Let $\mathcal{A}$ be a unital algebra over a number field $\mathbb{F}$. A linear mapping $\varphi$ from $\mathcal{A}$ into itself is called a Jordan-centralized mapping at a given point $G\in \mathcal{A}$ if $\varphi (AB+BA)=\varphi \left(A\right)B+\varphi \left(B\right)A=A\varphi \left(B\right)+B\varphi \left(A\right)$ for all $A$, $B\in \mathcal{A}$ with $AB=G$. In this paper, it is proved that each Jordan-centralized mapping at a given point of triangular algebras is a centralizer. These results are then applied to some non-self-adjoint operator algebras.

We consider a sequence of composite bivariate Bernstein operators and the cubature formula associated with them. The upper bounds for the remainder term of a cubature formula are described in terms of moduli of continuity of order two. Also, we include some results showing how nonmultiplicative the integration functional is.

The set of all absolute normalized norms on ${\mathbb{R}}^2$ (denoted by $A{N}_2$) has a convex structure with respect to the usual operation. In a previous article, N. Komuro, K.-S. Saito, and K.-I. Mitani calculated the James constants of $({\mathbb{R}}^2,\Vert \cdot \Vert )$ when $\Vert \cdot \Vert$ is an extreme point of $A{N}_2$. In this article, we calculate the James constant of its dual space.

We study the Schatten-class membership of generalized Volterra companion integral operators on the standard Fock spaces ${\mathcal{F}}_{\alpha }^2$. The Schatten ${\mathcal{S}}_p\left({\mathcal{F}}_{\alpha }^2\right)$ membership of the operators are characterized in terms of $L^{p/2}$-integrability of certain generalized Berezin-type integral transforms on the complex plane. We also give a more simplified and easy-to-apply description in terms of $L^p$-integrability of the symbols inducing the operators against super-exponentially decreasing weights. Asymptotic estimates for the ${\mathcal{S}}_p\left({\mathcal{F}}_{\alpha }^2\right)$ norms of the operators have also been provided.

The uniform structure of a descriptive normed space $(X,\Vert \cdot \Vert )$ always admits a description with an $\left(F\right)$-norm $\Vert \cdot {\Vert }_1$ such that weak and norm topologies coincide on

$$\{x\in X:\Vert x{\Vert }_1=\rho \}$$ for every $\rho >0$.

We provide a version for operators of the Bishop–Phelps–Bollobás theorem when the domain space is the complex space ${\mathcal{C}}_0\left(L\right)$. In fact, we prove that the pair $\left({\mathcal{C}}_0\right(L),Y)$ will satisfy the Bishop–Phelps–Bollobás property for operators for every Hausdorff locally compact space $L$ and any $\mathbb{C}$-uniformly convex space. As a consequence, this holds for $Y=L_p\left(\mu \right)$ ($1\le p<\infty$).

In this article, we derive several multidimensional Hilbert-type inequalities, including certain differential operators. Further, we determine the conditions under which the constants appearing on the right-hand sides of the established inequalities are the best possible. As an application, some particular examples are also studied.

In this paper, we consider square functions (also called Littlewood–Paley $g$-functions) associated to Hankel convolutions acting on functions in the Bochner–Lebesgue space $L^p\left(\right(0,\infty ),\mathbb{B})$, where $\mathbb{B}$ is a UMD Banach space. As special cases, we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator ${\Delta }_{\lambda }=-x^{-\lambda }\frac{d}{dx}{x}^{2\lambda }\frac{d}{dx}{x}^{-\lambda }$, $\lambda >0$. We characterize the UMD property for a Banach space $\mathbb{B}$ by using $L^p\left(\right(0,\infty ),\mathbb{B})$-boundedness properties of $g$-functions defined by Bessel–Poisson semigroups. As a by-product, we prove that the fact that the imaginary power ${\Delta }_{\lambda }^{i\omega }$, $\omega \in \mathbb{R}\setminus \left\{0\right\}$, of the Bessel operator ${\Delta }_{\lambda }$ is bounded in $L^p\left(\right(0,\infty ),\mathbb{B})$, $1<p<\infty$, characterizes the UMD property for the Banach space $\mathbb{B}$. As applications of our results for square functions, we establish the boundedness in $L^p\left(\right(0,\infty ),\mathbb{B})$ of spectral multipliers $m\left({\Delta }_{\lambda }\right)$ of Bessel operators defined by functions $m$ which are holomorphic in sectors ${\Sigma }_{\vartheta }$.

In this paper, we describe the powers of maximal ideals in the measure algebra of some locally compact Abelian groups in terms of the derivatives of the Fourier–Laplace transform of compactly supported measures. We show that if the locally compact Abelian group has sufficiently many real characters, then all derivatives of the Fourier–Laplace transform of a measure at some point of its spectrum completely characterize the measure. We also show that the derivatives of the Fourier–Laplace transform of a measure can be used to describe the powers of the maximal ideals corresponding to the points of the spectrum of the measure on discrete Abelian groups with finite torsion-free rank.

Given that $A$ and $B$ are two nonempty subsets of the convex metric space $(X,d,\mathcal{W})$, a mapping $T:A\cup B\to A\cup B$ is noncyclic relatively nonexpansive, provided that $T\left(A\right)\subseteq A$, $T\left(B\right)\subseteq B$, and $d(Tx,Ty)\le d(x,y)$ for all $(x,y)\in A\times B$. A point $(p,q)\in A\times B$ is called a best proximity pair for the mapping $T$ if $p=Tp$, $q=Tq$, and $d(p,q)=dist(A,B)$. In this work, we study the existence of best proximity pairs for noncyclic relatively nonexpansive mappings by using the notion of nonconvex proximal normal structure. In this way, we generalize a main result of Eldred, Kirk, and Veeramani. We also establish a common best proximity pair theorem for a commuting family of noncyclic relatively nonexpansive mappings in the setting of convex metric spaces, and as an application we conclude a common fixed-point theorem.

In this paper, we prove the following: let $\omega \left(t\right)$ be a continuous function with $\omega (+0)=0$ and increasing in $[0,\infty )$. Then there exists a series of the form

$${\sum }_{k=1}^{\infty }{c}_k{\psi }_k\left(x\right)\mathrm{with}{\sum }_{k=1}^{\infty }{c}_k^2\omega \left(\right|c_k\left|\right)<\infty$$with the following property: for each $\epsilon >0$ a weight function $\mu \left(x\right)$, $0<\mu \left(x\right)\le 1$, $\left|\right\{x\in [0,1):\mu \left(x\right)\ne 1\left\}\right|<\epsilon$ can be constructed so that the series is universal in the weighted space $L_{\mu }^1[0,1)$ both with respect to rearrangements and subseries.

In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb{R}}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that Hilbert transforms along some curves are completely $L^p$-bounded in the terminology from operator space theory. Moreover, we obtain the $L^p({\mathbb{R}}^n;X)$-boundedness of anisotropic singular integrals by using the “method of rotations” of Calderón–Zygmund. All these results extend preexisting related ones.