Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một sự bất bình đẳng khu vực cực cho hyponormal phổ. | J. Operator theory 4 1980 191-200 Copyright by INCREST 1980 A POLAR AREA INEQUALITY FOR HYPONORMAL SPECTRA c. R. PUTNAM 1. INTRODUCTION Let Ộ be a separable Hilbert space and let T be a bounded operator with the rectangular representation T A B where A ị táE . Then T is said to be hyponormal if T T TT D 0 equivalently AB - BA - 4 iZ . 2 in this case ơ A and ơ B are the real projections of ff T onto the real and imaginary axes see 7 p. 46. If A is any open interval of the real line then E A TE A is hyponormal on E A ệi and hence its spectrum as an operator on E A lies in the closure of the strip z Re z e d . It is known that ff d T d c ff T and that 7Ĩ D ắ meas2 ơ T . These relations were proved in 8 and in the special case when D is compact in 1 . In 9 it was shown that ơ d T d n z Re z eA ơ T n z Re z eA . It follows from or even from that if is any Borel set of the real line and if F t is the Lebesgue linear measure of the vertical cross section ơ T n 192 c. R. PUTNAM n z t of ff T then 1-6 7r Ị2WW U dz. fl In particular follows by choosing oo oo . As was shown in 9 p. 701 relation can be generalized to ơ j8 W z a T . Several results were obtained in 12 relating the spectra of a hyponormal operator T and its polar factors in case T has a polar factorization 1-8 2n-0 T UP u unitary ịe dGt ị j p ằ 0. 0 In this paper there will be proved certain polar analogues of and . For later use observe that by T T TT p2 UP2U D ằ 0 It may be noted that if T is any bounded operator on Ộ and if 0 ị a T then T has a factorization 1 8 see Wintner 15 A generalization was given by von Neumann 6 p. 307. Further it was shown by Hartman 4 using von Neumann s result that an arbitrary bounded T has a representation if and only if the null spaces of T and T have the same dimension. Also if this common dimension is zero so that 0 is not in the point spectra of T and T then the representation is unique. A hyponormal .
