An extension of the Fuglede commutativity theorem modulo the Hilbert-Schmidt class to operators of the form $\sum M_{n}XN_{n}$

Abstract:

We study the operators $\Delta (X) = \sum \nolimits _1^n {{M_n}X{N_n}}$ and ${\Delta ^{\ast }}(X) = \sum \nolimits _1^n {M_n^{\ast }XN_n^{\ast }}$ which map the algebra of all bounded linear operators on a separable Hubert space to itself, where $\langle {M_n}\rangle _1^m$ and $\langle {N_n}\rangle _1^m$ are separately commuting sequences of normal operators. We prove that (1) when $m \leqslant 2$, the Hilbert-Schmidt norms of $\Delta (X)$ and ${\Delta ^{\ast }}(X)$ are equal (finite or infinite); (2) for $m \geqslant 3$, if $\Delta (X)$ and ${\Delta ^{\ast }}(X)$ are Hilbert-Schmidt operators, then their Hilbert-Schmidt norms are equal; (3) if $\Delta ,{\Delta ^{\ast }}$ have the property that for each $X,\Delta (X) = 0$ implies ${\Delta ^{\ast }}(X) = 0$, then for each $X$, if $\Delta (X)$ is a Hilbert-Schmidt operator then ${\Delta ^{\ast }}^2(X)$ is also and the latter has the same Hilbert-Schmidt norm as ${\Delta ^2}(X)$. Note that Fuglede’s Theorem is immediate from $(1)$ in the case $m = 2,{M_1} = {N_2}$ and ${N_1} = I = - {M_2}$. The proofs employ the duality between the trace class and the class of all bounded linear operators and, unlike the early proofs of Fuglede’s Theorem, they are free of complex function theory.