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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đặc điểm chức năng và dilations noncontractions. | J. OPERATOR THEORY 3 1980 71 87 Copyright by INCREST 1980 CHARACTERISTIC FUNCTIONS AND DILATIONS OF NONCONTRACTIONS BRIAN w. McENNIS 1. INTRODUCTION In a series of papers in Acta Sei. Math between 1953 and 1966 B. Sz.-Nagy and c. Foias developed a theory of contractions on Hilbert space. This theory is presented in the book 19 where references to these papers can be found. The original paper 18 by Sz.-Nagy proved the existence of a unitary dilation of a contraction and this forms the basis of the Sz.-Nagy and Foias theory. In 1970 Ch. Davis 8 proved that every closed operator T has a dilation which is unitary with respect to an indefinite inner product see Sec. 2 below and.in 9 Davis and Foias study the relationship between this dilation and the characteristic function see Sec. 6 below . We continue this study in this paper generalizing some of the work of Sz.-Nagy and Foias for contractions. 2. KREIN SPACES. DILATIONS Here is a summary of some of the notation and results that will be used in this paper see 3 13 14 15 . An indefinite inner product space is a complex vector space on which is defined an inner product . . that is not assumed to be positive i.e. it is possible for h h to be negative for some h e . We call a Krein space if there is an operator J on Jf such that J2 I J J i.e. Jh k h Jk and the J-inner product 2.1 h k Jh k makes X a Hilbert space. Such an operator J is called a fundamental symmetry. See 3 Chapter V. In Krein spaces the emphasis is always on the indefinite inner product with the J-norm Jh hỴ12 serving mainly to define the topology the strong topology . Accordingly if A is a continuous operator between Krein spaces and Jf we use A to denote the adjoint of A with respect to the indefinite inner products. 72 B. w. McENNIS Different fundamental symmetries J on a Krein space define different J-norms but the strong topologies obtained coincide see 13 Sec. 1.4 3 Corollary IV.6.3 Theorem v.1.1 . Thus we can talk about the strong topology on a .