TAILIEUCHUNG - Báo cáo toán học: "On domains of powers of closed symmetric operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên lĩnh vực quyền hạn của nhà khai thác đóng đối xứng. | 1. OPERATOR THEORY 9 1983 53-75 Copyright by INCREST 1983 ON DOMAINS OF POWERS OF CLOSED SYMMETRIC OPERATORS KONRAD SCHMŨDGEN INTRODUCTION Let T be a densely defined closed symmetric operator in a Hilbert space 3 c. As M. A. Naimark has observed the domain ÍT2 of its square need not to be dense in iff. Naimark 5 even proved using a r esult of J. von Neumann the existence of a closed symmetric operator T with .@ T2 0 . In this paper we investigate some properties of the domains .@ T of the powers of T. To be more precise we are mainly but not only concerned with the following two questions When 2 Tn is dense ìn When 3 Tn r is a core for Trl Except from Naimark s Doklady notes 5 published in 1940 this problems appearently have not yet been further studied as far as the author is aware . Let us briefly describe the content of this paper and some of the results. In 1 we investigate some general properties of powers of a closed symmetric operator T. Among other things we prove that if at least one of the deficiency indices of T is finite then .@00 7 Q Tn is a core for each operator Tk ke N and neN hence S afT is dense in . Consequently in our further study of the problems mentioned above we may restrict ourselves to the case where both deficiency indices are infinite. If the underlying Hilbert spaced is separable then Thas a self-adjoint extension in ff. This enables US at least in principle to investigate both problems for restrictions T of a given unbounded self-adjoint operator A in 3ff. Throughout the remaining parts of the paper we will adopt this view point. In 2 we give general answers to the questions from above. The criterions depend on the self-adjoint operator A and the size of the deficiency space of T- In 3 we prove some technical results which are needed in 4. 4 contains the main result Theorem concerning the two questions. Let A be an arbitrary unbounded self-adjoint operator in and 5R a set of positive integers. Then there exists a closed symmetric .

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