Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên phổ ràng buộc của các máy phát điện của các nhà khai thác tích cực semigroups. | J. OPERATOR THEORY 5 1981 257-266 Copyright by INCREST 1981 ON THE SPECTRUM OF HYPONORMAL OR SEMIHYPONORMAL OPERATORS DAOXING XIA 1 Let . be a complex separable Hilbert space be the algebra of all linear bounded operators in Xf. An operator T e is called semi-hyponormal 10 11 if T T 1. 2 7 7 1 2 0 and T is called hyponormal if J _ TT 0 If T is semi-hyponormal then there is an isometric operator u such that T t7 r 7 1 2 10 . Let UM U and u n for n 1 2 3 . . By results in 10 the polar symbols T st-lim UMTUt ni exist. The operator T is normal and the operator T is subnormal. However if u is unitary then T is also normal. If T X ÌY is hyponormal X and Y are self-adjoint then the symbols 2 13 T st-lim éx TQixt t- co exist and are normal. We construct the operators Tk k7 1 - k T_ Tik kT k T 0 k 1. 8-2843 258 DAOXlNG XIA It is easy to verify that these operators are normal when the operator u in the polar decomposition T U T Tflĩ is unitary in the semi-hyponormal case. In a previous paper 11 the author proved that if T is in a special subclass of semi-hyponormal operators then 1 ữ 7 Ụơ m The aim of the present paper is to prove that 1 is true for all semi-hyponormal operators and 2 r T Ự o Tk if T is hyponormal. 2 We shall consider the singular integral model of a hyponormal operator. Lemma 1. 8 9 6 . IfT X- -iY is completely non-normal hyponormal operator X and Y are self-adjoint Xi is the o-algebra of all Borel sets in ơ X m is the Lebesgue measure on a and Í2 o X XI m then there are an auxiliary complex separable Hilbert space 3t a strongly measurable projectionvalued function Q - with Q x eẩ S a uniformly bounded strongly measurable S -valued function on Í2 o X gs m satisfying oQ Q x a. ỊÌQ QP p ỵ a p p a unitary operator W XY Hi Xf where xe is the Hilbert space of all strongly measurable square integrable Qt-valued functionsf satisfying Qf f and an operator T in 3 if x x ip x f x ia .r P xf for feXf where P g st-lim e- 0 2ni Ja X .V s -f- ie such that T WTWX In
