Schwartz operators

TL;DR

This paper introduces Schwartz operators as a non-commutative analogue of Schwartz functions, exploring their properties, duals, and applications in quantum harmonic analysis, operator moment problems, and convergence of fluctuation operators.

Contribution

It defines Schwartz operators, develops their topological and duality properties, and applies these concepts to quantum analysis and operator moment problems.

Findings

Schwartz operators form a Fréchet space with multiple seminorms.

Non-commutative tempered distributions can be identified with quadratic forms.

Applications include operator moment problems and convergence analysis of fluctuation operators.

Abstract

In this paper we introduce Schwartz operators as a non-commutative analog of Schwartz functions and provide a detailed discussion of their properties. We equip them in particular with a number of different (but equivalent) families of seminorms which turns the space of Schwartz operators into a Frechet space. The study of the topological dual leads to non-commutative tempered distributions which are discussed in detail as well. We show in particular that the latter can be identified with a certain class of quadratic forms, therefore making operations like products with bounded (and also some unbounded) operators and quantum harmonic analysis available to objects which are otherwise too singular for being a Hilbert space operator. Finally we show how the new methods can be applied by studying operator moment problems and convergence properties of fluctuation operators.