عنوان مجله
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Complex Analysis and Operator Theory
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چکیده
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In this paper, we study the extended limit set $J_{\{T_n\}}(x)$ for
a sequence of bounded linear operators $\{T_n\}_{n=1}^{\infty}$ on a
separable Banach space $X$, to describe the dynamics of a sequence
of linear operators. First, it is shown that a mutually commuting
sequence $\{T_n\}_{n=1}^{\infty} \subseteq \mathcal{B}(X)$ is
hypercyclic if and only if the closed and connected set $H=\{x\in X:
J_{\{T_n\}}(x)=X\}$ is dense in $X$ or equivalently has nonempty
interior.
% Later, we show that a sequence
% $\{T_n\}_{n=1}^{\infty}$ with $T_{m+n}=T_mT_n$
% is hypercyclic if and only if there exists a cyclic vector
%$x\in X$ such that $J_{\{T_n\}}(x)=X$. Furthermore, for such a
%sequence we prove that it is $J$-class if and only if
%$J_{\{T_n\}}(0)=X$ or equivalently, $J_{\{T_n\}}(x)=X$ for some
% nonzero periodic vector $x$.
Later, we prove that a mutually commuting sequence
$\{T_n\}_{n=1}^{\infty}$ with dense ranges,
is hypercyclic if there exists a cyclic vector
$x\in X$ for some term of $\{T_n\}_{n=1}^{\infty}$ such that
$J_{\{T_n\}}(x)=X$. Furthermore, it is proved that an arbitrary
sequence $\{T_n\}_{n=1}^{\infty}$ is $J$-class if and only if
$J_{\{T_n\}}(0)=X$ or equivalently, $J_{\{T_n\}}(x)=X$ for some
nonzero mixed periodic vector $x$.
Finally, some examples are presented to
illustrate these results.
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