Subspace-hypercyclic conditional type operators on $L^p$-spaces

A conditional weighted composition operator $T_u:L^p(\mathrm{\Sigma })\to L^p(\mathcal{A})$ ($1\le p<\mathrm{\infty }$), is defined by $T_u(f):=E^{\mathcal{A}}(uf\circ \phi )$, where $\phi :X\to X$ 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 $\phi$ 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 $\phi$ is non-singular and finitely non-mixing. For the sufficient conditions, the normality of $\phi$ 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.