Algebraic Elementary Operators

Abstract

A Banach space operator $A$ is algebraic
if there exists a non-trivial polynomial $p(.)$ such that $p(A)=0$.
Equivalently, $A$ is algebraic if $\sigma(A)$ is a finite set
consisting of poles. The sum of two commuting Banach space algebraic
operators is  algebraic, and the generalized derivation
$\delta_{AB}=L_A-R_B$ (and, for non-nilpotent $A$ and $B$, the left
right multiplication operator $L_AR_B$) is algebraic if and only if
$A$ and $B$ are algebraic. We prove: If $\asc(d_{AB}-\lambda)\leq 1$
for all complex $\lambda$, and if $A^*, B$ have SVEP, then
$d_{AB}-\lambda$ has closed range for every complex $\lambda$ if and
only if $A, B$ are algebraic;  if $A, B$ are simply polaroid, then
$d_{AB}-\lambda$ has closed range for every
$\lambda\in\iso\sigma(d_{AB})$; and  if $A, B$ are normaloid, then
$L_AR_B-\lambda$ has closed range at every $\lambda$ in the
peripheral spectrum of $L_AR_B$ if and only if $L_AR_B$ is left
polar at $\lambda$