Subspace-hypercyclic conditional type operators on $L^p$-spaces
A conditional weighted composition operator $T_u: L^p(\Sigma)\rightarrow L^p(\mathcal{A})$ ($1\leq p<\infty$), is defined by $T_u(f):= E^{\mathcal{A}}(u f\circ \varphi)$, where $\varphi: X\rightarrow X$ is a measurable transformation, $u$ is a weight function on $X$ and $E^{\mathcal{A}}$ is the c...
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| Main Authors | , |
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| Format | Journal Article |
| Language | English |
| Published |
15.11.2022
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| Subjects | |
| Online Access | Get full text |
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| Summary: | A conditional weighted composition
operator $T_u: L^p(\Sigma)\rightarrow L^p(\mathcal{A})$ ($1\leq p<\infty$),
is defined by $T_u(f):= E^{\mathcal{A}}(u f\circ \varphi)$, where $\varphi:
X\rightarrow X$ is a measurable transformation, $u$ is a weight function on $X$
and $E^{\mathcal{A}}$ is the conditional expectation operator with respect to
$\mathcal{A}$. In this paper, we study the subspace-hypercyclicity of $T_u$
with respect to $L^p(\mathcal{A})$. First, we show that if $\varphi$ is a
periodic nonsingular transformation, then $T_u$ is not
$L^p(\mathcal{A})$-hypercyclic. The necessary conditions for the
subspace-hypercyclicity of $T_u$ are obtained when $\varphi$ is non-singular
and finitely non-mixing. For the sufficient conditions, the normality of
$\varphi$ is required. The subspace-weakly mixing and subspace-topologically
mixing concepts are also studied for $T_u$. Finally, we give an example which
is subspace-hypercyclic while is not hypercyclic. |
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| DOI: | 10.48550/arxiv.2211.07939 |