TAILIEUCHUNG - Báo cáo toán học: "Limits of spectra of strongly converging compressions "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Giới hạn của quang phổ nén mạnh mẽ hội tụ. | J OPERATOR THEORY 12 1984 199-212 Copyright by INCREST 1984 LIMITS OF SPECTRA OF STRONGLY CONVERGING COMPRESSIONS ANDRZEJ POKRZYWA 0. PRELIMINARIES Let H denote a complex separable Hilbert space with scalar product and norm II . H H stands for the set of bounded compact linear operators acting in H. For A e ơ A denotes the spectrum of A. By a projection we shall mean an operator p p p2 e Ỉ H . SPfiH denotes the set of finite-dimensional projections. For aprojectionPandan operator 7Í Ap PA lPH the restriction of PA to the range of p is called the compression of A to PH. For two compact nonempty subsets M N of the plane c of complex numbers the Hausdorff distance is defined as follows dist Af N inf e 0 such that M c N fiB N c M eB where B denotes the closed unit disc in c. We are interested in answering the following Question. Given a sequence p of projections in H converging strongly to the identity operator in H P Itl what is the asymptotical behaviour of a Ap for A e This problem is equivalent to the characterization of the family of sets S A defined below. Definition. For A e S y4 denotes the family of all compact nonvoid subsets of c such that for every ữ e S A there is a sequence p of projections in H such that p In and dist Í2 a Ap 0. We shall show that S 4 can be characterized with the use of ơ A and WC A the essential numerical range of A . We A Pl W A 1_P where W A x x e H x 1 is the numerical range of A. Note that We A 0 if and only if A e H cf. 7 . It follows from 6 Theorem 2 that each Ấ e ơ A We A is an isolated point of a A and 2 is an eigenvalue of a finite-multiplicity . the spectral projection E . A is finite-dimensional. 200 ANDRZEJ POKRZYWA It was shown in 3 Theorem 1 that for each A E Q e S A we have 1 ơ A We A íĩ c ơ A u We A and ÍĨ n WfA 0- The set We A is always convex and closed. It was shown in 4 that if IKe zl contains an interior point then a closed set Í2 belongs to S A if and only if 1 holds. If WC A has an empty interior it is

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