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Báo cáo toán học: "On the spectral semi-distance and quasi-nilpotent equivalence for systems of operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên bán khoảng cách và bán lũy linh quang phổ tương đương của hệ thống cho các nhà khai thác. | J. OPERATOR THEORY 10 1983 57-60 Copyright by INCREST 198 ON THE SPECTRAL SEMI-DISTANCE AND QUASI-NILPOTENT EQUIVALENCE FOR SYSTEMS OF OPERATORS TEFAN FRUNZĂ The main result of this paper Theorem 2 is that the joint spectral semi-distance is uniformly equivalent to the cartesian spectral semi-distance on any spectrally bounded set of commuting nt-tuples of operators. As a corollary we get that the joint quasi-nilpotent equivalence coincides with the cartesian or separate quasi-nilpotent equivalence. We also discuss other possible spectral semi-distances and quasi-nilpotent equivalences for systems of operators which may be also reduced to cartesian notions. 1. Let X be a complex Banach space and T s 6 L X be two continuous linear operators on X. For any positive integer n e N denote T S fỉ ị ÍẠ Y kTn kSk. k 0 k J Then T is said to be quasi-nilpotent equivalent to s denoted T s if lim T 5 fn 1 n 0 lim S T W 1 see 3 . The spectral semi-distance between T and s is defined by d T S max dSp T S dSp S T where dSp T S lim supII T S M p n see 6 and 2 . It is obvious that T s if and only if d T S 0. In 5 we generalized the notion of quasi-nilpotent equivalence to commuting systems of operators. For technical reasons we choose the following definition. Denote by L X the set of all commuting m-tuples in L X m. For any system T 7 . Tm e L X and any multi-index p pn . . p of positive inte- 58 ST1FAN FRLNZA gers denote r Tfi.7 and ipi-A-i- . 4-pra. For any two systems T Tj . Tm and 5 Si . Sm in L X Ĩ and any multiindex n . nm denote T 5 W . y i M . . M k Tn-kSk ki kmJ where k kx . fcm and n k 1 kỵ . nm km . Tis said to be quasi--nilpotent equivalent to s if lim max i T S Wi V 0 - lim maxỊ S T W Ị Vfc. Ịn -k k J Zc With that notion at hand we were able to prove the invariance of Taylor spectrum and other joint spectra with respect to quasi-nilpotent equivalence 5 Section 4 . The spectral semi-distance between the systems T and s may be defined by d T S max dsp T S dsp S T where d