In this article, we introduce quasi-MIXING OPERATORS and construct various examples. We prove that quasi-MIXING OPERATORS exist on all finite-dimensional and infinite-dimensional  Banach spaces. We also prove that an invertible operator $T$ is quasi-MIXING if and only if $T^{-1}$ is quasi-MIXING. We state some sufficient conditions  under which an operator is quasi-MIXING. Moreover, we prove that the direct sum of two OPERATORS is quasi-MIXING if and only if any of them is quasi-MIXING.

A continuous operator $T$ between two Banach lattices $E$ and     $F$ is called almost order-WEAKLY compact, whenever for each almost order bounded subset $A$ of $E$,  $T(A)$ is a relatively WEAKLY compact subset of $F$. We show that the positive operator $T$ from  $E$ into a  Dedekind complete Banach lattice $F$  is almost order-WEAKLY compact iff  $T(x_n) \xrightarrow{\|.\|}0$ in $F$ for each disjoint almost order bounded sequence $\{x_n\}$ in $E$. In this manuscript, we study some properties of this class of OPERATORS and its relationships with the others known classes of OPERATORS.

Let $E$ be a sublattice of a vector lattice $F$. A continuous operator $T$ from $E$ into a normed vector space $X$ is said to be $\tilde{o}$rder-norm continuous if $x_\alpha\xrightarrow{Fo}0$ implies $T(x_\alpha)\xrightarrow{\Vert.\Vert}0$ for every $(x_{\alpha})_{\alpha \in A}\subseteq E$. This paper aims to investigate the properties of this new class of OPERATORS and explore their relationships with existing classifications of OPERATORS. We introduce a new class of OPERATORS called $\tilde{o}$rder WEAKLY compact OPERATORS. A continuous operator $T: E \rightarrow X $ is considered $\tilde{o}$rder WEAKLY compact if $ T(A) $ in $X$ is a relatively WEAKLY compact set for every $Fo$-bounded $A\subseteq E$. In this manuscript, we examine various properties of this class of OPERATORS and explore their connections with $\tilde{o}$rder-norm continuous OPERATORS.

An operator T on a Banach space X is said to be WEAKLY hypercyclic if there exists a vector x Î X whose orbit under T is WEAKLY dense in X. We show that every WEAKLY hypercyclic composition operator on classic Hardy space H2 is norm hypercyclic.

In this paper, WEAKLY supercyclicity and WEAKLY hypercyclicity of composition OPERATORS on some Hilbert spaces of analytic functions, especially on some weighted Hardy spaces are investigated.