tailieunhanh - Báo cáo toán học: "On operators wich almost commute with the shift "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Người điều khiển gần như đi làm với sự thay đổi. | J. OPERATOR THEORY 11 1984 365 377 Copyright by INCREST 1984 ON OPERATORS WHICH ALMOST COMMUTE WITH THE SHIFT I. D. BERG INTRODUCTION The aim of this paper is to determine when weighted shifts on which almost commute with the canonical left shift can be perturbed so as to commute with a perturbation of the left shift. Our analysis encompasses weighted right and left shifts as well as their products including for example any multiplication operator. We are interested both in questions of compact perturbations and pertubations of small norm. It turns out that some of the wonted parallelism between compact perturbations and small norm perturbations breaks down itself a phenomeon which is of quite some interest. There has been a great deal of interest in the general phenomenon of lifting from the Calkin algebra most of it tending towards an abstract approach. These abstractions have been interesting it is possible that in general they are actually unavoidable because of the non-constructive nature of the realization of the Calkin algebra as an operator algebra. However it is satisfying to find a situation that yields to constructive methods. Our entire analysis is in the context of where r 4. We are able to present an essentially complete analysis in the case of shifts with real weights. There are phenomena in the complex case which still elude US. We should mention immediately that the perturbations contemplated do not leave either operator as a weighted shift commutativity of an operator with a weighted shift is a triviality. We are concerned with shifts with real weights on . We denote the canonical basis vector of as Ỗị i 1 2 . . We define the left shift of index r with weights t by 1 T 5 tnỗ r for II r 2 T S -- 0 for II r. 366 I. D. BERG T SJ Ổ TS - ST 4 T SJ Ỗ - TS - ST ỏn We will usually suppress the obvious condition 2 when describing a shift. Contrary to the usual convention we will allow weights of 0. We also consider the multiplicative operator r 0 as an .