TAILIEUCHUNG - Báo cáo toán học: "On restrictions of unbounded symmetric operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về các hạn chế của các nhà khai thác không bị chặn đối xứng. | J. OPERATOR THEORY 11 1984 379 -393 Copyright by INCREST 1984 ON RESTRICTIONS OF UNBOUNDED SYMMETRIC OPERATORS KONRAD SCHMUDGEN 1. INTRODUCTION AND MAIN RESULTS . Let A1 and A2 be densely defined unbounded symmetric operators in separable Hilbert spaces resp. 2- Suppose that AsS As c 3 AS for s e 1 2 . We write A1 A2 if there exist an isometry u of onto 4 2 and a dense linear subspace c S Ax of such that AỵSi10 c .@10 á 20 U2W i2 and Aỵ p U A2U p for all pe u. This clearly implies that . 20 is dense in f2 20 - 20 and hence A2 Aỵ. In other words the relation Ai A2 means that A and A2 have restrictions to dense invariant domains which are unitarily equivalent. In this paper we show that if @ AS Pl @ A for s e 4 2 then Tj A2 if and only if Aỵ and A2 are both strongly unbounded from above or strongly unbounded from below for a definition of these notions see . In the case where As is essentially self-adjoint and âĩ As Pl S As for se l 2 this n l result means that Aỵ A2 if and only if Aỵ and A2 are both unbounded from above or unbounded from below. This answers a question raised by the author in 1 p. 335 Problem 10 in the affirmative. For diagonal operators the latter was already proved in 3 Theorem by a different method. Note that an error in the proof or Corollary in 3 has been corrected in 4 Section 7. Of course the result mentioned above shows the pathology of unbounded operators in the spirit of 2 . In case of diagonal operators we used it in 4 to construct non-integrable representations of the Heisenberg commutation relation PQ QP iz by essentially self-adjoint operators p Q. The theorems and the corollaries stated below have similar applications but they seem to be of some interest in its own right as well. In a forthcoming paper they will be the main tool for the construction of -representations of unbounded operator algebras. 380 KONRAD SCHMỮDGEN . Before stating our results we introduce some more notation. We let No 0 1 2 . and N 1 2 . . .

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