Methods of ascent and descent in multivariable spectral theory

Abstract

In this dissertation the theory of ascent and descent for a linear operator acting on a vector space is extended to arbitrary sets of operators and applied to the study of joint spectra for finite commuting systems of bounded operators acting on a complex Banach space. To each member of a large class of joint spectra containing the Taylor spectrum, Harte spectrum, two-sided Slodkowski spectra and split versions we associate a Browder joint spectrum. We show that these Browder joint spectra are non-empty, compact-valued, have the projection property and consequently satisfy a spectral mapping theorem. In the case of the Taylor spectrum, the associated Browder joint spectrum agrees with that of R.E. Curto and A.T. Dash ([21]).