Spectral properties of pseudo-differential operators over the compact group of $p$-adic integers and compact Vilenkin groups

TL;DR

This paper investigates the spectral and boundedness properties of pseudo-differential operators on compact and non-compact Vilenkin groups, extending H"ormander classes and providing explicit spectral formulas.

Contribution

It extends pseudo-differential calculus and spectral analysis to Vilenkin groups, including new definitions and explicit spectral formulas for non-compact cases.

Findings

Established $L^r$-boundedness and compactness criteria.

Derived explicit Fredholm spectrum formulas.

Extended H"ormander classes to non-compact Vilenkin groups.

Abstract

In this paper. we study properties such as $L^r$-boundedness, compactness, belonging to Schatten classes and nuclearity, Riesz spectral theory, Fredholmness, ellipticity and Gohberg's lemma, among others, for pseudo-differential operators over the compact group of $p$-adic integers $\mathbb{Z}_p^d$, where the author in a recent paper introduced a notion of H\"ormander classes and pseudo-differential calculus. We extend the results to compact Vilenkin groups which are essentially the same as $\mathbb{Z}_p^d$. Also, we provide a new definition of H\"ormander classes for pseudo-differential operators acting on non-compact Vilenkin groups and an explicit formula for the Fredholm spectrum in terms of the associated symbol.