‎‎‎In this article‎, ‎we present a Chlodowsky type variation of Szász operators defined by means of the Sheffer type polynomials‎. ‎We established convergence properties and the order of‎ ‎convergence through a classical approach‎, ‎the second order modulus of continuity‎, ‎Peetre's $K$-functional‎, ‎and a new type of weighted modulus of continuity‎. ‎Furthermore‎, ‎$A$-statistical approximation of Korokin type for the operators is also shown and the rate of convergence of operators for functions having derivatives of bounded variation is also obtained‎. ‎Moreover‎, ‎some numerical and graphical examples are also given to support our results‎.

‎‎‎We introduce the definition of Lie centralizers and investigate the additivity of Lie centralizers on triangular rings‎. ‎We also present characterizations of both centralizers and Lie centralizers on triangular rings and nest algebras‎.

‎We prove a general spectral theorem for an arbitrary densely defined closed linear operator $T$ between complex Hilbert‎ ‎spaces $H$ and $K$‎. ‎The corresponding operator measure is partial isometry valued and has properties similar to those of the resolution of‎ ‎the identity of a non-negative self-adjoint operator‎. ‎The main method is the use of the canonical factorization (polar decomposition) obtained‎ ‎by v‎. ‎Neumann and Murray‎. ‎The uniqueness of the generalized resolution of the identity is studied together with the properties of a (non-multiplicative)‎ ‎functional calculus‎. ‎The properties of this generalized resolution of the identity are also investigated‎.

‎The concept of $p$-$L$-limited sets and Banach spaces with the $p$-$L$-limited property ($p\in [1, \infty)$) are studied‎. ‎Some characterizations of limited $p$-convergent operators are obtained‎. ‎The complementability of some spaces of operators in the space of limited $p$-convergent operators is also investigated‎.

‎In this paper‎, ‎we define an intersection space between weighted classical Lebesgue spaces and weighted Sobolev spaces with variable exponent‎. ‎We consider the basic properties of the space‎. ‎Also‎, ‎we investigate some inclusions‎, ‎continuous embeddings‎, ‎and compact embeddings under some‎ ‎conditions‎.

‎In this paper‎, ‎we formulate a version of Fundamental Theorem for the Itô-Henstock integral of an operator-valued stochastic process with respect to a Hilbert space-valued Wiener process‎. ‎This theorem will give a descriptive definition of the Itô-Henstock integral for the operator-valued stochastic process‎.

‎Let $\pi‎ :‎\mathcal{E}$ $\rightarrow X$ be a bundle of Banach algebras‎, ‎where $X$ is a completely regular Hausdorff space‎. ‎We identify the sets of irreducible representations of several topological subalgebras of $\Gamma(\pi ),$ the space of continuous sections of $\pi‎ .‎$ The results unify recent and older work of various authors regarding representations on algebra-valued function spaces‎.

‎ ‎The paper is devoted to the characterization and weighted shift representation of regular homogeneous polynomials between vector lattices admitting a decomposition into a sum of monomials in lattice homomorphisms‎. ‎The main tool is the factorization theorem for order bounded disjointness preserving multilinear operators obtained earlier by the authors‎.

‎‎In multicentric holomorphic calculus‎, ‎one represents the function $\varphi$‎, ‎using a new polynomial variable $w=p(z),$ $z\in \mathbb{C},$ in such a way that when it is evaluated at the operator $T,$ then $p(T)$ is small in norm‎. ‎Usually it is assumed that $p$ has distinct roots‎. ‎In this paper we aim to extend this multicentric holomorphic calculus to $n$-tuples of commuting operators looking in particular at the case when $n=2$‎.

‎‎Let $X$ be a real locally uniformly convex reflexive Banach space with the locally uniformly convex dual space $X^*$‎, ‎and let $K$ be a nonempty‎, ‎closed‎, ‎and convex subset of $X$‎. ‎Let $T‎: ‎X\supseteq D(T)\to 2^{X^*}$ be maximal monotone‎, ‎let $S‎: ‎K\to 2^{X^*}$ be bounded and of type $(S_+)$‎, ‎and let $C‎: ‎X\supseteq D(C)\to X^*$ with $D(T)\cap D(\partial \phi)\cap K\subseteq D(C)$‎. ‎Let $\phi‎ : ‎X\to (-\infty‎, ‎\infty]$ be a proper‎, ‎convex‎, ‎and lower semicontinuous function‎. ‎New existence theorems are proved for solvability of variational inequality problems of the type $\rm{VIP}(T+S+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $C$ is compact and $\rm{VIP}(T+C‎, ‎K‎, ‎\phi‎, ‎f^*)$ if $T$ is of compact resolvent and $C$ is bounded and continuous‎. ‎Various improvements and generalizations of the existing results for $T+S$ and $\phi$ are obtained‎. ‎The theory is applied to prove existence of solution for nonlinear constrained variational inequality problems‎.

‎For a matrix $A \in \mathbb{R}^{n \times n}$ whose off-diagonal entries are nonpositive‎, ‎there are several well-known properties that are equivalent to $A$ being an invertible $M$-matrix‎. ‎One of them is the positive stability of $A$‎. ‎A generalization of this characterization to partially ordered Banach spaces is considered in this article‎. ‎Relationships with certain other equivalent conditions are derived‎. ‎An important result on singular irreducible $M$-matrices is generalized using the concept of $M$-operators and irreducibility‎. ‎Certain other invertibility conditions of $M$-operators are also investigated‎.

‎‎In this paper‎, ‎we prove the existence of infinitely many solutions of a system of boundary value problems involving flux boundary conditions in anisotropic variable exponent Sobolev spaces‎, ‎by applying a critical point variational principle obtained by Ricceri as a consequence of a more general variational principle and the theory of the anisotropic variable exponent Sobolev spaces‎.

‎ ‎In this paper‎, ‎we investigate the existence of weak solutions for a boundary value problem of an infinite system of second order differential equations‎. ‎As the main tool‎, ‎a new Krasnosel'skii type fixed point theorem in Fréchet spaces is established in conjunction with the technique of measures of weak noncompactness‎.

‎We present a self-contained proof of the following famous extension theorem due to Carl Herz‎. ‎A closed subgroup $H$ of a locally compact group $G$ is a set of $p$-synthesis in $G$ if and only if‎, ‎for every $u\in A_p(H)\cap C_{00}(H)$ and for every $\varepsilon > 0$‎, ‎there is $v\in A_p(G)\cap C_{00}(G)$, an extension of $u$, such that \[\|v\|_{A_p(G)} < \|u\|_{A_p(H)}+\varepsilon.\]

‎ ‎‎In this paper‎, ‎we study a nonlocal eigenvalue problem involving variable exponent growth conditions‎, ‎on a bounded domain $\Omega \subset \mathbb{R}^n$‎. ‎Using adequate variational techniques‎, ‎mainly based on Ekeland's variational principle‎, ‎we establish the existence of a continuous family of eigenvalues lying in a neighborhood at the right of the origin‎.