On commutant lifting with finite defect. II (Q674734)

For \(j= 1,2\), \(T_j\) is a contraction in a Krein space \(H_j\), \(W_j\) is an isometry in a Krein space \(G_j\supset H_j\), such that \(G_j\ominus H_j\) is a Hilbert space, and \(P_j W_j= T_j P_j\), where \(P_j\) denotes the orthogonal projection onto \(H_j\) in \(G_j\). Let the operator \(A: H_1\to H_2\) be such that \(AT_1= T_2A\), and the negative index \(k\) of \(I-A^*A\) is finite. Then for some \(W_1\)-invariant subspace \(E\subset G_1\) with \(\text{codim }E= k\) a contraction \(\widetilde A: E\to G_2\) exists such that \(\widetilde A W_1|_E= W_2\widetilde A\) and \(P_2\widetilde A= AP_1|_E\). [For part I see the authors and \textit{S. A. M. Marcantognini}, J. Oper. Theory 35, No. 1, 117-132 (1996; Zbl 0853.46021)].