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Cím:Operators with regular norm-sequences.
Szerző:K\'erchy, L.
Forrás:Acta Sci. Math. 63, No.3-4, 571-605 (1997).
Nyelv:English
Absztrakt:Applying appropriate normalizing gauge functions and using the concept of almost convergence, the author extends several results connected with the asymptotic behaviour of power bounded operators to a much larger class of operators acting on Banach spaces. \par Section 2 is devoted to the study of properties of Banach limits, almost convergence and almost convergence in the strong sense. In Section 3, operators with regular norm-sequences are defined and some sufficient conditions are given for an operator to belong to this class. Those conditions show that the operators in question form a really large class including power bounded operators. A classification of operators with regular norm-sequences based on the asymptotic behaviour of vector orbits is given in Section 4. Isometries canonically associated to the operators in question are constructed in Section 5. The construction is different in the Banach space and Hilbert space settings, and in the separable and non-separable cases as well. An extension of a theorem of Sz.-Nagy on the similarity of a power bounded invertible operator with power bounded inverse to a unitary operator is presented. In Section 6, hyperinvariant subspace theorems are given which extend a well-known theorem of Sz.-Nagy and Foias, and a result of Atzmon, respectively. As a consequence, a generalization of the Arendt-Batty stability theorem is obtained. An extension of a stability theorem for supercyclic operators due to Ansari and Bourdon is presented in Section 7. Finally, in Section 8, the well-known Katznelson-Tzafriri theorem is generalized for operators with regular norm-sequences.} \RV{Lajos Moln\'ar (Debrecen) 
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