The spectrum of an interpolated operator and analytic multivalued functions

Thomas Joseph Ransford

Vol. 121 (1986), No. 2, 445–466

DOI: 10.2140/pjm.1986.121.445

Abstract

Let [B₀,B₁] be a complex interpolation pair and T : B₀ + B₁ → B₀ + B₁ be a linear map whose restriction to each interpolation space [B₀,B₁]_s is a bounded operator on that space with spectrum Sp_sT. Under mild conditions on T it is shown that the set-valued map λ → Sp_{(Re λ)}T is an analytic multivalued function. This fact is used to unify and generalise a number of previously known results about the spectrum of an interpolated operator, and also to motivate some new ones.

Mathematical Subject Classification 2000

Primary: 46M35

Secondary: 47A10

Milestones

Received: 28 March 1984

Revised: 12 November 1984

Published: 1 February 1986