tailieunhanh - Báo cáo toán học: "Seminormal composition operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Seminormal thành phần các nhà khai thác. | J. OPERATOR THEORY 11 1984 Copyright by INCREST J 984 SEMINORMAL COMPOSITION OPERATORS DAVID J. HARRINGTON and ROBERT WHITLEY 1. INTRODUCTION A composition operator CT on L2 S z p S z ụ sigma-finite throughout this paper is a bounded linear operator given by composition with a map T s - 5 via CTf s f T sỴ . The classical necessary and sufficient conditions for this are 4 p. 39 and 2 pp. 663 5 T measurable p o T 1 absolutely continuous with respect to p and h ảpoT 1 dp in IFfS X p . Various properties of these operators can be characterized in measure theoretic terms. For example answering a question raised by Nordgren in his interesting lectures 4 see 6 and 7 0. Theorem a CT is normal iff i T-1 is essentially all of z and ii h h T . where h d i r-1 d . b If p S is finite then CT is normal iff i T-1 Z is essentially all of z. and ii h 1 . . T is measure preserving. In this case CT is unitary. An operator A is said to be hyponormal in case A A AA 0 A is seminormal if either A or A is hyponormal. A is called quasinormal when A commutes with A A. It is our purpose here to obtain characterizations analogous to Theorem 0 for c f hyponormal Cf quasinormal and CT hyponormal cr quasinormal being characterized in 7 . We will often consider a sigma-algebra sd of subsets of 5 and the subspace of L s z p consisting of all -measurable functions. Strictly speaking this subspace consists of all functions in L2 which are equal . to an -measurable function or alternately consists of all equivalence classes each of which contains an -measurable function. To avoid circumlocutions like these we will take sd to be relatively complete with respect to z that is we will pass to the sigma-algebra generated by sd and F in z p F 0 . AAF A in sd 126 DAVID J. HARRINGTON and ROBERT WHITLEY F in z F 0 once again calling it sú. In keeping with this practice 7-1 2 will denote the relative completion of the sigma-algebra T-1 t4 A in 2 . With this convention condition i in .