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J-class weighted shifts on the space of bounded sequences of complex numbers
We provide a characterization of -class and -class unilateral
weighted shifts on in terms of their weight sequences.
In contrast to the previously mentioned result we show that a bilateral
weighted shift on cannot be a -class operator.Comment: We correct some of the statements and the proof
Expansivity and Shadowing in Linear Dynamics
In the early 1970's Eisenberg and Hedlund investigated relationships between
expansivity and spectrum of operators on Banach spaces. In this paper we
establish relationships between notions of expansivity and hypercyclicity,
supercyclicity, Li-Yorke chaos and shadowing. In the case that the Banach space
is or (), we give complete characterizations
of weighted shifts which satisfy various notions of expansivity. We also
establish new relationships between notions of expansivity and spectrum.
Moreover, we study various notions of shadowing for operators on Banach spaces.
In particular, we solve a basic problem in linear dynamics by proving the
existence of nonhyperbolic invertible operators with the shadowing property.
This also contrasts with the expected results for nonlinear dynamics on compact
manifolds, illuminating the richness of dynamics of infinite dimensional linear
operators
Szemeredi's theorem, frequent hypercyclicity and multiple recurrence
Let T be a bounded linear operator acting on a complex Banach space X and
(\lambda_n) a sequence of complex numbers. Our main result is that if
|\lambda_n|/|\lambda_{n+1}| \to 1 and the sequence (\lambda_n T^n) is
frequently universal then T is topologically multiply recurrent. To achieve
such a result one has to carefully apply Szemer\'edi's theorem in arithmetic
progressions. We show that the previous assumption on the sequence (\lambda_n)
is optimal among sequences such that |\lambda_n|/|\lambda_{n+1}| converges in
[0,+\infty]. In the case of bilateral weighted shifts and adjoints of
multiplication operators we provide characterizations of topological multiple
recurrence in terms of the weight sequence and the symbol of the multiplication
operator respectively.Comment: 18 pages; to appear in Math. Scand., this second version of the paper
is significantly revised to deal with the more general case of a sequence of
operators (\lambda_n T^n). The hypothesis of the theorem has been weakened.
The numbering has changed, the main theorem now being Th. 3.8 (in place of
Proposition 3.3). The changes incorporate the suggestions and corrections of
the anonymous refere
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