The inverse of nonsymmetric two-level Toeplitz operator matrices (Q448362)

A theorem of \textit{I. C. Gohberg} and \textit{A. A. Semencul} [Mat. Issled. 7, No.~2(24), 201--223 (1972; Zbl 0288.15004)] states that, given a Toeplitz matrix \(T_n=\|t_{i-j}\|_{i,j=0}^n\), the inverse \(T_n^{-1}\) can be reconstructed from its first and last columns, provided that both \(T_n\) and \(T_{n-1}\) are invertible. The algorithm of such reconstruction is available. In the present paper, the authors extend this result to operator-valued two-level Toeplitz matrices \(T_n=\|t_{i-j}\|_{i,j\in\Lambda}\). Here, \(\Lambda\) is a finite set in the lattice \(\mathbb{Z}_+\times\mathbb{Z}_+\), ordered in a certain way, and \(t_k\) is a Hilbert space operator. Precisely, NEWLINE\[NEWLINE \sum_{(k_1,k_2)}t_{(k_1,k_2)} z_1^{k_1}z_2^{k_2}=\frac1{\overline{P(z_1,z_2)}\,Q(z_1,z_2)}\,, NEWLINE\]NEWLINE where \(P\) and \(Q\) are stable polynomials of two variables.