Polynomially compact derivations on Banach algebras
Volume 190 / 2009
Studia Mathematica 190 (2009), 185-191
MSC: 47B47, 47B48, 47B07, 46H15, 46H20.
DOI: 10.4064/sm190-2-6
Abstract
We consider a continuous derivation $D$ on a Banach algebra ${\cal A}$ such that $p(D)$ is a compact operator for some polynomial $p$. It is shown that either ${\cal A}$ has a nonzero finite-dimensional ideal not contained in the radical $\mathop{\rm rad}({\cal A})$ of ${\cal A}$ or there exists another polynomial $\tilde{p}$ such that $\tilde{p}(D)$ maps ${\cal A}$ into $\mathop{\rm rad}({\cal A})$. A special case where $D^n$ is compact is discussed in greater detail.
