On Putnam-Fuglede theorems of Hilbert-Schmidt type (Q2639313)

The paper contains the following Putnam-Fuglede type theorems. (a) Assume \(A_ 1A_ 2-A_ 2A_ 1=B_ 1B_ 2-B_ 2B_ 1=0\) and \(A_ 1XB_ 1=A_ 2XB_ 2\) for quasi-normal operators \(A_ 1\) and \(B^*_ 1\), invertible normal operators \(A_ 2,B_ 2\), and a Hilbert- Schmidt operator X. Then \(A^*_ 1XB^*_ 1=A^*_ 2XB^*_ 2.\) (b) Let A be a contraction operator whose c.n.u. part is \(C_{\cdot 0}\) and whose pure part has no point spectrum. Assume \((1-A^*A)^{1/2}\) and AX-XA are Hilbert-Schmidt operators for some operators and that every normal part of A is reducing. Then \(A^*X-XA^*\) is a Hilbert-Schmidt operator. The above mentioned results are special cases of more technical results of the paper under review.