×

Representing measures and infinite-dimensional holomorphy. (English) Zbl 1131.46033

The authors consider some applications of the Bishop–De Leeuw theorem about representing measures to the situation of certain algebras of analytic functions on unit balls of Banach spaces. In particular, the Hardy spaces \(H^2(\mu)\) are studied. Let \(B\) be the unit ball of a complex Banach space and let \(A_a(B)\) be the Banach algebra of holomorphic functions on \(B\) generated by the continuous linear forms. Applying the Bishop–De Leeuw theorem to a homomorphism \(\varphi\) on \(A_a(B),\) one gets a representing measure \(\mu\) on the maximal ideal space \({\mathcal M}(A_a(B)).\) One next constructs \(H^2_a(\mu)\) as being the completion of \(A_a(B)\) relative to \(\| f\| _\mu = \int_{{\mathcal{M}}(A_a(B))} \sqrt{| {f}| ^2 \,d\mu }.\)
The authors study \(H^2_a(\mu)\) for certain measures \(\mu\) for which \(\| \cdot \| \) is a norm. In addition, conditions are studied under which \(H^2_a(\mu)\) is a reproducing kernel Hilbert space and when that reproducing kernel is determined by an analytic function. Examples of representing measures and corresponding Hardy spaces \(H^2\) for \(c_0\) and \(\ell_p, \;1 < p < \infty,\) are given. Useful references include [B.Cole and T.W.Gamelin, Proc.Lond.Math.Soc.53, 112–142 (1986; Zbl 0624.46032); D.Pinasco and I.Zaldendo, J. Math.Anal.Appl.308, No.1, 159–174 (2005; Zbl 1086.46033); the authors, Ann.Pol.Math.81, No.2, 111–122 (2003; Zbl 1036.46030)].

MSC:

46G20 Infinite-dimensional holomorphy
32A65 Banach algebra techniques applied to functions of several complex variables
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alencar, R.; Aron, R.; Galindo, P.; Zagorodnyuk, A., Algebras of symmetric holomorphic functions on \(\ell_p\), Bull. London Math. Soc., 35, 55-64 (2003) · Zbl 1026.46035
[2] Arenson, E. L., Gleason parts and the Choquet boundary of a function algebra on a convex compactum, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 113, 204-207 (1981) · Zbl 0472.46039
[3] Aron, R. M.; Choi, Y. S.; Louren, M. L.; Paques, O. W., Boundaries for algebras of analytic functions on Banach spaces, (Contemp. Math., vol. 144 (1993)), 15-22 · Zbl 0805.46032
[4] Aron, R. M.; Cole, B. J.; Gamelin, T. W., Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math., 415, 51-93 (1991) · Zbl 0717.46031
[5] Aron, R. M.; Cole, B. J.; Gamelin, T. W., Weak-star continuous analytic functions, Canad. J. Math., 47, 673-683 (1995) · Zbl 0829.46034
[6] Aron, R. M.; Prolla, J. B., Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math., 313, 195-216 (1980) · Zbl 0413.41022
[7] Bishop, E.; De Leeuw, K., The representations of linear functionals by measures on sets of extreme points, Ann. Inst. Fourier, 9, 305-331 (1959) · Zbl 0096.08103
[8] Chae, S. B., Holomorphy and Calculus in Normed Spaces, Monogr. Textbooks Pure Appl. Math., vol. 92 (1985), Marcel Dekker: Marcel Dekker New York · Zbl 0571.46031
[9] Cole, B.; Gamelin, T. W., Representing measures and Hardy spaces for the infinite polydisk algebra, Proc. London Math. Soc., 53, 112-142 (1986) · Zbl 0624.46032
[10] Dineen, S., Complex Analysis on Infinite-Dimensional Spaces, Monogr. Math. (1999), Springer: Springer New York · Zbl 1034.46504
[11] Gamelin, T. W., Uniform Algebras (1984), Chelsea: Chelsea New York · Zbl 0213.40401
[12] Lopushansky, O.; Zagorodnyuk, A., Hilbert spaces of analytic functions of infinitely many variables, Ann. Polon. Math., 81, 2, 111-122 (2003) · Zbl 1036.46030
[13] Mujica, J., Complex Analysis in Banach Spaces (1986), North-Holland: North-Holland Amsterdam
[14] Neeb, K.-H.; Ørsted, B., Hardy spaces in an infinite dimensional setting, (Doebrer, H.-D.; Dobrev, V. K.; Hilgert, J., Proceedings of II International Workshop “Lie Theory and Its Application in Physics” (1998), Clausthal: Clausthal Germany), 3-27
[15] Pełczyński, A., A property of multilinear operations, Studia Math., 16, 173-182 (1957) · Zbl 0080.09701
[16] Pinasco, D.; Zaldendo, I., Integral representation of holomorphic functions on Banach spaces, J. Math. Anal. Appl., 308, 159-174 (2005) · Zbl 1086.46033
[17] Rudin, W., Some representing measures for the ball algebra, Michigan Math. J., 27, 315-320 (1980) · Zbl 0454.60004
[18] Saitoh, S., Integral Transforms, Reproducing Kernels and Their Applications, Pitman Res. Notes Math. Ser., vol. 369 (1997), Longman · Zbl 0891.44001
[19] Zagorodnyuk, A., Spectra of algebras of entire functions on Banach spaces, Proc. Amer. Math. Soc., 134, 2559-2569 (2006) · Zbl 1158.46038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.