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 form f(z1,z2)=1P(z1,z2)¯Q(z1,z2) where P(z1,z2) and Q(z1,z2) 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.