One of the aims of the present paper is to obtain some properties about logarithmic Bergman-type space on the unit ball. As some applications, the bounded and compact operators ${\mathfrak{S}}_{\overrightarrow{u},\phi }^m=\sum _{i=0}^m{M}_{u_i}{C}_{\phi }{\mathrm{\Re }}^i$ from logarithmic Bergman-type space to weighted-type space on the unit ball are completely characterized.

The aim of the paper is to obtain a interesting result about the atomic decomposition of Bloch-type space in the polydisk. The existing similar results have been applied many times to the atomic decompositions of Bloch-type and weighted Bergman spaces in the unit ball.

Let $u_j$ be the holomorphic functions on the open unit ball $\mathbb{B}$ in ${\mathbb{C}}^n$, $j=\overline{0,m}$, $\phi$ a holomorphic self-map of $\mathbb{B}$, and ${\mathrm{\Re }}^j$ the $j$th iterated radial derivative operator. In this paper, the boundedness and compactness of the sum operator ${\mathfrak{S}}_{\overrightarrow{u},\phi }^m=\sum _{j=0}^m{M}_{u_j}{C}_{\phi }{\mathrm{\Re }}^j$ from the mixed-norm space $H(p,q,\varphi )$, where $0<p,q<+\mathrm{\infty }$, and $\varphi$ is normal, to the weighted-type space $H_{\mu }^{\mathrm{\infty }}$ are characterized. For the mixed-norm space $H(p,q,\varphi )$, $1\le p<+\mathrm{\infty }$, $1<q<+\mathrm{\infty }$, the essential norm estimate of the operator is given, and the Hilbert-Schmidt norm of the operator on the weighted Bergman space $A_{\alpha }^2$ is also calculated.

Let ${\mathbb{U}}^n$ be the unit polydisc in the complex vector space ${\mathbb{C}}^n$. We defined the $\alpha$-Bloch-Orlicz space on ${\mathbb{U}}^n$ by using Young's function and showed that its norm is equivalent with a special $\mu$-Bloch space. We also characterized the boundedness and compactness of the weighted composition operator on $\alpha$-Bloch-Orlicz space. Our results generalized the corresponding results on the unit disk.

One major aim of this paper is to characterize the self-adjointness of some special weighted composition operators on Fock space, and another major aim is to characterize the hyponormality of some composition operators on such space.

The main goal of this paper was to completely characterize complex symmetric difference of the weighted composition operators induced by three type symbols on weighted Bergman space of the right half-plane with the conjugations $\mathcal{J}f(z)=\overline{f(\overline{z})}$, ${\mathcal{J}}_{s}f(z)=\overline{f(\overline{z}+is)}$, and ${\mathcal{J}}_{\ast }f(z)=\frac{1}{z^{\alpha +2}}\overline{f(\frac{1}{\overline{z}})}$. The special phenomenon that we focus on is that the difference is complex symmetric on weighted Bergman spaces of the half-plane with the related conjugation if and only if each weighted composition operator is complex symmetric.