Interpolation of the essential spectrum and the essential norm
Volume 68 / 2005
Banach Center Publications 68 (2005), 9-18
MSC: 46B70, 47A10, 47A30.
DOI: 10.4064/bc68-0-1
Abstract
The behavior of the essential spectrum and the essential norm under (complex/real) interpolation is investigated. We extend an example of Albrecht and Müller for the spectrum by showing that in complex interpolation the essential spectrum $\sigma _e(S_{[\theta ]})$ of an interpolated operator is also in general a discontinuous map of the parameter $\theta$. We discuss the logarithmic convexity (up to a multiplicative constant) of the essential norm under real interpolation, and show that this holds provided certain compact approximation conditions are satisfied. Some evidence supporting a counterexample is presented.
