Two remarks about nilpotent operators of order two

TL;DR

This paper investigates nilpotent operators of order two, revealing they are indestructible complex symmetric operators and unitarily equivalent to truncated Toeplitz operators, thus advancing understanding of their structure and symmetry properties.

Contribution

It proves nilpotent order two operators are indestructible complex symmetric and characterizes them as unitarily equivalent to truncated Toeplitz operators.

Findings

Nilpotent order two operators are indestructible complex symmetric.

Such operators are unitarily equivalent to truncated Toeplitz operators.

This characterizes their structure and symmetry properties.

Abstract

We present two novel results about Hilbert space operators which are nilpotent of order two. First, we prove that such operators are indestructible complex symmetric operators, in the sense that tensoring them with any operator yields a complex symmetric operator. In fact, we prove that this property characterizes nilpotents of order two among all nonzero bounded operators. Second, we establish that every nilpotent of order two is unitarily equivalent to a truncated Toeplitz operator.