عنوان مجله
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Advances in Operator Theory
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کلیدواژهها
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Hypercyclic, Subspace-supercyclic, Orbit, Conditional
expectation, Weighted translation, Locally compact group
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چکیده
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A bounded linear operator $T$ on a Banach space $X$ is called
subspace-supercyclic for a closed and non-zero subspace $M$ of $X$
if there exists a vector $x\in X$ whose projective orbit
$\mathbb{C}{orb}(T, x)\cap M= \{\beta T^n(x):~~\beta \in
\mathbb{C},n=0, 1, 2, \ldots\}\cap M$ is dense in $M$. Let $1\le
p<\infty$ and $\mathcal{B}$ is a Borel $\sigma$-algebra. For a
$\sigma$-subalgebra $\mathcal{A}$ of $\mathcal{B}$, the conditional
expectation with $\mathcal{A}$ is the mapping $f\to
E^{\mathcal{A}}f$. Also, for a second countable locally compact
group $G$ and $g\in G$, $\delta_g$ is the unit point mass measure at
$g$. A bounded Borel measurable function $v:G\to (0, \infty)$ is
called {\em weight}. In this paper, we characterize the
subspace-supercyclicity of the conditional weighted translation
$R_{g, v}:L^p(\mathcal{B})\to L^p(\mathcal{A})$
defined by
$R_{g, v}(f):=E^{\mathcal{A}}(v f*\delta_g)$.
% where the convolution
%$$(f*\delta_g)(x)=\int_G f(xy^{-1})d\delta_g(y)=f(xg^{-1})\qquad x\in G.$$
First, we prove that if $g\in G$ is a torsion element,
then $R_{g, v}$ cannot be subspace-supercyclic with respect to
$L^p(\mathcal{A})$. Besides,
%in the case that $g\in G$ is an
%aperiodic element,
we reveals that
$R_{g, v}$ is subspace-supercyclic if $g$ is an
aperiodic element.
For $\alpha\in \mathbb{C}\setminus \{0\}$, the
%The concepts of
subspace-hypercyclicity and subspace-mixing are also studied for
$\alpha R_{g, v}.$
%with $\alpha\in \mathbb{C}\setminus \{0\}$.
Also, we specify the Ces$\grave{a}$ro subspace-hypercyclicity for $R_{g,
v}$ and offer conditions for the adjoint of $R_{g, v}$
to be subspace-supercyclic with respect to $L^2(\mathcal{A}g^{-1})$.
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