On Algebras of Functions over Infinite Dimensions

TL;DR

This paper introduces a family of holomorphic function spaces over infinite-dimensional domains, constructed via Gaussian measures, and explores their algebraic structure and operator relations.

Contribution

It defines new reproducing kernel Hilbert spaces on infinite-dimensional domains, establishing conditions for their algebraic closure and analyzing related operator relations.

Findings

The spaces are closed under pointwise multiplication under certain conditions.

The constructed spaces form reproducing kernel Hilbert algebras.

Bounded creation and annihilation operators are studied within these algebras.

Abstract

We introduce a family of reproducing kernel Hilbert spaces $\mathcal A_\Lambda$ of holomorphic functions defined on an infinite--dimensional domain in a separable Hilbert space, $\mathbb{H}$. The reproducing kernel of $\mathcal A_\Lambda$ is constructed using the covariance operator associated with a Gaussian measure on $\mathbb{H}$, along with a holomorphic function $\Lambda$ on the unit disk. Under certain conditions on the kernel, $\mathcal A_\Lambda$ is closed under pointwise multiplication, giving it the structure of a reproducing kernel Hilbert algebra (RKHA). We also study twisted canonical commutation relations on these RKHAs, where the creation and annihilation operators are both bounded.