Krein spaces of analytic functions

Abstract

An invariant subspace theory for contractive transformations in Hilbert spaces which is due to L. de Branges and J. Rovnyak ( in “Perturbation Theory and Its Applications in Quantum Mechanics,” pp. 295–391, Wiley, New York, 1966 ) is generalized. Observable, contractive, and conjugate-isometric linear systems are considered whose state spaces and coefficient spaces are Krein spaces. Such linear systems have canonical models with state spaces chosen as Krein spaces whose elements are vector-valued analytic functions. The properties of these spaces are related to the theory of square summable power series with coefficients in a Krein space. Generalizations of a celebrated theorem of A. Beurling ( Acta Math. 81 (1949) , 209–255) and P. D. Lax ( Acta Math. 101 (1959) , 163–178) are obtained in that context. An underlying theme is the relation between factorization and invariant subspaces for continuous and contractive transformations in Krein spaces. Known results for Hilbert spaces ( L. de Branges, J. Math. Anal. Appl. 29 (1970) , 168–200) are generalized. Some of these results have previously been generalized to Pontryagin spaces by D. Alpay and H. Dym ( in “Operator Theory: Advances and Applications,” Vol. 18, pp. 89–159, Birkhaüser-Verlag, Basel, 1986 ).