Matrix representation of Magnetic pseudo-differential operators via tight Gabor frames

TL;DR

This paper presents a matrix representation of magnetic pseudo-differential operators using tight Gabor frames, enabling simplified proofs of classical theorems and new trace-class criteria.

Contribution

It introduces a novel matrix-based framework for magnetic pseudo-differential operators, facilitating easier analysis and proof of key properties.

Findings

Operators are represented as matrices localized near the diagonal.

Simplified proofs of Calderón-Vaillancourt theorem and Beals' criterion.

Established local trace-class criteria for these operators.

Abstract

In this paper we use some ideas from \cite{FG-97, G-06} and consider the description of H\"{o}rmander type pseudo-differential operators on $\mathbb{R}^d$ ($d\geq1$), including the case of the magnetic pseudo-differential operators introduced in \cite{IMP-1, IMP-19}, with respect to a tight Gabor frame. We show that all these operators can be identified with some infinitely dimensional matrices whose elements are strongly localized near the diagonal. Using this matrix representation, one can give short and elegant proofs to classical results like the Calder{\'o}n-Vaillancourt theorem and Beals' commutator criterion, and also establish local trace-class criteria.