The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I

Abstract:

We prove the following statements about bounded linear operators on a separable, complex Hilbert space: (1) Every normal operator N that is similar to a Hilbert-Schmidt perturbation of a diagonal operator D is unitarily equivalent to a Hilbert-Schmidt perturbation of D; (2) For every normal operator N, diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX - XD$ and ${N^{\ast }}X - X{D^{\ast }}$ are equal; (3) If $NX - XN$ and ${N^{\ast }}X - X{N^{\ast }}$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and N is a normal operator so that $NX - XN$ is a trace class operator, then Trace$\left ( {NX - XN} \right ) = 0$; (5) For every normal operator N that is a Hilbert-Schmidt perturbation of a diagonal operator, and every bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of $NX - XN$ and ${N^{\ast }}X - X{N^{\ast }}$ are equal. The main technique employs the use of a new concept which we call ’generating functions for matrices’.