Continuous analogs of Schur extension problems and bitangential generalizations

Abstract

The classical Schur extension problem is to describe the set of contractive holomorphic functions s(ζ) in the open unit disk of the form It is well-known that this set is nonempty if and only if the (n + 1) × (n + 1) lower triangular Toeplitz matrix A with entries aij = ai − j is contractive, i.e., if and only if In + 1 − AA* ≥ 0; see e.g., p. 79 of 5. In this paper the analogue of this problem for mvf's (matrix valued functions) that are holomorphic and contractive in the open upper half plane and bitangential generalizations thereof are studied.