tailieunhanh - Báo cáo toán học: "Quasisimilarity and hyponormal operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Gần như tương tự và các nhà khai thác hyponormal. | J. OPERATOR THEORY 5 1981 127-139 Copyright by INCREST 1981 QUASISIMILARITY AND HYPONORMAL OPERATORS L. R. WILLIAMS s. Clary proved in 8 that quasisimilar hyponormal operators have equal spectra and he asked whether quasisimilar hyponormal operators have equal essential spectra. The present author studied this question in 26 and proved in 27 that quasisimilar quasinormal operators have equal essential spectra. The purpose of this note is to study further the above mentioned question of Clary and to present some other results that relate to quasisimilarity and hyponormal operators. If is a Hilbert space let JZ denote the algebra of all bounded linear operators on ye. In this note we shall use the term operator to mean an element of JỄ for some complex Hilbert space ye. If Tis an operator and TT T T then T is said to be hyponormal. If and ye are Hilbert spaces and X yex - ye2 Is a bounded linear transformation having trivial kernel and dense range then X is called a quasiaffinity. If T e ee yeY and e iffiye. and there exist quasiaffinities X .ye-L - ye2 and - yet satisfying XTị T x and TỵY YT2 then Tỵ and T2 are said to be quasisimilar. If T is an operator let ơ T denote the spectrum of T ye T the kernel of T and ẩỉ. T the range of T. If T 6 and ye is an infinite dimensional Hilbert space letoựT denote the essential spectrum of T . the spectrum of T where T - T is the natural quotient map of e ye onto the Calkin algebra e yy ffi. denotes the norm-closed ideal of all compact operators in e ye . As stated earlier our primary interest is in the class of hyponormal operators. However many of the results in this note are valid for the more general class of dominant operators. Recall that an operator Tis said to be dominant if Ổ T 2 Ố T 2 for each 2 in a T . The study of dominant operators was introduced by Stampfli and Wadhwa in 25 It follows easily from Theorem 1 of 11 that every hyponormal operator is dominant. Some of the properties of hyponormal operators .