tailieunhanh - Báo cáo toán học: "On derivation ranges and the identity operator"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một dẫn xuất phạm vi và điều hành bản sắc. | Copyright by INCREST 1982 J. OPERATOR THEORY 7 1982 139-148 ON DERIVATION RANGES AND THE IDENTITY OPERATOR DOMINGO A. HERRERO 1. INTRODUCTION Let yyyi be the algebra of all bounded linear operators acting on the complex separable infinite dimensional Hilbert space ye. Given A e let ỖA be the inner derivation induced by A defined by ỎA X AX XA Xs y and let ran 5J j 1 .2 j be the range of ỖA . In 1 Joel H. Anderson proved that JzfQF contains a c -algebra ci l generated by A and the identity operator 1 such that its intersection with J ACT .ỡe M f 1 e ran 5a - the upper bar denotes norm-closure is a Ga-dense subset of C A in particular JA .r 0 . Furthermore Anderson s proof strongly suggests that every operator of the form T 1 e X2 ye such an operator will be called an ampliation is the norm limit of a sequence of operators in JA J X yf . It will be shown here that this is indeed the case If s ye A e .yyye A is unitarily equivalent to an ampliation then S ye fl JA .j is a C 5-dense subset of sr ye . Let ƠC T denote the essential spectrum of T e ee ye . the spectrum of the canonical projection n T of T in the quotient Calkin algebra f ye iye yy where Jf F denotes the ideal of all compact operators then ƠO T z. e ơ T ơe T Ả is an isolated point of r T is the set of all normal eigenvalues of T ơB T ơ T ơ0 T is the Browder spectrum of T and ee ye can be written as the disjoint union of w T 6 r0 T 0 which is open and dense in ee ye 15 and r B T e pf ƠO F 0 Te ỈỂựe ơ T 7B T which of course is closed and nowhere dense . 140 DOMINGO A. HERRERO It follows from 16 Lemma 2 that JA J so that JA . is nowhere dense in ỈẾựẾ . Furthermore it is well-known that in a certain sense 1 tends to be far from ran 5r for any T in see 9 Chapter 19 so that JA bf is a very small subset of y ị yẾ in many senses. A straightforward matrix computation shows that if A B c then 1 dist 1 ran 5zl max dist l ran 5fl dist l ran 5c . Thus if ị J A - 0 n - co A B c and dist l ranfig 1 for all n ----- 1