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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 81 (1981), 240-242
MSC: Primary 47B20; Secondary 47A05, 47B10
DOI: https://doi.org/10.1090/S0002-9939-1981-0593465-4
MathSciNet review: 593465
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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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DOI: https://doi.org/10.1090/S0002-9939-1981-0593465-4
Keywords: Subnormal operator, hyponormal operator, Hilbert-Schmidt class
Article copyright: © Copyright 1981 American Mathematical Society

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