Glasgow Mathematical Journal

Research Article

Local spectral theory and spectral inclusions

Kjeld B. Laursena1 and Michael M. Neumanna2

a1 Mathematics Institute, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

a2 Department of Mathematics and Statistics, Mississippi State University, P.O. Drawer Ma, Mississippi State, MS 39762, U.S.A.

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

(Received February 16 1993)

(Revised June 23 1993)