A generalized commutativity theorem (Q1188359)

Summary: Let \(H\) be a complex separable Hilbert space, \({\mathcal C}\) the class of contractions with \(C_{\cdot 0}\) completely non-unitary parts, \({\mathcal C}_ 0\) the class of \(A\in {\mathcal C}\) which satisfy the property (called property (P2)) that if the restriction of \(A\) to an invariant subspace \(M\) is normal, then \(M\) reduces \(A\), and let \({\mathcal C}_ 1\) be the class of \(A\in {\mathcal C}_ 0\) with defect operator \(D_ A\) being of the Hilbert-Schmidt class \(C_ 2\) and which are such that either the pure part of \(A\) has empty point spectrum or the eigenvalues of \(A\) are all simple. It is known that if \(A\in{\mathcal C}_ 0\) and \(B^*\in {\mathcal C}_ 1\), then \(AX=XB\) implies \(A^*X=XB^*\). This implication fails to hold for the case in which \(A\in{\mathcal C}\). It is shown here that if \(A\in {\mathcal C}\) and \(B^*\in {\mathcal C}_ 1\), then \(AX=XB\) implies either (i) \(A| \overline{\text{ran}} X\) and \((B^*|\ker^ \perp X)^*\) are quasi-similar \(C_ 0\) contractions (with \(B^*|\ker^ \perp X\) normal), or (ii) \(A^* X=XB^*\). Let \({\mathcal C}^ 1\) denote the class of contractions \(E\) satisfying property (P2), the inclusion \(D_ E\in C_ 2\) and which are such that the pure part of \(E\) has empty point spectrum. Choosing the intertwining operator \(X\) to be compact it is shown that \(AX=XB\) implies \(A^*X=XB^*\) for \(A\in{\mathcal C}_ 0\) and \(B^*\in{\mathcal C}^ 1\). Recall that quasi-similar operators need not be unitarily equivalent (or, even, similar). We show that if \(A\in{\mathcal C}_ 0\) and \(B\in{\mathcal C}^ 1\) are quasi-similar with one of the implementing quasi-affinities compact, then \(A\) and \(B\) are unitarily equivalent normal contractions. Also it is shown that a compact operator \(A\in{\mathcal C}^ 1\) is normal.