A Converse to a Theorem of Adamyan, Arov and Krein

A classical theorem of Pick gives a criterion for interpolation by analytic functions in the open unit disc D subject to an H∞-norm bound. This result has substantial generalizations in two different directions. On the one hand, one can replace H∞ by the multiplier algebras of certain Hilbert function spaces, some of them having no connection with analyticity [Ag1, Ag2]. On the other hand, one can obtain a criterion for interpolation by meromorphic functions with a prescribed number of poles in D and with an L∞-norm bound on the unit circle T; this is a classical result of Akhiezer [Ak], now better known in the form of the far-reaching generalizations due to Adamyan, Arov and Krein [AAK]. In each case the criterion is in terms of the signature of a “Pick matrix” constructed from the interpolation data and the reproducing kernel of the appropriate Hilbert function space (i.e. H in the case of the AAK theorem). It is therefore conceivable that there might be a common generalization which would hold for a significant class of function spaces. After all, the analogue of Pick’s theorem is true for the Dirichlet space D of analytic functions in D with finite Dirichlet integral [Ag1] and for the space W [a, b] of L functions f on [a, b] for which f ′ ∈ L(a, b) [Ag2]. Might not an analogue of the Akhiezer-Adamyan-Arov-Krein theorem hold for these spaces? This natural question was posed in [Q2]. Pick’s theorem has long held the attention of analysts as one of the most elegant of all interpolation results. However, there are grounds beyond the aesthetic ones for its continued prominence. Knowledge that the analogue of Pick’s theorem holds for a particular Hilbert function space gives a powerful tool for the conversion of L∞-type problems to Hilbert space problems; this principle is brought out in [MS], where among other things it is used to give a relatively simple proof of Carleson’s theorem on interpolation sequences for H∞ and also to characterize geometrically the interpolation sequences for multipliers of the Dirichlet space. Since about 1980 the Pick property has played an important role in linear control theory. The feedback controllers which internally stabilize a given linear system can be described by analytic functions satisfying a finite set of interpolation conditions, so