Canonical integral operators on the Fock space II

Abstract

In \cite{DZ3} we introduced and studied a two-parameter family of integral operators $T^{(s,t)}$ on the Fock space $F^2$ of the complex plane. Under the inverse Bargmann transform, these operators include the classical {\it linear canonical transforms} in mathematical physics as special cases, so we called $T^{(s,t)}$ {\it canonical linear operators} on the Fock space. In this paper we continue the study of these operators. We show that when a canonical linear operator $T^{(s,t)}$ is compact, it actually belongs to the Schatten class $S_p$ for all $p>0$. In this case, we find all singular values, determine the $S_p$ norm, and obtain a trace formula for $T^{(s,t)}$. We also show that the boundedness (and a natural version of compactness) of $T^{(s,t)}$ on $F^p$ for any given $p\in(0,\infty]$ is equivalent to the boundedness (and compactness) of $T^{(s,t)}$ on $F^2$. Our analysis is based on estimates and computations with the integral kernel of $T^{(s,t)}$, which also yield some interesting results about the Berezin transform and the bivariate Berezin transform of $T^{(s,t)}$.