On some inequalities for the approximation numbers in Banach algebras

Abstract

In this paper, we generalize some inequalities for the approximation numbers of an element in a normed (Banach) algebra $X$ and, as an application, we present inequalities for the quasinorms of some ideals defined by means of the approximation numbers.

In particular, if $X=L(E)$ - the algebra of linear and bounded operators $T:E \to E$, where $E$ is a Banach space, we obtain inequalities for certain quasinorms of operators.