The Fuglede Commutativity Theorem modulo the Hilbert-schmidt Class and Generating Functions for Matrix Operators. I

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 A', diagonal operator D and bounded operator X, the Hilbert-Schmidt norms (finite or infinite) of NX XD and N*X XD* are equal; (3) If NX XN and N*X XN* are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (4) If X is a Hilbert-Schmidt operator and A is a normal operator so that NX — XN is a trace class operator, then Trace(NX XN) = 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*X — XN* are equal. The main technique employs the use of a new concept which we call 'generating functions for matrices'. Let H denote a separable, complex Hilbert space and let L(H) denote the class of all bounded linear operators acting on H. Let K(H) denote the class of compact operators in L(H) and let Cp denote the Schatten /7-class (0 < p < oo) with || • \\p (1 < p < oo) denoting the associatedp-norm. Hence C2 is the Hilbert-Schmidt class and C, is the trace class. Consider the following statements: (1) For every normal operator N and e > 0, there exist a diagonal operator D and a Hilbert-Schmidt operator Ke with \\Ke\\2 < e for which N st D + Ke (» denotes unitary equivalence). (2) For every normal operator N, there exist a diagonal operator D and a K E C2 for which N = D + K. (3) For every normal operator A^ and bounded operator X, \\NX — XN\\2 = \\N*X XN*\U. Received by the editors March 3, 1977 and, in revised form, August 29, 1977. AMS (MOS) subject classifications (1970). Primary 47A05, 47A55, 47B10, 47B15, 47B47; Secondary 05A15, 05B20.