Triangularizability of Families of Polynomially Compact Operators

TL;DR

This paper extends the triangularizability results of certain matrix pairs to families of polynomially compact operators on Banach spaces, including special cases involving normal and algebraic operators, revealing new structural insights.

Contribution

It generalizes Shemesh's matrix results to polynomially compact operators on Banach spaces and explores special cases like normal and algebraic operators.

Findings

Triangularizability of polynomially compact operator families established.

Structure results for finite algebraic operator families obtained.

Conditions for commutativity and triangularization extended to Banach space operators.

Abstract

A recent paper of Shemesh shows triangularizability of a pair $\{A, B\}$ of complex matrices satisfying the condition $A [A,B]=[A,B] B=0$, or equivalently, the matrices $A$ and $B$ commute with their product $A B$. In this paper we extend this result to polynomially compact operators on Banach spaces. The case when the underlying space is Hilbert and one of operators is normal is also studied. Furthermore, we consider families of polynomially compact operators whose iterated commutators of some fixed length are zero. We also obtain a structure result in the special case of a finite family of algebraic operators.