Abstract:
A weighted Hilbert space $F^2_{\varphi}$ of entire functions of $n$ variables will be considered in the talk. It is constructed with a help a weight function $\varphi$ on ${\mathbb R}^n$. $\varphi$ is a semicontinuous from below function on ${\mathbb R}^n$ depending on modules of variables, growing at infinity faster than $a \ln (1 + \Vert x \Vert)$ for each positive $a$ and such that its restriction on $[0, \infty)^n$ is nondecreasing in each variable. Properties of the space $F^2_{\varphi}$ will be described. The main part of the talk is devoted to concrete operators acting on $F^2_{\varphi}$ (among them Toeplitz operators, weighted composition operators).
Under some additional conditions on $\varphi$ (convexity, growth conditions) the space of the Laplace transforms of linear continuous functionals on $F^2_{\varphi}$ is described.
