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 T T T T - TT r p p for some p in The notation p p in is the standard notation for the one dimensional operator p p f f p p where is the inner product of Xf. The condition 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 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 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 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 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
