On some linear operators on Fock type space

Abstract

We consider a lower semi-continuous function $\varphi$ in $\mathbb{R}^n$ depending on the absolute values of the variables and growing faster than $a \ln (1 + \Vert x \Vert)$ for each positive $a$. In terms of this function, we define a Hilbert space $F^2_{\varphi}$ of entire functions in $\mathbb{C}^n$. This is a natural generalization of a classical Fock space. In this paper we provide an alternative description of the space $F^2_{\varphi}$ in terms of the coefficients in the power expansions for the entire functions in this space. We mention simplest properties of reproducing kernels in the space $F^2_{\varphi}$. We consider the orthogonal projector from the space $L^2_{\varphi}$ of measurable complex-valued functions $f$ in $\mathbb{C}^n$ such that $$ \Vert f \Vert_{\varphi}^2 = \int_{\mathbb{C}^n} \vert f(z)\vert^2 e^{- 2 \varphi (\mathrm{abs}\, z)} \ d \mu_n (z) < \infty , $$ where $z =(z_1, \ldots , z_n)$, $\mathrm{abs}\, z = (\vert z_1 \vert, \ldots , \vert z_1 \vert)$, on its closed subspace $F^2_{\varphi}$, and for this projector we obtain an integral representation. We also obtain an integral formula for the trace of a positive linear continuous operator on the space $F^2_{\varphi}$. By means of this formula we find the conditions, under which a weighted operator of the composition on $F^2_{\varphi}$ is a Hilbert–Schmidt operator. Two latter results generalize corresponding results by Sei-Ichiro Ueki, who studied similar questions for operators in Fock space.