On generalized Fuglede-Putnam theorems of Hilbert-Schmidt type

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.