The well-known von Bahr--Esseen bound on the absolute $p$th moments of martingales with $p\in(1,2]$ is extended to a large class of moment functions, and now with a best possible constant factor (which depends on the moment function). As an application, measure concentration inequalities for separately Lipschitz functions on product spaces are obtained. Relations with $p$-uniformly smooth and $q$-uniformly convex normed spaces are discussed.

The following theorem on the circle group $\mathbb{T}$ is due to Norbert Wiener: If $f\in L^{1}\left(\mathbb{T}\right)$ has non-negative Fourier coefficients and is square integrable on a neighbourhood of the identity, then $f\in L^{2}\left( \mathbb{T}\right) $. This result has been extended to even exponents including $p=\infty$, but shown to fail for all other $p\in\left( 1,\infty\right]$. All of this was extended further (appropriately formulated) well beyond locally compact abelian groups. In this paper we prove Wiener's theorem for even exponents for a large class of commutative hypergroups. In addition, we present examples of commutative hypergroups for which, in sharp contrast to the group case, Wiener's theorem holds for all exponents $p\in\left[1,\infty\right]$. For these hypergroups and the Bessel-Kingman hypergroup with parameter $\frac{1}{2}$ we characterise those locally integrable functions that are of positive type and square-integrable near the identity in terms of amalgam spaces.

Let $X, Y$ be Banach spaces, $A, D: X\rightarrow Y$ and $B, C: Y\rightarrow X$ be the bounded linear operators satisfying operator equation set $$\left\{ \begin{aligned} ACD=DBD~ \\ DBA=ACA. \\ \end{aligned} \right.. $$ The concept of regularity was firstly introduced by Kordula and M$\ddot{u}$ller. In this paper, we investigate the common properties of $AC$ and $BD$ in viewpoint of regularity when $A, B, C$ and $D$ all satisfy the operator equation set above.

We investigate maps between normed spaces which preserve the orthogonality defined by the norm derivative. These maps are showed to be a scalar multiple of an isometry.

In this paper, we give a sufficient and necessary condition for an analytic function $f(z)$ on the unit disc $\mathbb{D}$ with Hadamard gaps, that is, $f(z)=\sum\limits_{k=1}^{\infty}a_kz^{n_k}$, where $\frac{n_{k+1}}{n_k}\geq\lambda>1$ for all $k\in \mathbb{N}$, belongs to the Bloch--Orlicz space $ \mathcal{B}^{\varphi}$. As an application of our results, the compactness of composition operator are discussed.

This paper deals with some classes of bounded linear operators on Hilbert spaces. The main emphasis is put onto the classes $\mathcal{A}(k^{*})$ and $\mathcal{A}_{(k^{*})}P,\, k>0$. Some additional results are given for other classes, like $P\mathcal{A}(k^{*})$, $M-\mathcal{A}(k^{*})$ and spectral properties of operators belonging to $\mathcal{A}(k^{*})$ are considered. We also describe under what conditions a matrix-operator $T_{A,B}$ belongs to $\mathcal{A}(k^{*})$, $\mathcal{A}_{(k^{*})}P$ or $P\mathcal{A}(k^{*})$.

Ekeland's variational principle and the existence of critical points of dynamical systems, also known as multiobjective optimization, have been proved in the setting of locally complete spaces. In this article we prove that these two properties can be deduced one from the other under certain convexity conditions.

Weakly singular Volterra integral equations of the different types are considered. The construction of accuracy-optimal numerical methods for one-dimensional and multidimensional equations is discussed. Since this question is closely related with the optimal approximation problem, the orders of the Babenko and Kolmogorov $n$-widths of compact sets from some classes of functions have been evaluated. In conclusion we adduce some numerical illustrations for 2-D Volterra equations.

Let $(\mathcal{X},d,\mu)$ be a geometrically doubling metric space and assume that the measure $\mu$ satisfies the upper doubling condition. In this paper, the authors, by invoking a Cotlar type inequality, show that the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$ is bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p}(\mu)$ for any $p_{i}\in(1,\infty]$ and bounded from $L^{p_{1}}(\mu)\times L^{p_{2}}(\mu)$ into $L^{p,\infty}(\mu)$ for $p_{1}=1$ or $p_{2}=1$, where $p \in [1/2,\infty)$, $1/{p_{1}}+1/{p_{2}}=1/{p}$. Moreover, if $\vec{w}=(w_{1},w_{2})$ belongs to the weight class $A_{\vec{p}}^{\rho}(\mu)$, using the John-strömberg maximal operator and the John-strömberg sharp maximal operator, the authors obtain a weighted weak type estimate $L^{p_{1}}(w_{1}) \times L^{p_{1}}(w_{2}) \rightarrow L^{p,{\infty}}(v_{\vec{w}})$ for the maximal bilinear Calderón--Zygmund operators of type $\omega(t)$. By weakening the assumption of $\omega\in \mathrm{Dini}(1/2)$ into $\omega\in \mathrm{Dini}(1)$, the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces $\mathbb{R}^{n}$.

We find necessary and sufficient conditions for the validity of weighted Hardy-type inequalities for a class of semiadditive operators.

It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analyzing subsystems of the tensor product of product systems. A relation with cluster systems as introduced by B.V.R. Bhat, M. Lindsay and M. Mukherjee, is established.

In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

We study the boundedness of Fourier convolution operators $W^0(b)$ and the compactness of commutators of $W^0(b)$ with multiplication operators $aI$ on some Banach function spaces $X(\mathbb{R})$ for certain classes of piecewise quasicontinuous functions $a\in PQC$ and piecewise slowly oscillating Fourier multipliers $b\in PSO_{X,1}^\diamond$. We suppose that $X(\mathbb{R})$ is a separable rearrangement-invariant space with nontrivial Boyd indices or a reflexive variable Lebesgue space, in which the Hardy--Littlewood maximal operator is bounded. Our results complement those of Isaac De La Cruz--Rodríguez, Yuri Karlovich, and Iván Loreto Hernández obtained for Lebesgue spaces with Muckenhoupt weights.

Recently, a new geometric constant called generalized von Neumann--Jordan constant was introduced. In this paper, the relationships between above constant and generalized García--Falset coefficient are given. In terms of this constant, the lower bounds for the weakly convergent sequence coefficient of a Banach space $X$ are also shown. Moreover, some sufficient conditions which imply normal structure and uniform normal structure are presented.

Given $\varphi:\mathbb{U}\rightarrow\mathbb{U}$, an analytic self-map of the open unit disc in complex plane, the composition operator $C_{\varphi}$ is defined by $C_{\varphi}f=f\circ\varphi$ for $f$ belonging to some Hilbert space of analytic functions on $\mathbb{U}$. In the present paper, we introduce a generalization of the composition operators and reproducing kernel functions on the weighted Hardy spaces. We also obtain the adjoints of generalized composition operators with rational symbol acting on the Hardy, Bergman and Dirichlet spaces.

In this article, we study the geometry and operator theory on quaternionic Hilbert spaces. As it is well-known, Cowen--Douglas operators are a class of non-normal operators related to complex geometry on complex Hilbert spaces. Our purpose is to generalize this concept on quaternionic Hilbert spaces. At the beginning, we study a class of complex holomorphic curves which naturally induce complex vector bundles as sub-bundles in the product space of the base space and a quaternionic Hilbert space. Then we introduce quaternionic Cowen--Douglas operators and give their quaternion unitarily equivalent invariant related to the geometry of the holomorphic curves.

Based on some numerical calculations, S.M. Ulam has conjectured that the ergodic theorem holds true for any quadratic stochastic operator acting on the finite dimensional simplex. However, M.I. Zakharevich showed that Ulam's conjecture is false in general. Later, N.N. Ganikhodjaev and D.V. Zanin have generalized Zakharevich's example in the class of quadratic stochastic Volterra operators acting on 2D simplex. In this paper, we provide a class of Lotka--Volterra operators for which any order Cesàro mean diverges. This class of Lotka--Volterra operators encompasses all previously presented operators in this context.

The aim of this paper is to study properties of Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the $\varphi$-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces.

Let $\mathbb{A}=(A_1,\cdots,A_n)$ and $\mathbb{B}=(B_1,\cdots,B_n)$ be operator distributions, that is, $A_i, B_i>0$ $(1\leq i \leq n)$ and $\sum_{i=1}^n A_i =\sum_{i=1}^n B_i =I$. We give a new relative operator entropy of two operator distributions as follows: For $t,s \in \mathbb{R} \setminus \{0\}$, \[ K_{t,s}(\mathbb{A}|\mathbb{B}) \equiv \dfrac{(\sum_{i=1}^n A_i \sharp_{t} B_i)^s -I}{ts}, \] where \( A \sharp_{t} B = A^{\frac12} (A^{\frac{-1}2} B A^{\frac{-1}2})^{t} A^{\frac12}. \) This includes relative operator entropy $S(\mathbb{A}|\mathbb{B})$, Rényi relative operator entropy $I_{t}(\mathbb{A}|\mathbb{B})$ and Tsallis relative operator entropy $T_{t}(\mathbb{A}|\mathbb{B})$. In this paper, firstly, we discuss fundamental properties of $K_{t,s}(\mathbb{A}|\mathbb{B})$. Secondly, we obtain Shannon type operator inequalities by using $K_{t,s}(\mathbb{A}|\mathbb{B})$, which include previous results by Furuta, Yanagi--Kuriyama--Furuichi and ourselves.

We study concave trace functions of several operator variables and formulate and prove multivariate generalisations of the Golden-Thompson inequality. The obtained results imply that certain functionals in quantum statistical mechanics have bounds of the same form as they appear in classical physics.