Let D be the open unit disc in the complex plane C. A sandwich WEIGHTED COMPOSITION OPERATOR S,' takes an analytic map f on the open unit disc D to the map (: f 0o')0, where ' is an analytic map of D into itself and is an analytic map on D. In this paper, we compute the adjoint of a sandwich WEIGHTED COMPOSITION OPERATOR S,' on WEIGHTED Hardy spaces.

WEIGHTED COMPOSITION OPERATORs appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of WEIGHTED COMPOSITION OPERATORs, are GENERALIZED WEIGHTED COMPOSITION OPERATORs which in special cases of their inducing functions give different types of well-known OPERATORs like: WEIGHTED COMPOSITION OPERATORs, COMPOSITION OPERATORs, multiplication OPERATORs and COMPOSITION OPERATORs followed by differentiation OPERATORs. In this paper we study GENERALIZED WEIGHTED COMPOSITION OPERATORs and give estimates for the essential norm of such OPERATORs on certain Banach spaces of analytic functions into WEIGHTED type spaces. The underlying Banach spaces of analytic functions include Bloch spaces, Zygmund spaces and WEIGHTED type spaces. Our estimates for the essential norms of GENERALIZED WEIGHTED COMPOSITION OPERATORs imply necessary and sufficient conditions for the compactness of such OPERATORs. As another application of our results, we obtain essential norm estimates of certain well-known OPERATORs which are special cases of GENERALIZED WEIGHTED COMPOSITION OPERATORs.

Let D be the open unit disk in the complex plane C and H(D) be the set of all analytic functions on D. Let u,v 2 H(D) and ' be an analytic self-map of D. A class of OPERATOR related WEIGHTED COMPOSITION OPERATORs is de, ned as follow Tu, v, 'f(z) = u(z)f('(z)) + v(z)f0('(z)),f 2 H(D),z 2 D: In this work, we obtain some new characterizations for boundedness and essential norm of OPERATOR Tu, v, ' between Zygmund space.

Investigating the mean ergodicity of COMPOSITION OPERATORs on various Banach Spaces has always been of interest to mathematicians and many authors studied this topics intensively, in many different spaces, such as, the space of all holomorphic functions on unit disk, Hardy space and Bloch space. In this paper, for a self map of the unit disk, φ,and λ, ∈, ℂ, , we consider WEIGHTED COMPOSITION OPERATOR, (λ, 𝐶, φ, )𝑓, =λ, 𝑓, 𝑜, φ, , for every 𝑓,in Bloch space and Little Bloch space and inquiry the conditions under which the WEIGHTED COMPOSITION OPERATOR 𝜆, 𝐶, 𝜑, , is mean ergodic or uniformly mean ergodic on the Bloch and Little Bloch Space. In fact, we will show, if |λ, |>1, 𝜆, 𝐶, 𝜑, , cannot be power bounded, mean ergodic or uniformly mean ergodic, in contrast, if |λ, |<1, 𝜆, 𝐶, 𝜑, , is always power bounded, mean ergodic or uniformly mean ergodic. In the case, |λ, |=1, we will see that it depends directly to the Denjoy-Wolff point of 𝜑, .

‎In this paper‎, ‎we first study the boundedness of GENERALIZED‎ WEIGHTED COMPOSITION OPERATORs from minimal Möbius invariant‎ ‎subspace to the $n$th WEIGHTED type space‎. ‎Then‎, ‎we obtain‎ ‎equivalence conditions for its boundedness by using the Bell‎ ‎polynomials‎. ‎We also find some estimates for the essential‎ ‎norm of this OPERATOR and use them to present equivalence‎ ‎conditions for the compactness of such an OPERATOR‎.

‎‎‎‎‎Let $H(\mathbb{D})$ be the space of all analytic functions on $\mathbb{D}$‎, ‎$u,v\in H(\mathbb{D})$ and $\varphi,\psi$ be self-map $(\varphi,\psi:\mathbb{D}\rightarrow \mathbb{D})$‎. Difference of WEIGHTED COMPOSITION OPERATOR is denoted by $uC_\varphi‎ -‎vC_\psi$ and defined as follows‎ ‎\begin{align*}‎ (uC_\varphi‎ -‎vC_\psi)f(z) = u(z) f{(\varphi(z))}‎- ‎v(z) f(\psi(z))‎ ,‎\quad f\in H(\mathbb{D} )‎, ‎\quad z\in \mathbb{D}‎. ‎\end{align*}‎ ‎In this paper‎, ‎boundedness of difference of WEIGHTED COMPOSITION OPERATOR from Cauchy transform into Dirichlet space will be considered and ‎an ‎equivalence condition for boundedness of such OPERATOR will be given‎.‎‎ Then the norm of COMPOSITION OPERATOR between the mentioned spaces will be studied and it will be shown ‎that‎‎ $\|C_\varphi\|\geq 1$ ‎and‎ there is no COMPOSITION isometry from Cauchy transform into Dirichlet ‎space‎.‎

Let $H(\mathbb{D})$ be the space of all analytic functions on the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. In this paper, we investigate the boundedness and compactness of the GENERALIZED integration OPERATOR$$I_{g,\varphi}^{(n)}(f)(z)=\int_0^z f^{(n)}(\varphi(\xi))g(\xi)\ d\xi,\quad z\in\mathbb{D},$$ from Zygmund space into WEIGHTED Dirichlet type space, where $\varphi$ is an analytic self-map of $\mathbb{D}$, $n\in\mathbb{N}$ and $g\in H(\mathbb{D})$. Also we give an estimate for the essential norm of the above OPERATOR.