Резольвента положительного элемента упорядоченной банаховой алгебры

Abstract

Let $A$ be an ordered Banach algebra with a unit $e$. If $z\in A$, $z\ge 0$, then an order idempotent $b$ (that is, $0\le b\le e$ and $b^2=b$) is called $z$-invariant whenever $(e-b)zb=0$. Formulas for the resolvent $R(.,z)$ of an element $z$ which has invariant order idempotents, are obtained. Some applications to the coeffcients of Laurent series expansion of $R(.,z)$ around the spectral radius $r(z)$ are given.