Polynomially compact derivations on Banach algebras

Matej Brešar ¹   ; Yuri V. Turovskii ²

¹ Department of Mathematics, FNM University of Maribor Koroška 160 2000 Maribor, Slovenia
² Institute of Mathematics and Mechanics National Academy of Sciences of Azerbaijan F. Agaev St. 9 Baku AZ1141, Azerbaijan

Studia Mathematica, Tome 190 (2009) no. 2, pp. 185-191

Résumé

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.

Détail

DOI : 10.4064/sm190-2-6

Keywords: consider continuous derivation banach algebra cal compact operator polynomial shown either cal has nonzero finite dimensional ideal contained radical mathop rad cal cal there exists another polynomial tilde tilde maps cal mathop rad cal special where compact discussed greater detail