On relaxation of normality in the Fuglede-Putnam theorem

Furuta, Takayuki

doi:10.1090/S0002-9939-1979-0545590-2

On relaxation of normality in the Fuglede-Putnam theorem
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Proc. Amer. Math. Soc. 77 (1979), 324-328 Request permission

Abstract:

An operator means a bounded linear operator on a complex Hilbert space. The familiar Fuglede-Putnam theorem asserts that if A and B are normal operators and if X is an operator such that $AX = XB$, then ${A^ \ast }X = X{B^ \ast }$. We shall relax the normality in the hypotheses on A and B. Theorem 1. If A and ${B^\ast }$ are subnormal and if X is an operator such that $AX = XB$, then ${A^ \ast }X = X{B^ \ast }$. Theorem 2. Suppose A, B, X are operators in the Hilbert space H such that $AX = XB$. Assume also that X is an operator of Hilbert-Schmidt class. Then ${A^ \ast }X = X{B^ \ast }$ under any one of the following hypotheses: (i) A is k-quasihyponormal and ${B^ \ast }$ is invertible hyponormal, (ii) A is quasihyponormal and ${B^\ast }$ is invertible hyponormal, (iii) A is nilpotent and ${B^\ast }$ is invertible hyponormal.

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