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) |