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.

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.

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.

In this paper we characterize Fredholm and invertible composition operators on harmonic BLOCH function SPACES. Indeed, we provide some necessary and sufficient conditions for Fredholmness of composition operators. Also, we investigate the relation between the Fredholm and invertible composition operators on harmonic BLOCH function 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.

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.

We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on BLOCH-TYPE SPACES and calculate its essential norm in terms of the n-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.

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.