Banach algebras of pseudodifferential operators and their almost diagonalization

TL;DR

This paper introduces new symbol classes for pseudodifferential operators linked to convolution algebras, demonstrating their almost diagonalization and Banach algebra properties, with implications for invertibility and classical operator classes.

Contribution

It defines a novel framework connecting convolution algebras to pseudodifferential operator calculus, providing new insights into their structure and invertibility.

Findings

Operators are almost diagonal with respect to wave packets.

The symbol classes form Banach algebras of bounded operators.

Invertibility is characterized by Wiener's lemma for convolution algebras.

Abstract

We define new symbol classes for pseudodifferntial operators and investigate their pseudodifferential calculus. The symbol classes are parametrized by commutative convolution algebras. To every solid convolution algebra over a lattice we associate a symbol class. Then every operator with such a symbol is almost diagonal with respect to special wave packets (coherent states or Gabor frames), and the rate of almost diagonalization is described precisely by the underlying convolution algebra. Furthermore, the corresponding class of pseudodifferential operators is a Banach algebra of bounded operators on $L^2 $. If a version of Wiener's lemma holds for the underlying convolution algebra, then the algebra of pseudodifferential operators is closed under inversion. The theory contains as a special case the fundamental results about Sj\"ostrand's class and yields a new proof of a theorem of Beals about the H\"ormander class of order 0.