TAILIEUCHUNG - Báo cáo toán học: "A hyponormal weighted shift whose spectrum is not a spectral set "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một sự thay đổi trọng hyponormal có phổ không phải là một tập hợp quang phổ. | J. OPERATOR THEORY 8 1982 401-403 Copyright by INCREST 1982 A HYPONORMAL WEIGHTED SHIFT WHOSE SPECTRUM IS NOT A SPECTRAL SET JIM HARTMAN Let .if be a separable Hilbert space with orthonormal basis e n e z . Also let denote the set of bounded linear operators on .yf. For T e âS ả let A T denote the spectrum of T in and let i T denote the spectral radius of T. A compact set K A T is said to be a spectral set for T if Ii T II sup z zgK k for all rational functions f with poles off K. An operator Te is called a bilateral weighted shift if there exists w n E z G c such that Te wnen 1 for all integers n. In this paper we will consider the weighted shift T with H - for n 0 and H - - 1 for n 0. The following facts are easy to establish see Shields 2 a T l b T is invertible and IT-1 II 2 c A T JzeC Ị- z 11 I 2 f d t T II Til and r T-i IIT-1 e T is hyponormal. We will answer the second part of Question 7 of Shields 2 by showing that A T is not a spectral set for T. We let -1 1 oo . -Z-14- y 2 n lzmn i 2 1 for m ỷ 1. We now have ll mIL T --i 1 p by the maximum modulus theorem. From this equality it is easy to see that i ll in 1 - ii ll mllj T - 1 as m co. 402 JIM HARTMAN However fm T -1 T-1 Ể 2 2 n 1 and IIÁẤDIi fm T e0 i. We then compute fni T eữ and get Hence If m 4 then ÁẤTX i -i 2 . n--l ll jnil 1 4 s 4- -1Ỵ 2. 13 12 1 2. n 1 and this inequality shows that d T is not a spectral set for T. One may ask whether T provides an answer to the question of whether every operator whose spectrum is a c-spectral set is similar to an operator whose spectrum is a spectral set. it was pointed out to me that the weighted shift T is similar to a subnormal weighted shift s whose weight sequence is constructed as follows. Let I a 1 2 4 1 - r2 da r 1 2 and v - pin Y IP n for n e z. Then the weighted shift 5 with weight sequence n 6 z is a subnormal weighted shift see Shields 2 Proposition 27 . If we let He h e for n e z where jM-1 1 2 - if n 0 if n 0 if n 0 then it is straightforward to .

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