In this paper, we introduce subspace-frequently Hypercyclic operators. We show that these operators are subspace-Hypercyclic and there are subspace-Hypercyclic  operators that are not subspace-frequently Hypercyclic. There is a criterion like to subspace-Hypercyclicity criterion that implies subspace-frequent Hypercyclicity and if an operator $T$ satisfies this criterion, then $T\oplus T$ is subspace-frequently Hypercyclic. Additionally, operators on finite spaces can not  be subspace-frequently Hypercyclic.

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.

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.

Let 1≤p<∞, and let G be a locally compact group. We characterize disjoint Hypercyclic weighted translation operators on the Lebesgue space Lp(G) in terms of the weight, the Haar measure, and the group element. Disjoint supercyclic, disjoint mixing, and dual disjoint Hypercyclic weighted translation operators are also characterized.