Complex symmetric partial isometries

TL;DR

This paper characterizes complex symmetric partial isometries on Hilbert spaces, showing all such operators are complex symmetric in spaces up to dimension 4, thus advancing understanding of their structure.

Contribution

It provides a concrete description of all complex symmetric partial isometries and proves that all partial isometries in spaces of dimension four or less are complex symmetric.

Findings

All partial isometries on Hilbert spaces of dimension ≤ 4 are complex symmetric.

Provides a concrete characterization of complex symmetric partial isometries.

Advances understanding of the structure of symmetric operators in finite-dimensional spaces.

Abstract

An operator $T \in B(\h)$ is complex symmetric if there exists a conjugate-linear, isometric involution $C:\h\to\h$ so that $T = CT^*C$. We provide a concrete description of all complex symmetric partial isometries. In particular, we prove that any partial isometry on a Hilbert space of dimension $\leq 4$ is complex symmetric.