## On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type

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- Proc. Amer. Math. Soc.
**88**(1983), 293-298 Request permission

## 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

- © Copyright 1983 American Mathematical Society
- 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