### 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 $\lambda$

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 $\lambda$

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