We consider a continuous derivation D on a Banach algebra 𝓐 such that p(D) is a compact operator for some polynomial p. It is shown that either 𝓐 has a nonzero finite-dimensional ideal not contained in the radical rad(𝓐) of 𝓐 or there exists another polynomial p̃ such that p̃(D) maps 𝓐 into rad(𝓐). A special case where Dⁿ is compact is discussed in greater detail.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-6,
author = {Matej Bre\v sar and Yuri V. Turovskii},
title = {Polynomially compact derivations on Banach algebras},
journal = {Studia Mathematica},
volume = {192},
year = {2009},
pages = {185-191},
zbl = {1156.47035},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-6}
}
Matej Brešar; Yuri V. Turovskii. Polynomially compact derivations on Banach algebras. Studia Mathematica, Tome 192 (2009) pp. 185-191. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm190-2-6/