In this dissertation we study the modern state space method for inverting semi-infinite block Toeplitz operators with rational matrix symbols explicitly from the representation of its symbol in realization form. The method for constructing explicit formulas for the inverse of a semi-infinite block Toeplitz operator with rational symbol is well-known for rational matrix functions that are analytic and invertible at infinity. However, in this work, we emphasize the case where the matrix function does not have these properties at infinity. In the main results in Chapter 2 we give necessary and sufficient conditions for the equivalence between block Toeplitz operators with rational symbol and discrete singular systems with boundary conditions. We also deal with the special case of finite block Toeplitz matrices. Different Fredholm characteristics are computed in Chapter 3, and a Riemann-Hilbert problem is solved as an application. This book should be useful to individuals who study Operator Theory in Mathematics and Control Sytems in Engineering.

Abraham Konegerie, Msc: Studied Mathematics at the University of the Western Cape, South Africa. Lecturer at the Cape Peninsula University of Technology, Belville, South Africa.