tailieunhanh - Báo cáo nghiên cứu khoa học: "Closure of similarity orbits of nilpotent operators I. Finite rank operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đóng quỹ đạo giống nhau của các nhà khai thác lũy linh của các nhà khai thác I. xếp hạng hữu hạn. | J. OPERATOR THEORY 1 1979 177 185 Copyright by INCREST 1979 CLOSURE OF SIMILARITY ORBITS OF NILPOTENT OPERATORS I. FINITE RANK OPERATORS JOSÉ BARRÍA and DOMINGO A. HERRERO 1. INTRODUCTION Let JiffT be the algebra of all continuous linear operators acting on a real or complex Banach space sc. The similarity orbit of T eC SC is the set f T IFTIF-1 w is invertible in X . Let .SC T denote the norm closure of f T . The main result of this article is the following one This result completes the answer ogiven in ref. 3 . Theorem . Let Teỉề SC be a necessarily algebraic finite rank operator with minimal monic polynomial p z JJ z Ằj kJ For sc a real Banach space j i it will be assumed that all the Ầfs are real then Cf T A E SC rank q A ỄỈ rank q T and dim Kerq A dim Ker ợ T for all q I where q I p denotes a monic polynomial q dividing p. Let Le CF X then C L C T if and only if rank q L rank q T for all q p if and only if L is similar to T. The real and complex case follow by the same proof so we shall only consider the complex case. The second statement of the theorem is a trivial consequence of the first one and suggests the following Definition . If A BeS 9C and Cf A Sf B A and B will be called asymptotically similar operators. By A B A B it will be meant that A is similar asymptotically similar resp. to B. Clearly is an equivalence relation in f .T . As in ref. 3 the equivalence class of A will be denoted by M . 3 9C I is a partially ordered set . where is the partial order induced by inclusion among the closures of similarity orbits. Let N SC denote the set of all nilpotent operators inJzfX . Theorem shows in particular that the section of the . CC 9C Ị corresponding to finite rank nilpotents is actually a lattice and moreover the structure of this lattice is independent of SC provided sc is an infinite dimensional space. 178 JOSÉ BARRIA and DOMINGO A. HERRERO Theorem has the following analog In what follows ye will always denote a complex Hilbert