Journal of Operator Theory

Volume 69, Issue 2, Spring 2013 pp. 435-461.

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

Authors: Nils Byrial Andersen (1) and Marcel de Jeu (2)
Author institution: (1) Department of Mathematics, Aarhus University, Ny Munkegade 118, Building 1530, DK-8000 Aarhus C, Denmark
(2) Mathematical Institute, Leiden University, P.O. Box 9512, 2300 RA Leiden, The Netherlands

Summary: We exhibit unbounded operators on Banach spaces having the single-valued extension property, for which the local spectrum can be determined at suitable points, and for which a local spectral radius formula holds, analogous to that for a bounded operator on a Banach space with the single-valued extension property. Such an operator can occur as (an extension of) a differential operator which, roughly speaking, can be diagonalized on its domain of smooth test functions via a discrete transform which is an isomorphism of topological vector spaces between the domain, in its own topology, and a sequence space. We give examples (constant coefficient differential operators on the $d$-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.

Keywords: unbounded operator, local spectrum, local spectral radius formula, differential operator, eigenfunction expansion, special functions

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