‎The weighted Cesáro operator $C_h$ in $\ell^2(h)$-spaces is‎ ‎investigated in terms of several concepts of normality‎, ‎where $h$‎ ‎denotes a positive discrete measure on $\mathcal{N}$‎. ‎We classify exactly‎ ‎those $h$ for which $C_h$ is hyponormal‎. ‎Two examples related to the‎ ‎Haar measures of orthogonal polynomials are discussed‎. ‎We show that‎ ‎the Cesáro operator is not always paranormal‎. ‎Furthermore‎, ‎we‎ ‎prove that the Cesáro operator is not quasinormal for any choice‎ ‎of $h$.

‎The main objective of the paper is to study the existence‎, ‎uniqueness and other properties of solution of certain dynamic integral equation on time scales‎. ‎The tools employed are based on the application of the Banach fixed point theorem and certain integral inequality with explicit estimates on time scales‎.

‎We introduce and study the family of Besov-Köthe spaces which‎ ‎is a generalization of the Besov spaces‎, ‎the Besov-Morrey spaces‎ ‎and the variable Besov spaces‎. ‎As an application of the general‎ ‎results for the Besov-Köthe spaces‎, ‎we identify a pre-dual of‎ ‎the Besov-Morrey space‎.

‎Let $X$ be a complex Banach space and let $\mathcal{L}(X)$ be the‎ ‎Banach algebra of all bounded linear operators on $X$‎. ‎We‎ ‎characterize surjective linear maps $\phi‎ :‎\mathcal{L}(X)‎ ‎\rightarrow \mathcal{L}(X)$ compressing or depressing any one of the‎ ‎range‎, ‎the hyper-range‎, ‎the analytic core and the kernel‎.

‎Let $B_{X}$ and $B_{Y}$ be the open unit balls of the Banach Spaces $X$ and $%‎ ‎Y$‎, ‎respectively‎. ‎Let $V$ and $W$ be two countable families of weights on $%‎ ‎B_{X}$ and $B_{Y}$‎, ‎respectively‎. ‎Let $HV\left( B_{X}\right) \left(‎ ‎\text{or }HV_{0}\left( B_{X}\right) \right) $ and $HW\left(‎ ‎B_{Y}\right) $ $\left( \text{or }HW_{0}\left( B_{Y}\right) \right) $‎ ‎be the weighted Fréchet spaces of holomorphic functions‎. ‎In this‎ ‎paper‎, ‎we investigate the holomorphic mappings $\phi‎ :‎B_{X}\rightarrow B_{Y}$ and $\psi‎ :‎B_{X}\rightarrow \mathbb{C}$‎ ‎which characterize continuous weighted composition operators between‎ ‎the spaces $HV\left( B_{X}\right) \left( \text{or }HV_{0}\left(‎ ‎B_{X}\right) \right) $ and $HW\left( B_{Y}\right) $ $\left( \text{or‎ ‎}HW_{0}\left( B_{Y}\right) \right)‎ .‎$ Also‎, ‎we obtained a (linear)‎ ‎dynamical system induced by multiplication operators on these‎ ‎weighted spaces‎.

‎It is shown that if $C$ is a nonempty convex and weakly compact subset of a‎ ‎Banach space $X$ with $M(X)>1$ and $T:C\rightarrow C$ satisfies condition $%‎ ‎(C)$ or is continuous and satisfies condition $(C_{\lambda })$ for some $%‎ ‎\lambda \in (0,1),$ then $T$ has a fixed point‎. ‎In particular‎, ‎our theorem‎ ‎holds for uniformly nonsquare Banach spaces‎. ‎A similar statement is proved‎ ‎for nearly uniformly noncreasy spaces‎.

‎In this paper‎, ‎we prove existence of optimal bundles for a countable‎ ‎set of data in a broad class of normed spaces‎, ‎which extend previous‎ ‎known results for a finite data set in a Hilbert space‎. ‎In addition‎, ‎we study the behavior of deviations and diameters for an increasing‎ ‎sequence of data sets.

We study an Ishikawa type algorithm for two multi-valued quasi-nonexpansive‎ ‎maps on a special class of nonlinear spaces namely hyperbolic metric spaces;‎ ‎in particular‎, ‎strong and $\triangle‎-‎$convergence theorems for the proposed‎ ‎algorithms are established in a uniformly convex hyperbolic space which‎ ‎improve and extend the corresponding known results in uniformly convex‎ ‎Banach spaces‎. ‎Our new results are also valid in geodesic spaces.

‎The paper is devoted to $2$-local derivations on the algebra $LS(M)$‎ ‎of all locally measurable operators affiliated with a type‎ ‎I$_\infty$ von Neumann algebra $M$‎. ‎We prove that every $2$-local‎ ‎derivations on any $*$-subalgebra $\mathcal{A}$ in $LS(M)$‎, ‎such‎ ‎that $M\subseteq\mathcal{A}$‎, ‎is a derivation‎.

‎In this paper a two weight criterion for multidimensional geometric mean operator in variable exponent‎ ‎Lebesgue space is proved‎. ‎Also‎, ‎we found a criterion on weight functions expressing one-dimensional Hardy‎ ‎inequality via a certain nonlinear differential equation‎. ‎In particular‎, ‎considered nonlinear differential‎ ‎equation is nonlinear integro-differential equation‎.

‎Starting from the classic definitions of resolvent set and spectrum‎ ‎of a linear bounded operator on a Banach space‎, ‎we introduce the‎ ‎local resolvent set and local spectrum‎, ‎the local spectral space and‎ ‎the single-valued extension property of a family of linear bounded‎ ‎operators on a Banach space‎. ‎Keeping the analogy with the classic‎ ‎case‎, ‎we extend some of the known results from the case of a linear‎ ‎bounded operator to the case of a family of linear bounded operators‎ ‎on a Banach space.

In this paper we present refinements and improvement of the Young inequality in the context of linear bounded‎ ‎operators‎.

‎In this paper‎, ‎we investigate the essential approximate point‎ ‎spectrum and the essential defect spectrum of a $3\times3$ block‎ ‎operator matrix with unbounded entries and with domain consisting of‎ ‎vectors which satisfy certain relations between their components by‎ ‎means of the Browder resolvent set‎. ‎Furthermore‎, ‎we apply the‎ ‎obtained results to three-Group transport operators in the Banach‎ ‎space $L_{p}( [-a,a]\times[-1,1])$ where $a\gt 0$ and $p \in [1,+\infty).$‎

‎In this paper we find conditions on the existence of bounded linear‎ ‎operators $A$ on the Bergman space $L_{a}^{2}(\mathbb{D})$ such that‎ ‎$A^{\ast}T_{\phi}A\geq S_{\psi}$ and $A^{\ast}T_{\phi}A\geq‎ ‎T_{\phi}$ where $T_{\phi}$ is a positive Toeplitz operator on‎ ‎$L_{a}^{2}(\mathbb{D})$ and $S_{\psi}$ is a self-adjoint little‎ ‎Hankel operator on $L_{a}^{2}(\mathbb{D})$ with symbols $\phi‎, ‎\psi\in L^{\infty}(\mathbb{D})$ respectively‎. ‎Also we show that if‎ ‎$T_{\phi}$ is a non-negative Toeplitz operator then there exists a‎ ‎rank one operator $R_{1}$ on $L_{a}^{2}(\mathbb{D})$ such that‎ ‎$\widetilde{\phi}(z)\geq \alpha^{2}\widetilde{R_{1}}(z)$ for some‎ ‎constant $\alpha\geq 0$ and for all $z\in \mathbb{D}$ where‎ ‎$\widetilde{\phi}$ is the Berezin transform of $T_{\phi}$ and‎ ‎$\widetilde{R_{1}}(z)$ is the Berezin transform of $R_{1}$‎.

‎In this paper we extend some theorems published lately on the relationship‎ ‎between convexity/concavity‎, ‎and subadditivity/superadditivity‎. ‎We also‎ ‎generalize inequalities of compound functions that refine Minkowski‎ ‎inequality‎.