A converse to a theorem of Adamyan, Arov and Krein

Abstract

A well known theorem of Akhiezer, Adamyan, Arov and Krein gives a criterion (in terms of the signature of a certain Hermitian matrix) for interpolation by a meromorphic function in the unit disc with at most m m poles subject to an L ∞ L^\infty -norm bound on the unit circle. One can view this theorem as an assertion about the Hardy space H 2 H^2 of analytic functions on the disc and its reproducing kernel. A similar assertion makes sense (though it is not usually true) for an arbitrary Hilbert space of functions. One can therefore ask for which spaces the assertion is true. We answer this question by showing that it holds precisely for a class of spaces closely related to H 2 H^2 .