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.

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, .

‎‎‎‎‎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.‎

We shall provide a brief account on new achievements in the study of COMPOSITION and COMPOSITION-differentiation OPERATORs acting on classical spaces of analytic functions on the unit disk, as well as on the polydisk and the unit ball. The emphasis is given to the Hardy space, the Bergman space, and the Dirichlet space on the unit disk in complex plane. In the last section, we shall provide some new results regarding the Hilbert-Schmidt OPERATORs in the setting of several complex variables.

The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an OPERATORvalued inner product, we discuss OPERATOR versions for the above inequalities and give simple conditions that the equalities hold.