Partial isometries and the conjecture of C. K. Fong and S. K. Tsui

TL;DR

This paper proves that certain classes of bounded linear operators on Hilbert spaces satisfy the Fong-Tsui conjecture, showing they are self-adjoint under specific conditions, thus advancing understanding of operator symmetry.

Contribution

It demonstrates that the Fong-Tsui conjecture holds for various classes of operators, including partial isometries and Brownian isometries, under the condition |T| ≤ |ReT|.

Findings

The conjecture holds for partial isometries and contractive quasi-isometries.

The condition |T| ≤ |ReT| implies self-adjointness in these classes.

The result extends to Brownian isometries of positive covariance.

Abstract

We investigate some bounded linear operators T on a Hilbert space which satisfy the condition |T | less or equal to |ReT |. We describe the maximum invariant subspace for a contraction T on which T is a partial isometry to obtain that, in certain cases, the above condition ensures that T is self-adjoint. In other words we show that the Fong-Tsui conjecture holds for partial isometries, contractive quasi-isometries, or 2-quasi-isometries, and Brownian isometries of positive covariance, or even for a more general class of operators.