When Nilpotence Implies Normality of Bounded Linear Operators

TL;DR

This paper explores conditions under which nilpotent bounded linear operators must be normal, showing that certain properties like positive real part prevent nilpotence, and also discusses quasinilpotence.

Contribution

It establishes new criteria linking nilpotence and normality in bounded linear operators, extending understanding of operator properties.

Findings

Nilpotent matrices under certain conditions are necessarily normal.

Operators with positive real part cannot be nilpotent.

The paper discusses implications for quasinilpotent operators.

Abstract

In this paper, we give conditions forcing nilpotent matrices (and bounded linear operators in general) to be null or equivalently to be normal. Therefore, a non-zero operator having e.g. a positive real part is never nilpotent. The case of quasinilpotence is also considered.