tailieunhanh - Báo cáo toán học: "On the distance to the set of compact perturbations of nilpotent operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Vào khoảng cách đến các thiết lập của các nhiễu loạn nhỏ gọn của các nhà khai thác lũy linh. | Copyright by INCREST 1980 J OPERATOR THEORY 3 1980 179-194 ON THE DISTANCE TO THE SET OF COMPACT PERTURBATIONS OF NILPOTENT OPERATORS NORBERTO SALINAS 1. INTRODUCTION Let be a separable infinite dimensional complex Hilbert space and be the algebra of all bounded linear operators on w. Also we denote by the ideal of compact operators on and by lĩ jS - 3ẮJÍ the canonical quotient map onto the Calkin algebra In this paper we deal with precise estimates of the distance from an arbitrary operator T in s j to the set jY of all nilpotent operators on If Theorem a . We also estimate the distance d T F X Theorem b and we establish a spectral formula for d 7 jV for the class of essentially Gj operators . those operators whose essential resolvent function has first order growth 12 whose semi-Fredholm domain contains no infinite indices Theorem . In particular we obtain a spectral characterization of d T yK jf for the class of essentially normal operators Remark a . In the last section of the paper we give an interesting example of an essentially Gj operator s in fact an essentially hyponormal operator S for which d S J jT is not obviously computable. We also pose some other problems concerning this operator s and we ask some questions related to the material presented in the previous sections. In what follows ơ T denotes the spectrum of an operator T in f J and E T denotes the essential spectrum of T . E T ơ tĩ T . Also we shall denote by E T and Et T the left and right essential spectra of T and we let Elr T Ef T n Er T . Finally Í2 T will denote the Weyl spectrum of T . Í2 T Ấe c ị X is not Fredholm of index 0 16 and 70 T will denote the set of normal eigen-values of T . UO T 2e C 2 is an isolated point of ơ T and T Ấ is Fredholm recall that the Browder spectrum B T is defined to be B T ơ T ơ0 T 17 . 180 NORBERTO SALINAS 2. A SPECTRAL ESTIMATE OF d T yr In order to state the main result of this section we need to introduce some non-standard .