TAILIEUCHUNG - Báo cáo toán học: "Analytic operator functions with compact spectrum. III. Hilbert space case: Inverse problem and applications "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Nhà điều hành với chức năng phân tích quang phổ nhỏ gọn. III. Trường hợp không gian Hilbert: ngược vấn đề và các ứng dụng. | J. OPERATOR THEORY 10 1983 219-250 Copyright by INCREST 1983 ANALYTIC OPERATOR FUNCTIONS WITH COMPACT SPECTRUM. III. HILBERT SPACE CASE INVERSE PROBLEM AND APPLICATIONS M. A. KAASHOEK c. V. M. van der MEE L. RODMAN 0. INTRODUCTION In the earlier papers 29 30 spectral linearizations and spectral pairs have been introduced to deal with classification and factorization problems for analytic operator functions with a compact spectrum. To explain the main results of the present paper we first recall the basic definitions of 29 30 . Let ữ be an open set in c and let W Q - be an analytic operator function whose values are bounded linear operators on the complex Hilbert space H. A bounded linear operator A G - G acting on a Hilbert space G is called a spectral linearization of w on Q if the spectrum a A of A is contained in Í2 and the operator functions W Ằ ZG and Ấ A IH are analytically equivalent cf. 10 11 on Í2 which means that the following identities hold E X F Ẳ e Í2 0 L 0 Ah where E 2. G H - H G and F a H G - G H are some invertible operators depending analytically on Ẵ ế Q. Here the symbol Iz stands for the identity operator on the space z. If a spectral linearization exists then the set Z IF Ấ ế ữ I ĨV Ằ is not invertible is compact in Í2 in fact as shows Z fF a A . The set Z W we call the spectrum of w. Conversely if E W is compact in ÍỈ in other words if W Ằ is invertible for Ả near the boundary of 12 then a spectral linearization of w exists and is unique up to similarity see 29 where this is proved in a Banach space setting . 220 M. A. KAASHOEK c. V. M. van der MEE 1 . RODMAN So the class of operator functions considered here as well as in 29 30 consists of analytic operator functions with a compact spectrum. A pair of Hilbert space operators C A is called a right spectral pair of w on ĨĨ if A G - G is a spectral linearization of w on Í2 and assuming holds the operator C G - H is given by c --1 Lf 2 -1t 2 - A _1d2. 2ni J Here 1Ĩ H G - H is the

• Báo cáo khoa học
• báo cáo của tạp chí Journal of Operator Theory
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học
• tài liệu báo cáo nghiện cứu khoa học