The main goal of this paper is the investigation of bounded-ness and compactness of a class of product-type operators Tm u, v, φ,from Besov spaces into nth weighted type spaces.

‎In this paper‎, ‎we first study the boundedness of generalized‎ weighted composition operators from minimal Möbius invariant‎ ‎subspace to the $n$th weighted type space‎. ‎Then‎, ‎we obtain‎ ‎equivalence conditions for its boundedness by using the Bell‎ ‎polynomials‎. ‎We also find some estimates for the essential‎ ‎norm of this operator and use them to present equivalence‎ ‎conditions for the compactness of such an operator‎.

Let $H(\mathbb{D})$ be the space of all analytic functions on the open unit disc $\mathbb{D}$ in the complex plane $\mathbb{C}$. In this paper, we investigate the boundedness and compactness of the generalized integration operator$$I_{g,\varphi}^{(n)}(f)(z)=\int_0^z f^{(n)}(\varphi(\xi))g(\xi)\ d\xi,\quad z\in\mathbb{D},$$ from Zygmund space into weighted Dirichlet type space, where $\varphi$ is an analytic self-map of $\mathbb{D}$, $n\in\mathbb{N}$ and $g\in H(\mathbb{D})$. Also we give an estimate for the essential norm of the above operator.

Let j be a holomorphic self-map of the open unit disk D on the complex plane and p, q > 0. In this paper, the boundedness and compactness of composition operator Cj from generally weighted Bloch space Bplog to Qq log are investigated.

‎‎‎‎‎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.‎

In this paper, we discuss matrix theoretic characterization for weighted conditional operators by properties of conditional expectation operator in some operator classes on $L^{2}(\Sigma)$-semi-Hilbertian space such as self-adjoint, isometry and normal classes of these type operators on this space. Also, we consider the matrix representation of the Moore-Penrose inverse for these types of operators. We also gave examples to show our results.

For 0 < r < 1 and 1£ p£q < ¥ we find necessary and sufficient conditions for the validity of the following inequality: (òba u(x) (òxa | g(x)-g (t)| r w(t)dt)q/r dx)1/q £ c(òba v(x)| g’ (x)|p dx)1/pwhere u(·), v(·), and w(·) are weight functions.

Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operators, composition operators, multiplication operators and composition operators followed by differentiation operators. In this paper we study generalized weighted composition operators and give estimates for the essential norm of such operators on certain Banach spaces of analytic functions into weighted type spaces. The underlying Banach spaces of analytic functions include Bloch spaces, Zygmund spaces and weighted type spaces. Our estimates for the essential norms of generalized weighted composition operators imply necessary and sufficient conditions for the compactness of such operators. As another application of our results, we obtain essential norm estimates of certain well-known operators which are special cases of generalized weighted composition operators.