TAILIEUCHUNG - Báo cáo toán học: "Characteristic functions and dilations of noncontractions "

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. 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 proved the existence of a unitary dilation of a contraction and this forms the basis of the 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 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 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 . 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 . Jh k h Jk and the J-inner product 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. 3 Corollary Theorem . Thus we can talk about the strong topology on a .

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