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In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi }$, when $\varphi$ is a linear-fractional self-map of $𝔻$. In this paper first, we investigate the essential normality problem for the operator $T_w{C}_{\varphi }$ on the Hardy space $H^2$, where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F\left(\varphi \right)$, $\varphi \in 𝒮\left(2\right)$, and $T_w$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal...

Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space $H^p$ or the Bergman space $L_a^p$ ($1<p<\infty$, $p\ne 2$) are not supercyclic....

In this paper, we discuss the hypercyclicity, supercyclicity and cyclicity of the adjoint of a weighted composition operator on a Hilbert space of analytic functions.