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Báo cáo toán học: "A Beurling-Lax theorem for the Lie group U(m,n) which contains most classical interpolation theory "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một định lý Beurling Lax cho nhóm Lie U (m, n), trong đó có hầu hết các lý thuyết cổ điển suy. | J. OPERATOR THEORY 9 1983 107-142 Copyright by INCREST 1983 A BEURLING-LAX THEOREM FOR THE LIE GROUP U m n WHICH CONTAINS MOST CLASSICAL INTERPOLATION THEORY JOSEPH A. BALL ano J. WILLIAM HELTON INTRODUCTION In this article we generalize the famous theorem of Beurling Lax and Halmos from the Hilbert space 7 2 C to a space with a signed Hermitian form. Our proof is an adaptation of Halmos wandering subspace proof of the theorem 14 and of McEnnis analysis of shifts on a space with an indefinite metric 23 . Our Beurling-Lax theorem for the Lie group U m n as opposed to the classical one where U n U n 0 has very strong consequences for Nevanlinna-Pick Carathéodory-Fejér etc. interpolation theory. We obtain directly from our theory a simple linear fractional parameterization of all solutions in ỖSHeo Mm or of the most general interpolation problem for a finite number of points and strong results for infinitely many points. Moreover we obtain a test to determine if any solution to a particular interpolation problem exists. Finally in the last section we apply an extended form of our Beurling-Lax theorem to the setting of the Sz.-Nagy Foias com-mutant lifting theorem. Here ỖSH ị Mm denotes the closed unit ball of m X n matrix valued functions on the unit circle with meromorphic continuations onto the unit disk with at most I poles there multiplicity must be counted carefully see 16 . As usual U m n denotes the group of m ri x m ri matrices g which leave the form x y X y x x cm - y y cn x yeC n xeCm yeC where is the usual Euclidean inner product invariant. The linear fractional parameterization and the test for existence was obtained for I -- 0 or for m n 1 by Adamjan Arov and Krein 1 2 The test for existence was obtained in general by Ball 5 and very refined results due to Arsene Ceausescu and Foia 3 when Ị 0 are also available. Also Nudelman has recently 108 JOSEPH A. BALL and J. WILLIAM hl X obtained such results 27 see also 26 . Fuither strong results aie also due to .