The solution of H∞ problems requires the determination of contractive operators that map certain input signals to certain output signals. Such operators, and tests for their contractiveness, arise naturally in a scattering formulation of the generalized Schur algorithm, which is an efficient procedure for the triangular factorization of matrices with displacement structure. In this paper we explain this connection and show how to reformulate H∞ problems, both for the finite and the infinite horizon cases, in terms of equivalent factorization problems for positive-definite matrices with structure.