A new characterization of the Hardy space and of other spaces of analytic functions
Natanael Alpay
TL;DR
The paper develops a unified kernel-based framework to characterize classical spaces of analytic functions by adjoint-differentiation identities in reproducing kernel Hilbert spaces. It shows that the Hardy space is, up to a positive factor, the unique RKHS on the unit disk with ${\partial_z^* = M_z\partial_z M_z}$ on kernel spans, and generalizes this to the family ${\mathfrak H}_\alpha$ with kernels ${\big(1 - z\overline{\omega}\big)^{-\alpha}}$ satisfying ${\partial_z^* = M_z\partial_z M_z -(1-\alpha)M_z}$ for $\alpha>0$, with Hardy and Bergman as special cases. The converse arguments rely on translating operator identities into PDEs for the kernel and solving for the kernel via its Taylor expansion, yielding the Hardy/Bergman kernels; for the Dirichlet case a pointwise kernel equation recovers the Dirichlet kernel ${-\log(1 - z\overline{\omega})}$. Together, these results provide a kernel-analytic route to identifying RKHSs associated with fundamental spaces and suggest broader operator-model connections between RKHS structure and polynomial characterizations.
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.