tailieunhanh - Báo cáo toán học: "Hyponormal operators are subscalar "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Hyponormal các nhà khai thác đang subscalar. | Copyright by INCREST 1984 J. OPERATOR THEORY 12 1984 385 -395 HYPONORMAL OPERATORS ARE SUBSCALAR MIHAI PUTINAR INTRODUCTION In this paper we construct a universal functional model for hyponormal operators. This shows in particular that every hyponormal operator is subscalar . is similar to the restriction to an invariant subspace of a generalized scalar operator in the sense of Colojoara-Foias 5 . Let H be a complex separable Hilbert space and let ẫ 7 denote the linear bounded operators on H. Recall that T e d H is called subnormal if there is a Hilbert space K containing H isometrically and a normal operator N on K such that Nh Th h 6 H in other words if is a closed invariant subspace for N and the restriction N H coincides with T. Interest in subnormal and related classes of operators has risen considerably since s. Brown 3 proved that every subnormal operator has a nontrivial invariant subspace. A larger class of operators related to subnormals is the following Te T H is called hyponormal if TT T T or equivalently if II THi 77z for every heH. There are classical examples of hyponormal non-subnormal operators see 7 Chapter 16 . As we shall prove below the distinction between hyponormal and subnormal operators lies only in two degrees of differentiability added to the admissible functional calculus of an extension. More precisely Theorem 1. Any hyponormal operator is subscalar of order 2. A linear bounded operator 5 on His called in 5 scalar of order m if it possesses a spectral distribution of order m . if there is a continuous unital morphism of topological algebras U CS C - H such that U z s where as usual z stands for the identical function on c and Cq C for the space of compactly supported functions on c continuously differentiable of order m 0 m oo. An operator is subscalar if it is the restriction of a scalar operator to an invariant subspace. It is necessary to point out the distinction between scalar and Dunford scalar operators which are .