tailieunhanh - Báo cáo toán học: "Some index theorems for subnormal operators"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một số chỉ số định lý cho các nhà khai thác dưới nhiệt độ. | Ĩ. OPERATOR THEORY 3 1980 115-142 Copyright by INCREST 1980 SOME INDEX THEOREMS FOR SUBNORMAL OPERATORS ROBERT F. OLIN and JAMES E. THOMSON I. INTRODUCTION Let N be a normal operator on a separable Hilbert space . An operator is pure if it has no reducing subspace on which it is normal. The set f N will denote the collection of subnormal operators that have N as their minimal normal extension . fp N denotes the pure operators in .P A . See 3 for the basic results concerning subnormal operators. An operator T is semi-Fredholm if the range of T is closed and either ker T or ker T is finite dimensional. If T is semi-Fredholm then the index of T denoted i T is defined to be the integer possibly oo dim ker T dim ker T . If s belongs to f N then ơ S the spectrum of S contains ơ A . In fact ơ S ơ N is either empty or equals the union of some of the bounded components of the complement of tr N . We refer to these latter components as the holes of ơ A . There has been considerable investigation of problems related to hole filling . See 8 and 13 for a history and some results concerning this subject. The major purpose of this paper is to describe an intrinsic relationship between each normal operator Ain a certain class and the index theory for the collection if p N . Before we discuss this matter further we present a result that is well-known. Suppose s e f N and let Í2 be a component of o- S ff A . Then for a p e Í2 it follows that a dim J S a J 0 and b dim Q S - a jf dim S - 0X . Here is the space on which s is defined. To see this observe that a p e Q implies that s a. and s p are bounded below but not invertible. Therefore S a J and S ppff are proper closed subspaces of . That establishes a . It also follows that s a and s p are semi-Fredholm operators because ker S a and ker S p are trivial. Hence to obtain b it suffices to show that i S a i S P . The latter now follows from the fact that the index is a continuous function from the semi-Fredholm operators to z u .