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Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một định lý tổng quát Weyl và $ L ^ p $-quang phổ của các nhà khai thác Schroedinger. | Copyright by INCREST 1985 J. OPERATOR THEORY 13 1985 119-129 A GENERALIZED WEYL THEOREM AND Z -SPECTRA OF SCHRODINGER OPERATORS I. M. SIGAL I. INTRODUCTION Weyl-type theorems occupy the unique place among tools of finding spectra of pseudo differential operators. The classical theorem of Weyl states see 8 vol. I that if A and B are self-adjoint and A B is compact then essspec X essspec B the definitions are given below . Known generalizations of this theorem see 8 vol. I IV 5 replace the compactness requirement on A B by the condition that A z -1 B z -1 is compact for z e p A fi p B and relax to various degrees the self-adjointness restriction on A. A generalization to closed operators which uses a different definition of the essential spectrum W zl defined below it coincides with our definition for self-adjoint operators was given in 5 p. 244 Theorem 5.35 The results below Theorems 1 and 5 go beyond these. We use also one of our results to solve a special case of one of the problems posed by B. Simon 12 13 14 we prove that the spectra of Schrodinger operators on the -spaces are independent of p. All operators below are densely defined. For a closed operator A on a Banach space we adopt the following definitions see 8 vol. I IV specx X the spectrum of A on X disc specx X the discrete spectrum of A on X the set of all isolated eigenvalues of finite algebraic multiplicities essspecx X the essential spectrum of A on X specx l discspecx 4 px X DX A the resolvent set and domain of A respectively on X. When the underlying space X is obvious from the context we omit the subindex X. W A the Weyl spectrum of A Ả e c I A A u - 0 for some sequence u e D y4 II u II 0 un 0 A X the dual operator and space respectively. It is a pleasure to thank w. Hunziker and Y. Kannai for discussions and J. Voigt for a very useful correspondence. 120 I. M. SIGA.L 2. GENERALIZED WEYL THEOREMS Theorem 1. Let L be a closed operator on a reflexive Banach space X and let G be an open complex set.
