Let $1\le p<\infty$, and let $G$ be a locally compact group. We characterize disjoint hypercyclic weighted translation operators on the Lebesgue space $L^p\left(G\right)$ in terms of the weight, the Haar measure, and the group element. Disjoint supercyclic, disjoint mixing, and dual disjoint hypercyclic weighted translation operators are also characterized.

Let $(\mathcal{X},d,\mu )$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón–Zygmund operator and let $\overrightarrow{b}:=(b_1,\dots ,b_m)$ be a finite family of $\backslash widetilde\{RBMO\}\left(\mu \right)$ functions. In this article, the authors establish the boundedness of the multilinear commutator $T_{\overrightarrow{b}}$, generated by $T$ and $\overrightarrow{b}$ from the atomic Hardy-type space ${\tilde{H}}_{\mathrm{fin},\overrightarrow{b},m,\rho }^{1,q,m+1}\left(\mu \right)$ into the Lebesgue space $L^1\left(\mu \right)$. The authors also prove that $T_{\overrightarrow{b}}$ is bounded from the atomic Hardy-type space ${\tilde{H}}_{\mathrm{fin},\overrightarrow{b},m,\rho }^{1,q,m+2}\left(\mu \right)$ into the atomic Hardy space ${\tilde{H}}^1\left(\mu \right)$ via the molecular characterization of ${\tilde{H}}^1\left(\mu \right)$.

In this article, we use fixed-point methods and measure of noncompactness theory to focus on the problem of establishing the existence of at least a solution for the following functional integral equation

$$u\left(t\right)=g(t,u(t\left)\right)+{\int }_0^{t}G(t,s,u(s\left)\right)ds,t\in [0,+\infty [,$$ in the space of all bounded and continuous real functions on ${\mathbb{R}}_{+}$, under suitable assumptions on $g$ and $G$. Also, we establish an extension of Darbo’s fixed-point theorem and discuss some consequences.

In this article, we study the Hausdorff operator, defined via a general linear mapping $A$, on weighted Herz spaces in the setting of the Heisenberg group. Under some assumptions on the mapping $A$, we establish its sharp boundedness on power-weighted Herz spaces and power-weighted Lebesgue spaces in the Heisenberg group. Our proof is heavily based on the block decomposition of the Herz space, which is quite different from any other function spaces. Our results extend and improve some existing theorems.

We generate a representation of the Toeplitz $C^{*}$-algebra ${\mathcal{T}}_{A_f}$ on a Hilbert space $H_x$ that encodes the orbit of an escape point $x\in I$ of a Markov interval map $f$, with transition matrix $A_f$. This leads to a family of representations of ${\mathcal{T}}_{A_f}$ labeled by points in all intervals $I$. The underlying dynamics of the interval map are used in the study of this family.

Let $\alpha$ be an orientation-preserving homeomorphism of $[0,\infty ]$ onto itself with only two fixed points at $0$ and $\infty$, whose restriction to ${\mathbb{R}}_{+}=(0,\infty )$ is a diffeomorphism, and let $U_{\alpha }$ be the isometric shift operator acting on the Lebesgue space $L^p\left({\mathbb{R}}_{+}\right)$ with $p\in [1,\infty ]$ by the rule $U_{\alpha }f=\left(\alpha \text{'}{)}^{1/p}\right(f\circ \alpha )$. We establish criteria of the two-sided and one-sided invertibility of functional operators of the form $$A={\sum }_{k\in \mathbb{Z}}{a}_k{U}_{\alpha }^k\text{where}\Vert A{\Vert }_W={\sum }_{k\in \mathbb{Z}}\Vert a_k{\Vert }_{L^{\infty }\left({\mathbb{R}}_{+}\right)}<\infty ,$$ on the spaces $L^p\left({\mathbb{R}}_{+}\right)$ under the assumptions that the functions $log\alpha \text{'}$ and $a_k$ for all $k\in \mathbb{Z}$ are bounded and continuous on ${\mathbb{R}}_{+}$ and may have slowly oscillating discontinuities at $0$ and $\infty$. The unital Banach algebra ${\mathfrak{A}}_W$ of such operators is inverse-closed: if $A\in {\mathfrak{A}}_W$ is invertible on $L^p\left({\mathbb{R}}_{+}\right)$ for $p\in [1,\infty ]$, then $A^{-1}\in {\mathfrak{A}}_W$. Obtained criteria are of two types: in terms of the two-sided or one-sided invertibility of so-called discrete operators on the spaces $l^p$ and in terms of conditions related to the fixed points of $\alpha$ and the orbits $\left\{{\alpha }^n\right(t):n\in \mathbb{Z}\}$ of points $t\in {\mathbb{R}}_{+}$.

In this article, we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number, and a sequence of Borel probability measures. By considering special cases of these parameters for particular convex compact subsets, we obtain the classical Kantorovich operators defined in the $1$-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, we discuss the preservation of Lipschitz-continuity and of convexity.

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^{\ast }$-algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem, or the uniform boundedness principle, are still available.

We are interested in the widest possible class of Orlicz functions $\Phi$ such that the easily calculable quasinorm $[f{]}_{\Phi ,p}:=\Vert f{\Vert }_E\{I_{\Phi }\left(\frac{f}{\Vert f{\Vert }_E}\right){\}}^{1p}$ if $f\ne 0$ and $[f{]}_{\Phi ,p}=0$ if $f=0$, on the Orlicz space $L^{\Phi }(\Omega ,\Sigma ,\mu )$ generated by $\Phi$, is equivalent to the Luxemburg norm $\Vert \cdot {\Vert }_{\Phi }$. To do this, we use a suitable ${\Delta }_2$-condition, lower and upper Simonenko indices $p_S^a(\Phi )$ and $q_S^a(\Phi )$ for the generating function $\Phi$, numbers $p\in [1,p_S^a(\Phi \left)\right]$ satisfying $q_S^a(\Phi )-p\le 1$, and an embedding of $L^{\Phi }(\Omega ,\Sigma ,\mu )$ into a suitable Köthe function space $E=E(\Omega ,\Sigma ,\mu )$. We take as $E$ the Lebesgue spaces $L^r(\Omega ,\Sigma ,\mu )$ with $r\in [1,p_S^l(\Phi \left)\right]$, when the measure $\mu$ is nonatomic and finite, and the weighted Lebesgue spaces $L_{\omega }^r(\Omega ,\Sigma ,\mu )$, with $r\in [1,p_S^a(\Phi \left)\right]$ and a suitable weight function $\omega$, when the measure $\mu$ is nonatomic infinite but $\sigma$-finite. We also use condition ${\nabla }_3$ if $p_S^a(\Phi )=1$ and condition ${\nabla }^2$ if $p_S^a(\Phi )>1$, proving their necessity in most of the considered cases. Our results seem important for applications of Orlicz function spaces.

In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of variational inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

Beside the triple product induced by ultrapowers on the bidual of a JB${}^{\ast }$-triple, we assign a triple product to the bidual, $E^{\ast \ast }$, of a JB-triple system $E$, and we show that, under some mild conditions, it makes $E^{\ast \ast }$ a JB-triple system. To study ternary $n$-weak amenability of $E^{\ast \ast }$, we need to improve the module structures in the category of JB-triple systems and their iterated duals, which lead us to introduce a new type of ternary module. We then focus on the main question: when does ternary $n$-weak amenability of $E^{\ast \ast }$ imply the same property for $E?$ In this respect, we show that if the bidual of a JB${}^{\ast }$-triple $E$ is ternary $n$-weakly amenable, then $E$ is ternary $n$-quasiweakly amenable. However, for a general JB-triple system, the results are slightly different for $n=1$ and $n\ge 2$, and the case $n=1$ requires some additional assumptions.

In this article, we construct a function $U\in L^1[0,1)$ with strictly decreasing Fourier–Walsh coefficients $\left\{c_k\right(U\left)\right\}\searrow$, and having a universal and strong $(L^1,L^{\infty })$-property with respect to the Walsh system.