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Báo cáo toán học: "Characterization of Hilbert space operators with unitary cross sections "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Mô tả đặc tính của các nhà khai thác không gian Hilbert với các bộ phận đơn nhất qua. | J. OPERATOR THEORY 2 1979 153 158 Copyright by INCREST 1979 CHARACTERIZATION OF HILBERT SPACE OPERATORS WITH UNITARY CROSS SECTIONS DON DECKARD and L. A. FIALKOW 1. INTRODUCTION The purpose of this note is to characterize the operators on a separable Hilbert space which have local unitary cross sections in the following sense. Let ã denote a c -algebra with identity and let LĨC ẽl denote the unitary group in t. For an element X in ã let cĨE y u xu u in Lĩt d denote the unitary orbit of X and let 71 denote the norm continuous mapping of fi d onto fit y given by 7i t u xu. local cross section for 71 is a pair cp cS such that is a relatively open subset of tĩí T that contains X and p Si qt d is a norm continuous function such that p X 1 and n p L6. If n admits a local cross section we say that X has a local unitary cross section and in this case X satisfies the following sequential unitary lifting property P If u Qt a and lim u xu X 0 then there exists a sequence w c T fi such that lim Ị JF 1 j 0 and such that w xw u xu for each n. Let JC denote a complex separable infinite dimensional Hilbert space and let X denote the algebra of all bounded linear operators on 3C. The fact that finite rank projections satisfy P relative to the algebra 3 X was proved in 5 Lemma 1 and in 3 it was shown that a normal or isometric operator satisfies P if and only if its spectrum is finite in which case a cross section exists. In 4 Section 3 this characterization was extended to hyponormal operators and the analogue for hyponormal elements of the Calkin algebra was obtained. 3 and 4 contain examples of non-normal operators with cross sections and in 4 Theorem 2.1 it was proved that a compact operator satisfies P if and only if it is of finite rank. In the present note we obtain the following characterization of operators with cross sections relative to the algebra fl 3C and our result affirms the conjecture of 4 . Theorem 1.1. For an operator T in the following are equivalent i T .