On H-Contractions and the Extension Problem for Hermitian Block Toeplitz Matrices

It is well known that any positive semi-definite Hermitian block Toeplitz matrix T_n(C₀, C₁, …, C_n) has positive semi-definite Toeplitz extensions T_n+1(C₀, …, C_n, C_n+1). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices T_n to matrices T_n+1 with the same number of negative eigenvalues as T_n. A necessary and sufficient condition for the existence of such T_n+1 is derived, and a parametrization of all corresponding C_n+1 is presented. Our approach to the Hermitian block Toeplitz extension problem is based on a formulation as a completion problem for contractions in indefinite inner product spaces. A complete solution of this more general problem is given.