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On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type


Author: Fuad Kittaneh
Journal: Proc. Amer. Math. Soc. 88 (1983), 293-298
MSC: Primary 47B20; Secondary 47B10
DOI: https://doi.org/10.1090/S0002-9939-1983-0695261-8
MathSciNet review: 695261
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Abstract: We prove the following statements about the bounded linear operators on a separable, complex Hilbert space: (1) If $ A$ and $ {B^ * }$ are subnormal operators, and $ X$ is an invertible operator such that $ AX - XB \in {C_2}$, then there exists a unitary operator $ U$ such that $ AU - UB \in {C_2}$. Moreover, $ {A^ * }A - A{A^ * }$ and $ {B^ * }B - B{B^ * }$ are in $ {C_1}$. (2) If $ A$ is a subnormal operator with $ {A^ * }A - A{A^ * } \in {C_1}$, then for any operator $ X$, $ AX - XA \in {C_2}$ implies $ {A^ * }X - X{A^ * } \in {C_2}$. (3) If $ A$ is a hyponormal contraction with $ 1 - A{A^ * } \in {C_1}$, then for any operator $ X$, $ AX - XA \in {C_2}$ implies $ {A^ * }X - X{A^ * } \in {C_2}$. (4) For every operator $ T$ for which $ {T^2}$ is normal and $ {T^ * }T - T{T^ * } \in {C_1}$, $ TX - XT \in {C_2}$ implies $ {T^ * }X - X{T^ * } \in {C_2}$ for any operator $ X$. Applications of a recent result of Moore, Rogers and Trent [8] are also given.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0695261-8
Keywords: Fuglede-Putnam theorem, isometry, unitary operator, normal operators, subnormal operators, hyponormal operators, compact operators, Hilbert-Schmidt operators, trace class operators, weighted shifts
Article copyright: © Copyright 1983 American Mathematical Society

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