OnH-contractions and the extension problem for hermitian block toeplitz matrices
Roland W. Freund, Thomas Huckle · Linear and Multilinear Algebra · 1989
It is well known that any positive semi-definite Hermitian block Toeplitz matrix Tn (C0 ,C1 ,…Cn ) has positive semi-definite Toeplitz extensions Tn+1 (C0 ,…Cn Cn+1 ). In this paper, we consider the more general problem of extending arbitrary Hermitian block Toeplitz matrices Tn to matrices Tn+1 with the same number of negative eigenvalues as Tn . A necessary and sufficient condition for the existence of such Cn+1 is derived, and a parametrization of all corresponding Cn+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.