Positivity results for Weyl's pseudodifferential calculus on the Wiener space

TL;DR

This paper explores positivity properties of Weyl's pseudodifferential calculus on infinite-dimensional Wiener space, revealing that positive symbols may not always produce positive operators, but certain conditions can guarantee positivity.

Contribution

It extends Weyl's calculus to infinite-dimensional Wiener space and establishes new positivity and Gårding's inequality results in this setting.

Findings

Positive symbols do not necessarily yield positive operators.

Gårding's inequality holds for the considered symbol classes.

Radial symbols with additional assumptions can produce positive operators.

Abstract

This paper deals with positivity properties for a pseudodifferential calculus, generalizing Weyl's classical quantization, and set on an infinite dimensional phase space, the Wiener space. In this frame, we show that a positive symbol does not, in general, give a positive operator. In order to measure the nonpositivity, we establish a G\r{a}rding's inequality, which holds for the symbol classes at hand. Nevertheless, for symbols with radial aspects, additional assumptions ensure the positivity of the associated operator.