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Báo cáo toán học: "Hilbert space operators with one dimensional self-commutators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác không gian Hilbert với một chiều tự-commutators. | J. OPERATOR THEORY 13 1985 265-289 Copyright by INCHEST 1985 HILBERT SPACE OPERATORS WITH ONE DIMENSIONAL SELF-COMMU TATORS KEVIN F. CLANCEY The intent is to present a report on the theory of bounded linear operators T acting on a separable Hilbert space whose self-commutators have the form 0.1 T T T T - TT r p p for some p in The notation p p in 0.1 is the standard notation for the one dimensional operator p p f f p p where is the inner product of Xf. The condition 0.1 only characterizes those operators with one dimensional self-commutators satisfying X y 0. Of course if X has one dimensional self-commutator then one of X or X will satisfy the latter semidefinite condition. By demanding 0.1 we have elected T to satisfy T T 0 and this will lead to surprizing differences in the spatial careers of T and T . As every operator theorist knows the unilateral shift satisfies 0.1 so that solitary examples from the class of operators having one dimensional self-commutators have been around for sometime. The first general study of the class appeared in 1962 32 . The main result in 32 is a singular integral model which can be summarized as follows Suppose T H J is the Cartesian form H l 2 r T J 1 2i T T of an operator satisfying 0.1 and H is unitarily equivalent to the operator Hjlf tf t acting on L2 a b . Then p is in L a b and J has the form 0-2 J t A t t i 0 1 dy 21 J t s a for some real valued i i in L a b . Without defining carefully the nature of the singular integral a Cauchy principal value the reader can formally verify that with H J as above T H ij satisfies T T 2i f J p cp. Until recent times singular integral models such as 0.2 and Cartesian or polar decompositions have been the mainstays of the theory of operators having 266 KEVIN F. CLANCEY one dimensional self-commutators. The references 1 4 10 13 15 24 26 28 34 are a representative variety of such techniques. Of course the Cartesian decomposition is not conducive to a natural analytic functional calculus. One