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Proceedings of the American Mathematical Society

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

 

 

On relaxation of normality in the Fuglede-Putnam theorem


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 77 (1979), 324-328
MSC: Primary 47B20
DOI: https://doi.org/10.1090/S0002-9939-1979-0545590-2
MathSciNet review: 545590
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0545590-2
Keywords: Quasinormal operators, subnormal operators, hyponormal operators, quasihyponormal operators, Hilbert-Schmidt operators
Article copyright: © Copyright 1979 American Mathematical Society