We introduce the generalized Bessel potential spaces in the Beurling’s distributions. We give the topological characterizations of the generalized Bessel potential spaces and consider multiplication and convolution operations in the generalized Bessel potential spaces.

For a finitely supported probability measure $\mu$ on a complete CAT(0) space $Y$ , Izeki and Nayatani defined an invariant $\delta(\mu) \in [0,1]$ in [1]. The supremum of those for all such measures on $Y$ is an invariant of $Y$ , called the Izeki-Nayatani invariant, which plays an important role in the study of fixed-point property of groups. In this paper, we establish continuity of $\delta$ on the space of finitely supported probability measures. We prove the lower-semicontinuity of $\delta$ with respect to the $(L^2-)$ Wasserstein metric, and continuity with respect to some metric which induces a stronger topology.

We introduce the strongly generalized difference $[V^\lambda,\Delta^m,p]$-summable sequences and give the relation between the spaces of strongly generalized difference $[V^\lambda,\Delta^m,p]$-summable sequences and strongly generalized difference $[V^\lambda,\Delta^m,p]$-summable sequences with respect to a sequence of moduli. Also we give natural relationship between strongly generalized difference $[V^\lambda,\Delta^m,p]$-convergence with respect to a sequence of moduli and strongly generalized difference $S^\lambda(\Delta^m)$-statistical convergence.

Let $\varpi : S \to \mathbb{P}^2$ be a non-Galois triple covering given by the cubic equation $\zeta^3+3u\zeta+2v=0$, where $u$ and $v$ denote inhomogeneous coordinates of $\mathbb{P}^2$. Let $\hat{\pi} : \hat{X} \to \mathbb{P}^2$ be a $D_6$-covering of $\mathbb{P}^2$ branched along a quintic. There are two possibilities for the ramification types of $\hat{\pi}$. One is that $\hat{\pi}$ has the ramification index 2 (resp. 3) along a conic (resp. a cubic), and the other is that $\hat{\pi}$ has the ramification index 2 (resp. 3) along a quartic (resp. a line). There exist 18 types in the latter case ([8]). For each $\hat{\pi}$ of the 18 types, there exists a non-Galois triple covering $\pi : X \to \mathbb{P}^2$ with the same branch locus as $\hat{\pi}$. In this article, we study rational maps $\Phi : \mathbb{P}^2 \to \mathbb{P}^2$ such that the pull-backs of $\varpi$ by $\Phi$ give rise to $\pi : X \to \mathbb{P}^2$.

In this paper, we study common invariant subspaces of $\mathbb{T}$ and $\mathbb{S}$ on $\ell^2 \otimes \mathbb{C}^2$ where $\mathbb{T} = T \otimes I_{\mathbb{C}^2}$ and $\mathbb{S} = I_{\ell^2} \otimes S$. We describe such invariant subspaces using $T$ and $S$.

Let $(\mathcal{M},\mathcal{H}, J,\mathcal{H}^+)$ be a standard form of a von Neumann algebra. We consider an order for operators preserving a self-dual cone $\mathcal{H}^+$. Let $A,B$ be positive semi-definite operators on $\mathcal{H}$ such that $A$ preserves $\mathcal{H}^+$ and $B$ belongs to a strong closure of the positive part of an order automorphism group on $\mathcal{H}^+$. We prove that if $A$ is majorized by $B$, then there exists a positive semi-definite operator $c$ in the center $Z(Q\mathcal{M}|_{Q\mathcal{H}})$ with $\| c \| \le 1$ such that $QA|_{Q\mathcal{H}} = cB|_{Q\mathcal{H}}$ where $Q$ is a support projection of $B$

A closed subspace $N$ of the Hardy space $H^2$ over the bidisk is said to be mixed invariant under $T_z$ and $T^\ast_w$ if $T_zN \subset N$ and $T^\ast_w N \subset N$. In this paper, we study unitary, similar and quasi-similar module maps for mixed invariant subspaces. We give some characterization of these maps. All unitary module maps are multiplication operators of unimodular functions. Under the condition $\dim(N \ominus zN)$ = 1, we can describe similar and quasi-similar module maps by outer functions.