An operator means here a bounded linear operator on a complex Hilbert space. An operator \(A\) is said to be a class \(A\) operator if and only if \(|A|^{2}\leq |A^{2}|\). For two normal operators \(A\) and \(B\), the famous Fuglede-Putnam’s theorem is known as follows: Let \(A\) and \(B\) be normal operators. If \(AX=XB\) holds for some operator \(X\), then \(A^{*}X=XB^{*}\). The authors extend this result to class A operators. Let \(A\) be a class \(A\) operator and \(B^{*}\) be an invertible class \(A\) operator. If \(AX=XB\) holds for some Hilbert-Schmidt class operator \(X\), then \(A^{*}X=XB^{*}\). Then the authors discuss a property of an operator \(\delta_{A,B}(X)=AX-XB\) on the set of all Hilbert-Schmidt class operators (Theorem 2.3 and Corollary 2.2).
Reviewer’s remark: Corollary 2.2 should be corrected as follows:
Corollary 2.2. Let \(A\) be a class \(A\) operator and \(B^{*}\) be an invertible class \(A\) operator. Then
\[ \|\delta_{A,B}(X)+S\|_{2}^{2}=\|\delta_{A,B}(X)\|_{2}^{2}+\|S\|_{2}^{2} \]
and
\[ \|\delta_{A,B}^{*}(X)+S\|_{2}^{2}=\|\delta_{A,B}^{*}(X)\|_{2}^{2}+\|S\|_{2}^{2} \]
holds for all \(X\in C_{2}(H)\) if and only if \(\delta_{A,B}(S)=0\).