# Hilbert-Schmidt and Trace Class Pseudo-differential Operators on the Abstract Heisenberg Group

## TL;DR

This paper introduces and characterizes pseudo-differential operators with operator-valued symbols on the abstract Heisenberg group, providing conditions for Hilbert-Schmidt and trace class properties along with trace formulas.

## Contribution

It establishes necessary and sufficient conditions for these operators to be Hilbert-Schmidt and trace class, including trace formulas, on the abstract Heisenberg group.

## Key findings

- Characterization of Hilbert-Schmidt operators via symbol conditions
- Trace formula for the $j$-Weyl transform with $L^2$ symbols
- Criteria for trace class pseudo-differential operators

## Abstract

In this paper we introduce and study pseudo-differential operators with operator valued symbols on the abstract Heisenberg group $\mathbb{H}(G):=G \times \widehat{G} \times \mathbb{T},$ where $G$ a locally compact abelian group with its dual group $\widehat{G}$. We obtain a necessary and sufficient condition on symbols for which these operators are in the class of Hilbert-Schmidt operators. As a key step in proving this we derive a trace formula for the trace class $j$-Weyl transform, $j \in \mathbb{Z}^*$ with symbols in $L^{2}(G\times \widehat{G}).$ We go on to present a characterization of the trace class pseudo-differential operators on $\mathbb{H}(G)$. Finally, we also give a trace formula for these trace class operators.