Local spectral radius formulas for a class of unbounded operators on Banach spaces

TL;DR

This paper establishes local spectral radius formulas for a class of unbounded operators on Banach spaces, including differential operators, showing how their spectra can be explicitly determined using a diagonalising transform.

Contribution

It introduces a broad class of unbounded operators with the single-valued extension property and derives local spectral radius formulas for them, extending spectral theory to practical differential operators.

Findings

Local spectral radius formulas hold for the class of operators.

Examples include differential operators on the d-torus and classical special function operators.

Operators can be diagonalized via a discrete transform, linking their domain to sequence spaces.

Abstract

We exhibit a general class of unbounded operators in Banach spaces which can be shown to have the single-valued extension property, and for which the local spectrum at suitable points can be determined. We show that a local spectral radius formula holds, analogous to that for a globally defined bounded operator on a Banach space with the single-valued extension property. An operator of the class under consideration can occur in practice as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform, such that the diagonalising transform establishes an isomorphism of topological vector spaces between the domain of the differential operator, in its own topology, and a sequence space. We give concrete examples of (extensions of) such operators (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.