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On Hadamard-Dirichlet algebras. (English) Zbl 1052.46016
Summary: S. Bhatt and R. Raina studied in [Acta Math. Univ. Comen., New Ser. 68, No. 1, 179–193 (1999; Zbl 0930.30016)] the behaviour of some fractional operators and Hadamard products on certain analytic functions on the unit disk. More generally, classes of analytic functions on the unit disk constitute a matter of actual intensive research. So it is desirable to dispose of an adequate theoretic frame which allows relatively simple and expeditious results on this subject. Recently, one of the authors considered topics on the structure of Hadamard algebras [C. C. Peña, J. Appl. Math. 4, No. 1, 23–26 (2000; Zbl 1070.46506); Matematiche 55, No. 1, 43–54 (2000; Zbl 1009.46031)].
In this article, our aim is to consider Dirichlet spaces, which constitute well-known Hilbert spaces, endowed with an abelian unitary Banach algebra structure induced by a Hadamard type product. The maximal ideal space, complex Hadamard homomorphisms, reproducing kernels, the generating function and spectra of their elements are determined.
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46G20 Infinite-dimensional holomorphy
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