We consider weighted Hardy spaces on polydisk ${\mathbf D}^n$ with $n\gt 1$. Let $z_1,z_2,\ldots,z_n$ be coordinate functions and $N_j \in \mathbf{N}$. In this paper, we determine common reducing subspaces of $M_{z_1}^{N_1},M_{z_2}^{N_2},\ldots,M_{z_n}^{N_n}$.

We discuss linear equations whose coefficients are bounded composition operators on the Hardy space over the unit disk. Some connections between those equations, Pick interpolation and de Branges-Rovnyak spaces are studied in detail.

In this paper, some generalization integral inequalities of Hermite-Hadamard type for functions whose derivatives are convex in modulus are given by using fractional integrals.

Normal del Pezzo surfaces of rank one with only rational log canonical singularities are studied. We classify such surfaces with unique singular points. Moreover, by using the classification result and the results in [11], we study the fundamental groups of their smooth parts.

Let $X$ be a topological space and $f:X\to X$ a bijection. Let ${\mathcal C}(X,f)$ be a set of integers such that an integer $n$ is an element of ${\mathcal C}(X,f)$ if and only if the bijection $f^n:X\to X$ is continuous. A subset $S$ of the set of integers ${\mathbb Z}$ is said to be realizable if there is a topological space $X$ and a bijection $f:X\to X$ such that $S={\mathcal C}(X,f)$. A subset $S$ of ${\mathbb Z}$ containing $0$ is called a submonoid of ${\mathbb Z}$ if the sum of any two elements of $S$ is also an element of $S$. We show that a subset $S$ of ${\mathbb Z}$ is realizable if and only if $S$ is a submonoid of ${\mathbb Z}$. Then we generalize this result to any submonoid in any group.

Generalized split feasibility problem governed by a widely more generalized hybrid mapping is studied. In particular, strong convergence of this algorithm is proved. As tools, resolvents of maximal monotone operators are technically maneuvered to facilitate the argument of the proof to the main result. Applications to iteration methods for various nonlinear mappings and to equilibrium problem are included.

In this paper we show that the extension of a surjective isometry on the unit sphere in a normed linear space is unique.

Recently, the author and two other researchers constructed a calculation method for the Dunkl-Williams constant $DW(X)$ of a normed linear space $X$. Using the method, we determined the constant of $\mathbb{R}^2$ with symmetric octagonal norms. In this paper, we calculate the Dunkl-Williams constant of its dual space. As the result, the space $\mathbb{R}^2$ with symmetric octagonal norm becomes an example for which the Dunkl-Williams constant of the own space and the dual space have same value.