x., where Tr f Ž x.,, in particular, 1 1 F x F x 1 1 F x 1 1 F x 10 or, equivalently, because F x n consequence, we have 1 1 f F x F x 1 1F x Ž r. FŽ x. 1 1, T f Ž x.. Ž.. Hence it follows that ž / 1 1 r FŽ x. 1 T Ž s., xh, sf Ž x.. This, together with the identity where is equivalent to ž Ž r / 1 1 T Ž s. TŽ r, s., TŽ r, s. 1² s, r: 1r 1s, Ž F x T r, s, sf x, rf z, xh..6 Moreover, by , ² :, z, z ² :, x, x 1 r P 1 z,, 1 s P 1 x,, 1

7 ANGULAR DERVATVES 7 1 1² s, r: P Ž 1² x,:.ž 1², z :.., x, z Thus 1, x, z Ž, z, z, x, x. T r, s P P P..7 Consequently, taking account of.6.7, we get.3. DEFNTON.1. f H is such that 1, then, for 1, define let D Ž. xh:1 ² x,:p Ž.8. V, X, X CŽ ; x. P 1² x, :. Ž.9., x, x For 1, we call D angular sets determined by. We require this easy fact: PROPOSTON.1. f 1, then D Ž. H. Moreoer, if x D Ž., then 1 Ž ² :. Q x 1 x, 1 or, equialently, C ; x 0 Ž.10. if only if f 1, then D Ž.. 1 Ž ² :. Q x 1 x, Proof. For f defined by 1. for 1, let 1 1 ½ ² : ž / 5 D xh: 1 f x, 1 f x..1 Of course, D Ž. H for all 1. When 1, this set is empty. Let us now observe that ² : f Ž x., 1 Ž ² x, :..

8 8 WŁODARCZYK AND SZAŁOWSKA Moreover, ž / ² : 1, x, x 1 f x 1 x, P. Thus Ž.1. Ž.8. are identical. Now, note that if x D Ž. f 1 Ž x.1, then, by Ž.1., 1 ² 1 : 1 f x, 0, which implies that s f Ž x.. Thus Ž.10. implies Ž.11.. The converse is obvious. We are now able to formulate one of our main results: THEOREM.. Let H be a complex Hilbert space let H where 1. Let F: H H be a map holomorphic in H. Ž. a Suppose there is a positie number L satisfying Re² FŽ x.,: LRe² x, : Ž.13. for all x H. f D Ž., 1, sts for an angular set such that, for any 0, there exists a point z D Ž. for which the inequality holds, then, for any 1, we hae Re² F z,: Re z, L.1 1 ² : ² : 1 ² : lim F x, x, L 0,.15 ² : 1 ² : lim Re F x, Re x, L 0.16 ² : ² : Ž ² :. lim L F x, DF x, x, 0.17 as CŽ ; x. 0, x D Ž.. Ž b. Suppose there is a positie number L satisfying Re² FŽ x.,: LRe² x, : Ž.18. for all x H. Then, for any 1, assertions Ž.15. Ž.17. hold as CŽ ; x. 0, x D Ž.. Here H, D Ž., C Ž ; x. are defined by Ž.1., Ž.8., Ž.9., respectiely. Proof. We define a holomorphic map M: H by the formula ² : ² : M x 1 f x, 1 f x,, xh..19

9 ANGULAR DERVATVES 9 Then, for all x H, we have ² : ² : 1 1 MŽ x. since Re M Ž x. 1 f Ž x., 1 f Ž x.,, for x H p H, ² : ² : Ž.0. D M x p 1 f x, Df x p,.1 ² : 1 Df x p, 1² x,: ² p, :.. Consequently, using 1.., from.19.1 we have, for x H p H, MŽ x. ² x, :, Re MŽ x. Re² x, : Ž.3. 1 Ž ² :. ² : D M x p x, p,,. respectively. Let 0 be arbitrary fixed. By Ž.1., there exists z D Ž. such that 1 ReŽ M F.Ž z. Re M Ž z. L. We define maps E G, holomorphic in H, by the formulae EŽ x. Ž MF.Ž x. L MŽ x. 1 GŽ x. Re EŽ z. EŽ x. i m EŽ z., Ž.5. respectively. Let us observe that, by Ž.13., Re EŽ x. 0 Re GŽ x. 0 for all x H, GŽ z. 1. Applying Ž.6. Ž.. to the map G, we get 1 1 G x T r, s 1² r,s: 1s 1r, xh,.6

10 10 WŁODARCZYK AND SZAŁOWSKA 1 1 where r f z, sf Ž x.. Now, from Ž.5. we obtain EŽ x. MŽ x. 1 1 Ž MF.Ž x. MŽ x. L 1 MŽ x. Re EŽ z. GŽ x. im EŽ z. 1 M x Re E z G x m E z ½ 1 1 M x Re M z Re E z Re M z G x Let 1 be arbitrary fixed. Since, by.19, m EŽ z.5. by.0, 1 1 MŽ x. 1² s,: 1² s, :, Re M Ž z. 1 ² r, : 1 ² r, : 1 1² r,: 1² r,: 1 ² r,: 1 Ž r., 1 1 1² s,: 1s, 1 ² r, : 1 r, by.6, 1 ² r,: 1 1² r, s: 1 EŽ x. MŽ x. ² : 1² r,: 1 s, 1r 1 MŽ x. m EŽ z.. Ž.7. Consequently, since the right-h side of inequality Ž.7. Žby Ž.1., Ž.10., Ž.11.. tends to 1 ² r,: 1² r,: 1² r,: Ž 1 r. 1² r,: 1r 1r 0 can be arbitrarily small, therefore 1 lim Ž M F.Ž x. MŽ x. L 0 1 as f Ž x., xd Ž., i.e., Ž.15. holds.

11 ANGULAR DERVATVES 11 Now, let us observe that 1 1 Re E x Re M x Re M F x Re M x L But from we get EŽ x. Re M Ž x E x M x M x Re M x. MŽ x. 1² s,: Ž 1s. 1 respectively. Thus 1 1 ² : Ž. Re M x 1 s, 1 s, 1 1 Re M F x Re M x L E x M x. Since, by.15, the right-h side of the above inequality tends to zero, we have 1 lim ReŽ M F.Ž x. Re M Ž x. L 0 1 as f Ž x., xd Ž., i.e., by Ž.3., we have Ž.16.. Let 1. We shall need the following relation between D Ž. D Ž.. Assume that f then 1 Ž 13.Ž 11. Ž ² : ž / x D, i.e., 1 f x, 1 f x. Ž.9. ² : 1 1 f x,,.30 ² : 1 1 f f x D, i.e., 1 f x, 1 ž / Ž. 1 f Ž x.. Ž.31.

12 1 WŁODARCZYK AND SZAŁOWSKA ndeed, from.8 we have,, 5 Ž.3. whenever is sufficiently small. From.9 we get ² : 1 1 f x 1 f x, Thus, using.3,.30,.33, we obtain ² : 1 1 f x 1 f x, ² : 1 1 f Ž x. 3 Ž. 1 f Ž x., Ž. ² : 1 1 f Ž x. 5 Ž. 1 f Ž x., ² : 1 1 f Ž x. Ž. 1 f Ž x., 1. This immediately yields Ž Now, we prove Ž.17.. By the Cauchy integral formula 1, Proposition, p. 1, 1 1 D L MF x M x 1 1 H Ž V. 1 ½ 5 Mf s i L Mf s MFf s 1 d, Ž.3. 1 it where s f x, e, Ž x. 1² s, :, t0;, f is defined by Ž But 1 1 Ž Mf.Ž s. 1 1² s,: 1² s,: 1 1 Ž 1. 1² s, :. 1 Since the right-h side of the above inequality tends to Ž 1 1,. from Ž.3. we get Ž.17. by using Ž.3., Ž.., Ž.15., Ž.8. Ž.31.. Ž b. f Ž.18. holds for all x H, let 0 be arbitrary fixed let be such that 0. Then ReŽ M F.Ž x. Re M Ž x. L Re M Ž x. 1

13 ANGULAR DERVATVES 13 for all x H. Moreover, obviously, there exists some z D Ž. for which the inequality 1 1 L Re M F z Re M z 1 holds. Now, we define maps E formulae G, holomorphic in H, by the 1 EŽ x. EŽ x. MŽ x., EŽ x. L Ž MF.Ž x. MŽ x. 1 G Ž x. Re E Ž z. E Ž x. im E Ž z., respectively. Let us note that Re E Ž x. 0 Re G Ž x. 0 for all xh, that G Ž z. 1. Using analogous considerations as in part Ž a., 1 1 we have, respectively, for r f z s f Ž x., E x M x L MF x M x 1 1 M x Re E z G x m E z. Ž. Thus, for any 1, using analogous arguments as in part a, we obtain 1 lim EŽ x. MŽ x. 1 as f Ž x., xd Ž.. This implies Ž.15.. Also, using analogous arguments as in part Ž. a, we prove that also Ž.16. Ž.17. hold as 1 f Ž x. Žin all angular sets D Ž., 1.. EXAMPLE.1. Let F: H H be a holomorphic map defined by the formula 1 F x f f x, xh, 1 where f f are defined by Ž 1.3. Ž 1.., respectively. Then we have FŽ x. x ² x,:, xh,, consequently, for L, we obtain ² : Re FŽ x., 1Re² x,: Re² x, :, xh,

14 1 WŁODARCZYK AND SZAŁOWSKA 1 Ž 1. Re² FŽ x.,: Re² x, : 1 1 Re x, 1 Re M x ² 1 : 1 ½ ² : ² : 5 ½ ² : f x, 1, f x 1 1Re 1 f x,, xh. Thus, by Proposition.1, F satisfies the assertions of Theorem.. 3. PCKJULA THEOREMS, CHARACTERZATON OF ANGULAR SETS, AND THE EXSTENCE OF ANGULAR LMTS AND DERVATVES OF HOLOMORPHC MAPS F: H H 0 0 n this case, we need some auxiliary results analogous to Theorem.1. THEOREM 3.1. Let H be a complex Hilbert space. f F: H0 is a holomorphic map such that FŽ z. 1 for some z H, then FŽ x. 1 T Ž x. 1 T Ž x. Ž 3.1. z for all x H. Here T is defined by the formula Ž z. Proof. Using analogous considerations as in the proof of Theorem.1 where the map f, defined by Ž 1.3., is replaced by the identity, we obtain that inequality Ž.. is identical with Ž DEFNTON 3.1. Let H be a complex Hilbert space let H 0. For 1, let D Ž. xh: 1² x,: Ž. 1x. Ž 3.. z 0 1 We call D Ž., 1, angular sets. Of course, D Ž. H0 for all 1. When 1, this set is empty. Define a holomorphic map M: H0 by the formula 1 MŽ x. 1² x,: 1² x, :, xh. Ž 3.3. Let us observe that Re M Ž x. 1 ² x, : 1 ² x, :, x H. Ž

15 ANGULAR DERVATVES 15 Obviously, M x for all x H. Moreover, for x H p H, ² : Ž ² :. D M x p p, 1 x,. As a consequence of Theorem 3.1 we derive the following THEOREM 3.. Let H be a complex Hilbert space, let F: H0 H0 be a map holomorphic in H 0, let H 0. Ž. a Suppose there is a positie number L satisfying L ReŽ M F.Ž x. Re M Ž x. Ž 3.5. for all x H. f D Ž. 0, 1, sts for an angular set such that, for any 0, there exists a point z D Ž. for which the inequality 1 L ReŽ M F.Ž z. Re M Ž z. 1 Ž 3.6. holds, then, for any 1, we hae 1 lim Ž M F.Ž x. MŽ x. L 0, Ž lim ReŽ M F.Ž x. Re M Ž x. L 0, Ž lim D M F x L M x as x, x D Ž.. Ž b. Suppose there is a positie number L satisfying L ReŽ M F.Ž x. Re M Ž x. Ž for all x H. Then, for any 1, assertions Ž 3.7. Ž hold as x, xd Ž.. Here M D Ž. are defined by Ž 3.3. Ž 3.., respectiely. Proof. Using arguments similar to those given in the proof of Theorem. where M defined by Ž.19. is replaced by that from Ž 3.3. Ži.e., the map f is replaced by the identity. but E is defined by the formula EŽ x. L Ž MF.Ž x. MŽ x., xh 0, we immediately get the assertions of Theorem 3.. EXAMPLE 3.1. Let F: H0 H0 be a holomorphic map defined by the formula FŽ x. Ž 1.Ž x., xh.

16 16 WŁODARCZYK AND SZAŁOWSKA Then, by 3., ReŽ M F.Ž x. 1² x,: 1² x, :, so, for L 1, we obtain Thus 3.5 holds. Moreover, L ReŽ M F.Ž x. Re M Ž x L ReŽ M F.Ž x. Re M Ž x. 1 Thus, 3.6 holds, too. L ReŽ M F.Ž x. Re M Ž x. Re M Ž x. Ž 1. Re MŽ x ² : ² : x, 1, x Re1² x, :, xh. 0. PCKJULA THEOREMS, CHARACTERZATON OF ANGULAR SETS, AND THE EXSTENCE OF ANGULAR LMTS AND DERVATVES OF HOLOMORPHC MAPS F: Let denote the identity map on H let X LŽ H, H.:ReX0, X LŽ H, H.: X1. We get counterparts of Theorems More precisely, we shall prove the following THEOREM.1. f F: is a holomorphic map such that FŽ Z. for some Z, then for all X. 1 1 F X Re Z XZ* Re X 1 Ž 1 Proof. Let R f Z here f is defined by Ž Applying the Schwarz lemma to the holomorphic map h:, hž , defined by the formula hž Y. f 1 Ff T 1 Ž Y., Y 0 R 0

17 ANGULAR DERVATVES 17 Žhere T is defined by Ž 1.8. Ž we obtain hy Y R,Y.n 0 Ž 1 particular, for Y T f.ž X., X, we get Thus R 1 1 f F X TRf X, X f F Ž X. * f F Ž X., where TR f Ž X.,, in particular, or, equivalently, 1 FŽ X.*FŽ X. Ž 1.Ž 1. FŽ X.* 1 Ž 1.Ž 1. FŽ X. 0 because F X n consequence, we have 1 1 f F X F X F X Ž R. FŽ X. 1 1, T f Ž X.. Ž.1. Hence it follows that FŽ X.1 Ž T Ž S.. 1, X, Sf 1 Ž X. R. This, together with the identity, formula Ž 18., p. 7 where is equivalent to ž R / 1 1 T Ž S. TŽ R,S., T R,S AR R*S AS S*R A R,. 1 1 F X T R,S, Sf X, Rf Z, X..3 Moreover, by , 1 1 A Z* Re Z Z, R 1 1 A X* Re X X, S 1 1 R*S Z* XZ* X.

18 18 WŁODARCZYK AND SZAŁOWSKA Consequently, Z T R,S W Re Z XZ* Re X Ž Z. 1 1 Ž ZX*.Ž Re Z. W *, Ž.. where W Z is a unitary operator of the form W Ž Re Z. Ž Z*. Ž Z*. Ž Re Z.Ž Z., Z which, by Ž.3. Ž.., yields the assertion of Theorem.1. DEFNTON 5.1. For 1, let let ½ D X L Ž H, H.: Ž X X XX* X* C X X Re X X*..6 We call D, 1, angular sets. We shall need the following PROPOSTON.1. f 1, then D. Moreoer, if X D, then if only if 1 Ž X.Ž X. 1 or, equivalently, CŽ X. 0 Ž.7. f 1, then D. Proof. For 1, let 1 Ž X.Ž X.. Ž ½ ž / 5 D X L Ž H, H.: f Ž X. Ž. 1 f Ž X. Ž.9. Ž 1 here f is defined by Ž Of course, D for all 1. When 1, this set is empty. Now, observe that 1 1 f X X.10 5

19 ANGULAR DERVATVES 19, using the spectrum, show that ž / ½ f X sup f X *f X 1 Ž 1. Ž X.Ž Re X. Ž X*.. Ž.11. Consequently, Ž.9. Ž.5. are identical. Now, note that if X D f 1 Ž X.1 or, by Ž.11., equiva- 1 lently C X 0, then, by.9, f Ž X. 0, which implies S 1 f Ž X.. n consequence, Ž.7. implies Ž.8.. The converse is obvious. An analogue of Theorems. 3. is THEOREM.. Let F: be a holomorphic map. Ž. a Suppose there is a Hermitian operator A L Ž H, H. satisfying 1 1 Re FŽ X. A Re XA Ž.1. for all X. f D, 1, sts for an angular set such that, for any 0, there exists an operator Z D for which the inequality Re Z A Re FŽ Z. A Re Z Ž.13. holds, then, for any 1, we hae lim X A FŽ X. A X 0, Ž lim Re X A Re FŽ X. A Re X 0, Ž lim A FŽ X. DFŽ X.Ž. FŽ X. A X 0 Ž.16. as CŽ X. 0, X D. Ž b. Suppose there is a Hermitian operator A L Ž H, H. satisfying 1 1 Re FŽ X. A Re XA Ž.17. for all X. Then for any 1, assertions Ž.1. Ž.16. hold as CŽ X. 0, X D. Here D CŽ X. are defined by Ž.5. Ž.6., respectiely. Proof. We define a holomorphic map M: L Ž H, H. by the formula M X f X f X, X..18

20 0 WŁODARCZYK AND SZAŁOWSKA Observe that Re M Ž X. f Ž X.* f Ž X.*f Ž X. f Ž X., 1 X. Ž.19. Obviously, the operator MŽ X. is invertible, i.e., MŽ X. 1 exists MŽ X. for all X. Now, let us notice that D M Ž X. Ž P. f Ž X. Df Ž X.Ž P. f Ž X., by, p. 501, 1 Ž Df X P X P X, X, P L H, H. Ž.1. Consequently, using 1.6, we obtain M Ž X. X, Re MŽ X. Re X, D MŽ X. Ž P. X PX. Ž.. Ž. a Let 0 be arbitrary fixed. By Ž.13., there exists Z D such that Re M Z A Re M F Z A Re M Z. We define maps E G, holomorphic in, by the formulae 1 1 E X A MF X A M X 1 1 G X Re E Z E X i m E Z Re E Z,.3 respectively. Let us observe that, by Ž.1., Re EŽ X. 0 Re GŽ X. 0 for all X, GŽ X.. Applying Theorem.1 to the map G, we get, by Ž.. Ž.3., G X T R,S AR R*S AS S*R A R, X, Ž..

21 ANGULAR DERVATVES where R f Z, Sf Ž X.. Now, from Ž.3. we obtain 1 1 M X E X M X M X A MF X A M X MŽ X. Re EŽ Z. GŽ X. Re EŽ Z. i m EŽ Z. 1 M X Re E Z G X m E Z MŽ X. Re M Ž Z. Re EŽ Z. Re M Ž Z. 1 Re M Ž Z. GŽ X. MŽ X. m EŽ Z.. Let 1 be arbitrary fixed. Since, by.19, we have 1 1 M X S S 1 1 Re M Z R* R*R R 1 R* R*R* R Ž. Ž. 1 R R* 1 R* 1 Ž R. R*, 1 1 S 1S, R 1 R, by., Ž.Ž. G X T R,S S*R 1 S 1 R, therefore, it follows that 1 1 M X E X M X 1 Ž. S S*R 1 R 1 1 S S m E Z..5 Consequently, since the right-h side of inequality Ž.5., by Ž.7., Ž.8., Ž 1.6., tends to R Ž 1 R. 3 R 1 R Ž 1 R. 0 can be arbitrarily small, therefore lim M X A MF X A M X 0 1 as f Ž X., XD, i.e., Ž.1. holds.

22 WŁODARCZYK AND SZAŁOWSKA But Now, let us observe that 1 1 Re M X Re E X Re M X 1 1 Re M X A Re M F X EŽ X. Re M Ž X. 1 1 A Re M Ž X MŽ X. EŽ X. MŽ X. MŽ X. Re M Ž X Ž. Ž. M X S 1 S 1 S 1 1 Re M X S 1 S 1S. Thus 1 1 Re M X Re E X Re M X 1 1 MŽ X. EŽ X. MŽ X.. Since, by.1, the right-h side of the above inequality tends to zero, we have lim Re M X A Re M F X A Re M X 0 1 as f Ž X., XD, i.e., Ž.15. holds. Let 1. We shall need the following relation between D D. Assume that f 1 Ž 13.Ž 11. Ž ž / X D, i.e., f Ž X. Ž. 1 f Ž X.. Ž.7. 1 f X,.8

23 ANGULAR DERVATVES 3 then 1 f f X D,.9 i.e,. f 1 Ž X. Ž.Ž1 f 1 Ž X... ndeed, from Ž.6. we have,, 5, Ž.30. whenever is sufficiently small. From.7 we get 1 1 f X f X Thus, using.30,.8,.31, we obtain 1 1 f Ž X. Ž. f Ž X. 1 1 f Ž X. 3 Ž. f Ž X. Ž. 1 1 f X 5 f X 1 1 f Ž X. Ž. f Ž X. 1. This immediately gives Ž.9.. Now, we prove Ž.16.. By Proposition. the Cauchy integral formula 1, Proposition, p. 1, D A MF X A M X 1 1 H Ž. r 1 1 ½ Mf S i Ž Mf.Ž S. A Ž MFf.Ž S Ž. 5 A Mf S 1 Mf S d,.3 1 it where S f X, re, r rž X. S, t0; f is defined by Ž But 1 Ž Mf.Ž S. S Ž S S.

24 WŁODARCZYK AND SZAŁOWSKA 1 Since the right-h side of the above inequality tends to Ž 1 1,. from Ž.3. we get Ž.16. by using Ž.1., Ž.6. Ž.9., Ž.0. Ž... Ž b. f Ž.17. holds for all X, let 0 be arbitrary fixed let be such that 0. Then 1 1 A Re M F X A Re M X Re M X for all X. Moreover, obviously, there exists some Z D the inequality for which Re M Z A Re MF Z A Re M Z holds. Now, we define maps E G, holomorphic in, by the formulae EŽ X. EŽ X. MŽ X., 1 1 E X A MF X A M X 1 1 G Ž X. Re E Ž Z. E Ž X. i m E Ž Z. Re E Ž Z., respectively. Note that Re E Ž X. 0 Re G Ž X. 0 for all X, that G Ž Z.. Using analogous considerations as in part Ž a., we 1 1 have, respectively, for R f Z S f Ž X., 1 1 M X E X M X M X A MF X A M X 1 M X Re E Z G X m E Z. Ž. Thus, for any 1, using analogous arguments as in part a, we obtain 1 1 lim M Ž X. EŽ X. MŽ X. 1 as f Ž X., XD. This implies Ž.1.. Using arguments similar to those given in part Ž. a, we prove that also Ž.15. Ž.16. hold as 1 f Ž X. in all angular sets D, 1. EXAMPLE.1. formula Let F: be a holomorphic map defined by the 1 F X f f X, X,

25 ANGULAR DERVATVES 5 1 where f f are defined by Ž 1.5. Ž 1.6., respectively. Then, for A, we get FŽ X. Ž X., Re FŽ X. A Re XA A Re XA, X. Thus.1 holds. We also get Re M X A Re M F X A Re M X Ž 1. Re MŽ X ½ ½ 5 Ž 1. f Ž X. f Ž X.* Ž 1. Re f Ž X., X, where M Re M are defined by Ž.18. Ž.19., respectively. Consequently, condition Ž.13. is satisfied in D for some 1if Asince 1 f Ž X., XD. 5. PCKJULA THEOREMS, CHARACTERZATON OF ANGULAR SETS, AND THE EXSTENCE OF ANGULAR LMTS AND DERVATVES OF HOLOMORPHC MAPS F: 0 0 Now, we can state the results in this case: THEOREM 5.1. f F: is a holomorphic map such that FŽ Z. 0 for some Z 0, then for all X. 0 FŽ X. 1 T Ž X. 1 T Ž X. Ž 5.1. Z Proof. Using analogous arguments as in the proof of Theorem.1 replacing map Ž 1.5. by the identity, we get that Ž.1. is identical with Ž We define a holomorphic map M: L H, H by the formula M X X X, X. 5. PROPOSTON 5.1. The operator MŽ X. is inertible MŽ X. for all X. Moreoer, for X P, D M X P X P X. 5.3 Z 1

26 6 WŁODARCZYK AND SZAŁOWSKA Proof. Observe that Ž.18. Ž.. when the map f is replaced by the identity map imply Ž DEFNTON 5.1. For 1, let D X: X Ž. 1X. Ž 5.. We call D, 1, angular sets. Of course, D 0 for all 1. When 1, this set is empty. THEOREM 5.. Let F: be a map holomorphic in Ž. a Suppose there is a Hermitian operator A satisfying 1 1 A Re M F X A Re M X 5.5 for all X 0. f D, 1, sts for an angular set such that, for any 0, there exists a point Z D for which the inequality Re M Z A Re M F Z A Re M Z holds, then, for any 1, we hae lim M Ž X. 1 1 A Ž MF.Ž X. 1 1 A MŽ X. 1 0, Ž lim Re M X A Re M F X 1 1 A Re M Ž X. 0, Ž lim D A MF X A M X as X, X D. b Suppose there is a Hermitian operator A L H, H satisfying 1 1 A Re M F X A Re M X for all X. Then, for any 1, assertions Ž 5.6. Ž hold as X, XD. Here M D are defined by Ž 5.. Ž 5.., respectiely. Proof. For M defined by Ž 5.., defining a map E by the formula E X A MF X A M X, X, using arguments similar to those given in the proof of Theorem., we immediately obtain the assertions of Theorem 5..

27 ANGULAR DERVATVES 7 EXAMPLE 5.1. Let F: be of the form 0 0 Then ReŽ M F.Ž X. FŽ X. Ž 1.Ž X., X X* X*X X* X X. Hence, for A 1, we get 1 1 A Re M F X A Re M X 1 Re M X, Thus condition 5.5 is satisfied. Moreover, we obtain Re M X A Re M F X A Re M X Ž 1. Re MŽ X X X* Re X, X. X. 0 Consequently, all the assumptions of Theorem 5. are satisfied for A 1. REFERENCES 1. T. Ando Ky Fan, Pick-Julia theorems for operators, Math. Z. 168 Ž 1979., 33.. R. B. Burckel, An ntroduction to Classical Complex Analysis, Vol., Birkhauser Verlag, BaselStuttgart, C. Caratheodory, Conformal Representations, Cambridge Tracts in Mathematics Mathematical Physics, Cambridge, C. Caratheodory, Uber die Winkelderivierten von beschrankten analytischen Functionen, Sitz. Ber. Preuss. Akad. Phys.-Math. Ž 199., C. Caratheodory, Theory of Functions, Vol., Chelsea, New York, B. G. Eke, On the angular derivative of regular functions, Math. Sc. 1 Ž 1967., Ky Fan, The angular derivative of an operator-valued analytic function, Pacific J. Math. 11 Ž 1986., J. L. Goldberg, Functions with positive real part in a half plane, Duke Math. J. 9 Ž 196.,

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