A new characterization of the Hardy space and of other Hilbert spaces of analytic functions

Year 2023, Volume: 1 Issue: 1, 1 - 11, 21.06.2023

Natanael Alpay

https://doi.org/10.26650/ijmath.2023.00005

Abstract

The Fock space can be characterized (up to a positive multiplicative factor) as the only Hilbert space of entire functions in which the adjoint of derivation is multiplication by the complex variable. Similarly (and still up to a positive multiplicative factor) the Hardy space is the only space of functions analytic in the open unit disk for which the adjoint of the backward shift operator is the multiplication operator. In the present paper we characterize the Hardy space and some related reproducing kernel Hilbert spaces in terms of the adjoint of the differentiation operator. We use reproducing kernel methods, which seem to also give a new characterization of the Fock space.

Keywords

Reproducing kernel, Hardy space, Fock space

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