tailieunhanh - Báo cáo toán học: "Spectra of compressions of an operator with compact imaginary part "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Spectra nén của một nhà điều hành với một phần tưởng tượng nhỏ gọn. | J. OPERATOR THEORY 3 1980 161-158 Copyright by INCREST 1980 SPECTRA OF COMPRESSIONS OF AN OPERATOR WITH COMPACT IMAGINARY PART ANDRZEJ POKRZYWA Let stands for a complex separable Hilbert space with the norm II II and the scalar product 1. denotes the set of all bounded compact linear operators on For an xe f jf ơ A denotes the spectrum of A p yf -the resolvent set W A We A - the essential numerical range see . 2 . S0CS2 denotes thes Macaev Hilbert-Schmidt ideal of all compact operators de such that 00 00 ML s 2n- 1 00 M 2 2 ỵ-si 00 n 1 M 1 where J1 0 2 . are the eigenvalues of Ị AA arranged in decreasing order and repeated according to multiplicity. For an operator A with the single valued extension property see . 1 and X e pA x is the union of all open subsets G of the complex plane c such that there exists the analytic function x - G which satisfies the relation A Ấ x Ấ X and we set C pA x tfXX . For any set F CL c we set xe J ơA x z Ff 3CA F . A is called a decomposable operator if for any finite open covering G L of ơ A FA Gn are closed subspaces and for any xe w there are x e 3CA G such that X x2 . xm. It is proved in 10 that an operator A whose imaginary part Im A A A 2i belongs to the Macaev ideal is decomposable. For any ẰeC F Gc c we define d 2 F inf 2 p neF dist F G max sup d z G sup d p F . 152 ANDRZEJ POKRZYWA Lemma 1. If 2 is an n-dimensional space Be IPCyf Im5 Ka Áọ where Ka Z 8 then for any Ae p B such that d A ơ B d 11 5 - A -1 2 Z exp 4 l 64 E2im . Proof. By the theorem on the triangular matrix form there are orthogonal projections 0 Qo Qx . Q 1 such that BQj QjBQj. Setting Qj Qj-X Ej we define the operators S s EjBEj 7 1 A- 2i È - Ế E Ka E W 7 1 Z 1 N2 2i ỵ 2 _x f G - f E K2Et Ej. 7 1 1 1 These operators have the following properties s is normal and ơ S -- ơ B B s -r Na Nt the operators Na and N2 are nilpotents. Using Theorems . 4 . 5 . 5 from the books of Gohberg and Krein we obtain the inequalities The operator S .