tailieunhanh - Báo cáo toán học: "On the form sum and the Friedrichs extention of Schroedinger operators with singular potentials "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên tổng hợp hình thức và mở rộng các nhà khai thác Friedrichs của Schrödinger có tiềm năng ít. | J. OPERATOR THEORY 6 1981 87 101 Copyright by INCREST 1981 A TOEPLITZ-HAUSDORFF THEOREM FOR MATRIX RANGES FRANCIS J. NARCOWICH and JOSEPH D. WARD 1. INTRODUCTION Given a bounded linear operator T defined on a complex Hilbert space the spatial numerical range D of T is the set 7 7r x x6 l x 1 . The Toeplitz-Hausdorff Theorem states that z 1 7 is convex 9 14 the closure of 1 7 is the numerical range FR1 7 p 7 ẹ eS1 where Slf the state space is the set of norm-one positive linear functionals defined on the set of all bounded linear operators on To simplify matters restrict to be finite dimensional note that 1 1 7 is closed in this case so Wi 7 Wi T . The Toeplitz-Hausdorff Theorem may be viewed as a consequence of the equality between iK T and Wị T . This equality comes about by virtue of the fact that not all functionals in s are needed to sweep out WjlT only those of the form - x x where xetf and x 1 all of which are extreme points in Si are needed. Thus to prove the Toeplitz-Hausdorff Theorem it suffices to show that the numerical range is swept out by the extreme points of Si- Such an approach will yield both a geometric proof of the Toeplitz-Hausdorff Theorem thus answering a question of p. R. Halmos 7 8 p. 110 and a generalization of the theorem to matrix ranges. Using completely positive maps Arveson 2 p. 300 generalized the concept of numerical range in defining matrix ranges. Recall that if si and Să are c -algebras and Mm is the set of complex m X m matrices with identity Im a linear map p sđ - á is said to be completely positive see 1 if the associated maps p Mm m 1 J Usage differs. Some authors use numerical range and algebra numerical range for what are here termed spatial numerical range and numerical range respectively. 88 FRANCIS J. NARCOWICH and JOSEPH D. WARD are all positive. The set of all such maps is denoted by CP daf Ố . If ỉđ has identity I then the subset of CP j ăẫ consisting of all p such that p I K K fixed and positive in ẩS is .