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 of an interpolated operator is also in general a discontinuous map of the parameter θ. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc68-0-1,
author = {A. G. Aksoy and H.-O. Tylli},
title = {Interpolation of the essential spectrum and the essential norm},
journal = {Banach Center Publications},
volume = {68},
year = {2005},
pages = {9-18},
zbl = {1092.46013},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc68-0-1}
}
A. G. Aksoy; H.-O. Tylli. Interpolation of the essential spectrum and the essential norm. Banach Center Publications, Tome 68 (2005) pp. 9-18. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc68-0-1/