Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Toàn cầu qua các phần của quỹ đạo đồng nhất và giống nhau của các nhà khai thác không gian Hilbert. | Copyright by INCREST 1984 J. OPERATOR THEORY 12 1984 265-283 GLOBAL CROSS SECTIONS OF UNITARY AND SIMILARITY ORBITS OF HILBERT SPACE OPERATORS MARTA PECUCH HERRERO INTRODUCTION In 2 D. Deckard and L. A. Fialkow characterized the operators on a separable Hilbert space which have local unitary cross sections. These are the operators of the form A B 1 where A and B are operators on finite dimensional spaces and 1 is the identity operator on a complex separable infinite dimensional Hilbert space yf. The purpose of this article is to characterize the operators which have global unitary cross sections or global similarity cross sections. To make these statements more precise it is necessary to introduce some standard notation which will be used throughout the paper denotes the algebra of all bounded linear ope- rators on iff denotes the group of unitary operators in WIT U TƯ ue is the unitary orbit of an operator T in and T z y - 4 7 is the norm continuous function defined by r C7 U - TU. A local unitary cross section for T is a pair p á such that Ổ is a relatively open subset of T which contains T and p Ổ - .J is a norm continuous function such that T o p 1 a and p T 1 p is a global unitary cross section when Ổ T . In Section 3 it is shown that T has a global unitary cross section if and only if T has the form B 1 where B is an operator on a finite dimensional space. Let T denote the canonical image of an operator T in the Calkin algebra. If T A B 1 A B acting on finite dimensional spaces then T admits a global unitary cross section only in the trivial case when f is a multiple of the identity. As a corollary it is shown that if T B 0 1 has a global unitary cross section p but T is not a multiple of the identity then it is impossible to construct p so that cplTj p T is a compact operator whenever Tỵ T2 e WIT and Tỵ T2 is compact. 266 MARTA PECUCII HERRERO The second part of the paper is devoted to the study of similarity cross sections. Let .5 denote the group of .
