Schatten-von Neumann classes of tensors of invariant operators

TL;DR

This paper investigates Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces, deriving spectral properties and traceability conditions, with applications to anharmonic oscillators and pseudo-differential operators.

Contribution

It provides new spectral analysis results for tensor products of invariant operators and establishes conditions for Dixmier traceability in specific tensor classes.

Findings

Spectral properties for tensors of anharmonic oscillators derived.

Formulae for Schatten-von Neumann classes in terms of symbols provided.

Sufficient conditions for Dixmier traceability of finite tensors established.

Abstract

In this work we study Schatten-von Neumann classes of tensor products of invariant operators on Hilbert spaces. In the first part we first deduce some spectral properties for tensors of anharmonic oscillators thanks to the knowledge on corresponding Schatten-von Neumann properties. In the second part we specialised on tensors of invariant operators. In the special case where a suitable Fourier analysis associated to a fixed partition of a Hilbert space into finite dimensional subspaces is available we also give the corresponding formulae in terms of symbols. We also give a sufficient condition for Dixmier traceability for a class of finite tensors of pseudo-differential operators on the flat torus.