The inverse of nonsymmetric two-level Toeplitz operator matrices

Abstract

The Gohberg-Semencul formula allows one to express the entries of the inverse of a Toeplitz matrix using only a few entries (the first row and the first column) of the inverse matrix, under some nonsingularity condition. In this paper we will provide a two variable generalization of the Gohberg-Semencul formula in the case of a nonsymmetric two-level Toeplitz matrix with a symbol of the formwhere P(z 1 , z 2 ) and Q (z 1 , z 2 ) are stable polynomials of two variables. We also consider the case of operator valued two-level Toeplitz matrices. In addition, we propose an equation solver involving two-level Toeplitz matrices. Numerical results are included.