Quasi-isometries associated to \(A\)-contractions (Q401205)

For \(A, T \in B(H)\) with \(A \geq 0\), \(\|A\|=1\), the operator \(T\) is called \(A\)-contraction if \(T^*AT\leq A\). Some various maximum invariant or reducing subspaces for such operators \(A\) and \(T\), or \(M =A^{1/2}T\) were studied in [the second author, Extr. Math. 21, No. 3, 221--247 (2006; Zbl 1179.47006)]. The authors of present paper investigate the maximum subspace of \(H\) which reduces \(M\) to a quasi-isometry, that is, on which the equality \(M^*M = M^{*2}M^2\) holds. This subspace has a concrete form if \(M\) is quasinormal, being just \(\ker(A-A^2)\) if \(T\) is an \(A\)-isometry, i.e., \(T^*AT=A\). In this case, one can decompose \(\ker(A-A^2)\) with respect to the maximum subspace of \(H\) which reduces \(M\) to a normal partial isometry. They also consider the case where \(T\) is a \(TT^*\)-contraction on \(H\), which implies that \(T\) is a contraction. In this case, some subspaces under consideration coincide with \(\ker(I -S_T) \oplus \ker(T^*)\), where \(S_T\in B(H)\) is defined as the strong limit of \(\{T^{*n}T^n\}\).