The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I
Gary M. Weiss · Transactions of the American Mathematical Society · 1978
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 N X − X D NX\, - \,XD and N ∗ X − X D ∗ {N^{\ast }}X\, - \,X{D^{\ast }} are equal; (3) If N X − X N NX\, - \,XN and N ∗ X − X N ∗ {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 N X − X N NX\, - \,XN is a trace class operator, then Trace ( N X − X N ) = 0 \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