TAILIEUCHUNG - Báo cáo toán học: "Quasisimilarity and closures of similarity orbits of operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: | Ỉ. OPERATOR THEORY Í4Í1985Ỉ 215-238 Copjrighi by ÍNCREST 1985 QUASISIMILARITY AND CLOSURES OF SIMILARITY ORBITS OF OPERATORS LAWRENCE A. FIALKOW . INTRODUCTION In the theory of Hilbert space operators there are several relations which play roles analogous to that played by similarity in the theory of operators on finite dimensional spaces. In the finite dimensional case the Jordan form models each operator up to similarity and this model preserves spectral and invariant subspace structures. In the infinite dimensional case such relations as quasisimilarity and asymptotic similarity have been successfully used to model special classes of operators but the general properties of these relations have not been fully described. In the present note we examine connections between quasisimilarity and asymptotic similarity as a step towards a better understanding of these relations. Let denote a separable complex infinite dimensional Hilbert space and let denote the algebra of all bounded linear operators on Xf. Operators T and S in J y are similar if there exists an invertible operator X e such that TX XS . s X-1TX. For T in let y T denote the similarity orbit of T . D s e i . s is similar to T and let y T denote the norm closure of in . Operators T and s are asymptotically similar if T S - equivalently T e 9 S and s e X T r . In 6 c. Apostol D. Herrero and D. Voiculescu described the closures of the similarity orbits of operators and they subsequently provided canonical models relative to asymptotic similarity see 5 Corollary . Operators T and s are quasisimilar T S if there exist operators X and y each injective and with dense range such that TX xs and YT SY for T e let T qs s e T s the quasisimilarity orbit of T. Quasisimilarity was introduced by and c. Foias in 35 where it was shown to be an effective tool in the model theory of contractions for example quasisimilarity models were given for cu contractions 35 Proposition and C0 N contractions 35

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