An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality

Furuta, Takayuki

doi:10.1090/S0002-9939-1981-0593465-4

An extension of the Fuglede-Putnam theorem to subnormal operators using a Hilbert-Schmidt norm inequality
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by Takayuki Furuta PDF

Proc. Amer. Math. Soc. 81 (1981), 240-242 Request permission

Abstract:

We prove that if $A$ and ${B^ * }$ are subnormal operators acting on a Hubert space, then for every bounded linear operator $X$, the Hilbert-Schmidt norm of $AX - XB$ is greater than or equal to the Hilbert-Schmidt norm of ${A^ * }X - X{B^ * }$. In particular, $AX = XB$ implies ${A^ * }X = X{B^ * }$. In addition, if we assume $X$ is a Hilbert-Schmidt operator, we can relax the subnormality conditions to hyponormality and still retain the inequality.

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Fuglede’s commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators