We characterize bounded and compact generalized composition operators between BLOCH type SPACES and WEIGHTED Dirichlet type SPACES. Then, we show that these results can be employed to characterize bounded and compact Volterra type operators between these SPACES. Finally, we give a connection between WEIGHTED and generalized composition operators to conclude some results about boundedness and compactness of them.

For an analytic self-map φ of the unit disk D in the complex plane C, a nonnegative integer n, and u analytic function on D, WEIGHTED di erentiation composition operator is de ned by (D n φ ; u f)(z) = (n) u(z)f (φ (z)), where f is an analytic function on D and z 2 D. In this paper, we study the boundedness and compactness of D n φ ; u, from WEIGHTED Bergman SPACES with admissible weights to BLOCH-type SPACES.

Let $\mathbb{D}= \{\upsilon\in\mathbb{C}:|\upsilon|<1\}$ be the open unit disk in the complex plane $\mathbb{C}$ and let $H(\mathbb{D})$ be the space of all holomorphic functions on  $\mathbb{D}$. For a non-negative integer $n$ and a function $f \in H(\mathbb{D})$, the $n^{th}-$ order differentiation operator is defined as $D^n f = f^{(n)}$. The WEIGHTED composition operator together with $n^{th}-$ order differentiation operator give rise to a new operator generally termed as generalized WEIGHTED composition operator denoted by $\mathcal{W}^{n}_{\phi,\xi}$ and is  defined by\begin{equation*}\mathcal{W}^{n}_{\phi,\xi}f(\upsilon)  =\phi(\upsilon)f^{(n)}(\xi(\upsilon)),\quad f\in H(\mathbb{D}); \upsilon\in%\mathbb{D},\end{equation*}where $\phi\in H(\mathbb{D})$ and $\xi$ is a holomorphic self-map of $\mathbb{D}$. This operator is basically the combination of multiplication operator $M_{\phi}$, composition operator  $C_{\xi}$ and $n^{th}-$ order differentiation operator $D^{n}$. We study the boundedness and compactness of this operator between Dirichlet-type SPACES and BLOCH-type SPACES.

We obtain lower and upper estimates for the essential norms of generalized composition operators from WEIGHTED Dirichlet SPACES or BLOCH type SPACES to QK type SPACES.

Norm and essential norm of differences of differentiation composition operators between BLOCH SPACES have been estimated in this paper. As a result, we find characterizations for boundedness and compactness of these operators.

Let X and Y be complete metric SPACES of analytic functions over the unit disk in the complex plane. A linear operator T:X®Y is a bounded operator with respect to metric balls if T takes every metric ball in X into a metric ball in Y. We also say that T is metrically compact if it takes every metric ball in X into a relatively compact subset in Y. In this paper we will consider these properties for composition operators from Nevanlinna type SPACES to BLOCH type SPACES.

Let j be a holomorphic self-map and g a fixed holomorphic function on the unit ball B. The boundedness and compactness of the Volterra composition operatorTg, j ¦ (z)=ò10 ¦ (j (tz) Rg(tz) dt/ton the logarithmic BLOCH space and little logarithmic BLOCH space are studied in this paper.