On the reduction of powers of self-adjoint operators

TL;DR

This paper investigates conditions under which powers of a bounded self-adjoint operator imply the original operator is self-adjoint, focusing on cases like $T^3=0$ and its implications for complex symmetry.

Contribution

It establishes new conditions linking powers of operators to their self-adjointness and explores the complex symmetry of operators when $T^3=0$.

Findings

If $T^n$ is self-adjoint for some $n eq 2$, then $T$ is self-adjoint under certain conditions.

When $T^3=0$, $T$ is shown to be complex symmetric.

The paper provides criteria connecting the algebraic properties of $T$ to its spectral properties.

Abstract

Let $T\in B(H)$ be such that $T^n$ is self-adjoint for some $n\in\mathbb{N}$ with $n\geq 3$. The paper's primary aim is to establish the conditions that lead to the self-adjointness of $T$. We pay particular attention to the case where $T^3=0$ and how it implies $T$ is complex symmetric.