Integral Equations and Operator Theory Vol. 13 (1990) 0378-620X/90/050701-2751.50+0.20/0 (c) 1990 Birkh~user Verlag, Basel UNITARY COLLIGATIONS OF OPERATORS IN KREIN SPACES Stefania Marcantognini e) A model for non-contractive, functions is given,according to which every bounded analytic function coincides with the characteristic function of a suitable unitary colligation. In our construction,the function is expressed in terms of a fundamentally reducible unitary dilation of the basic operator of the colligation I.KREIN SPACES A Krein space is a vector space R equipped with a symmetric sesquilinear form (.,.) such that there exist two subspaces R § and R- with the following properties: i) R is the algebraic direct sum of R§ and ~- ; ii) R* and ~- are orthogonal,in the sense that (x*,x-) = 0 for all x § E ~* and all x- E ~- ; iii) (~+,(.,.)) and (~-,-(.,.)) are Hilbert spaces In a Krein space all notions involving some kind of inner product such as orthogonal,symmetric, isometric etc. are to be understood,unless otherwise stated,to refer to the form (.,.) We write m) The results in this paper first appeared as part of the author's doctoral thesis [13] at the Universidad Central de Venezuela,under the supervision of Mischa Cotlar.The work has been supported by the . ConseJo Hacional de Investigaciones CientLficas y Tecnol~gicas " (CONICIT-Venezuela).I am grateful to Professor Aad Dijksma,who read an earlier version of this note and made several helpful criticisms. I also thank Professor Mischa Cotlar for his kind encouragement and Professor AleJandra Cabana for correcting the english text .