On extensions of Fuglede-Putnam theorem (Q1099385)

An operator means a bounded linear operator on a separable complex Hilbert space H. Let B(H), \(C_ 1\) and \(C_ 2\) denote respectively the Banach algebra of all operators, the trace class and the Hilbert-Schmidt class of operators on H. An operator A is called k-quasihyponormal if \(A^{*k}(A^*A-AA^*)A^ k\geq 0\), k being a non-negative integer and Q(k) denotes the set of all k-quasihyponormal operators. If \(A^{*k}A^ k\geq (A^*A)^ k\), then A is defined to be of class (M;k). The classical Fuglede-Putnam theorem states that if A, B are normal operators and \(AX=XB\) for some \(X\in B(H)\), then \(A^*X=XB^*\). This result was generalized in the cases where A, \(B^*\in Q(0)\) and \(X\in C_ 2\) by \textit{S. K. Berberian} [Proc. Am. Math. Soc. 71, 113-114 (1978; Zbl 0388.47019)], and where \(A\in Q(k)\), \(B^{*-1}\in Q(0)\) (or \(A^{- 1}\in Q(0)\), \(B^*\in Q(k))\) and \(X\in C_ 2\) by \textit{T. Furuta} [ibid. 77, 324-328 (1979; Zbl 0414.47024)]. In this paper the authors also generalized the result in the case where A, \(B^{*-1}\in\) class (M;k) (or \(A^{-1}\), \(B^*\in class\) (M;k)) and \(*X\in C_ 2\). And next they showed the condition \(A\in Q(0)\) in the result of \textit{F. Kittaneh} [ibid. 88, 293-298 (1983; Zbl 0521.47014)] that if \(A\in Q(0)\), \(\| A\| =1\) with \(I-AA^*\in C_ 1\), then \(AX-XA\in C_ 2\) for some \(X\in B(H)\) implies \(A^*X-XA^*\in C_ 2\) can be relaxed to \(A\in Q(1)\) by using the result gotten by themselves about the structure of \(A\in Q(k)\) [\textit{S. L. Campbell}, \textit{B. C. Gupta}, Math. Jap. 23, 185- 189 (1978; Zbl 0398.47014)].