2024 : 11 : 23
Mohammad Reza Azimi

Mohammad Reza Azimi

Academic rank: Associate Professor
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Education: PhD.
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Research

Title
subspace-hypercyclic conditional weighted composition operators on‎ ‎$L^p$-spaces‎
Type
JournalPaper
Keywords
Subspace-hypercyclic‎, ‎Orbit‎, ‎Subspace-weakly mixing‎, ‎Subspace-topologically mixing‎, ‎Measurable transformation‎, ‎Normal‎, ‎Radon-Nikodym derivative‎, ‎Conditional expectation‎, ‎aperiodic.
Year
2024
Journal MATHEMATICAL INEQUALITIES & APPLICATIONS
DOI
Researchers Mohammad Reza Azimi ،

Abstract

‎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.