tailieunhanh - Báo cáo toán học: "Singly generated hyponormal C*-algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đơn lẻ tạo ra C *- đại số hyponormal. | J. OPERATOR THEORY 11 1984 243 - 254 Copyright by INCREST 1984 SINGLY GENERATED HYPONORMAL C -ALGEBRAS c. R. PUTNAM 1. Let H be a separable infinite dimensional Hilbert space and T a completely hyponormal operator on H. Thus if T has the Cartesian representation T A iB then T T - TT 2i AB - BA D 0 and T has no nontrivial reducing subspace on which it is normal. It is known that meas2ff T see 15 and in the special case where D is compact 2 . Let A Re T have the spectral resolution 4 ịíd r and let A be any open interval of the real axis for which E A 0. Then Ta E A TE A regarded as an operator on ECA H with spectrum r 7j is also completely hyponormal on E A H and ơ TJ ơ T . See 15 and in case T is compact 2 Moreover ơ 7 j n z Re z e d ơ T n z Re z 6 A see 16 . For a recent survey of hyponormal operators including in particular a discussion of the cut down operators 7 j see 5 . It follows from that if p is any Borel subset of the real line and if F l denotes the Lebesgue linear measure of the vertical cross section O- T n z Re z 244 c. R. PUTNAM then n E P DE P W F f dt. i This paper will be concerned mainly with completely hyponormal operators for which equality holds in that is 1-7 7r Z meas2ơ 7 . In this case equality also holds in . For otherwise let a be a Borel set of the real line satisfying n E ot DE 0 a F r dr so that 7r1 2 ịỡ1- 2E a alị2 ã f I y F f dt I . If y oo oo a then for any X in H and in view of a with p y nl 2 l 2x - 7rl 2 DV2 a A- l ZJV2 y x i i Zf Xl 2 4 ữI 2 a x I I F t dt I E y x . V Since x 2 ỊE a x ị2 4- l E y x 2 and X is arbitrary an application of the Schwarz inequality yields Tĩ Z a y F r dr y F r dr y E r dr meas2ơ 7 in contradiction with . If T is completely hyponormal then A Re T and B Im T are absolutely continuous and ơ A and ơ B are the l real projections on the real and imaginary axes of ff T see 14 pp. 42 46. If A has the spectral resolution then iEfX i2 is an absolutely .