Advances in Operator Theory Advances in Operator Theory Thu, 10 Jan 2019 11:38:58 +0100 FeedCreator Advances in Operator Theory Feed provided by Advances in Operator Theory. Click to visit. Approximation by Chlodowsky variant of Szasz operators‎ ‎involving Sheffer polynomials ‎In this article‎, ‎we present a Chlodowsky type variation of Sz'{a}sz 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. Sun, 31 Mar 2019 19:30:00 +0100 Lie centralizers on triangular rings and nest algebras ‎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.‎ Sun, 31 Mar 2019 19:30:00 +0100 Partial isometries and a general spectral theorem 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 nonnegative 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. Sun, 31 Mar 2019 19:30:00 +0100 Some classes of Banach spaces and complemented subspaces of operators ‎The concept of $p$-$L$-limited sets and Banach spaces with the $p$-$L$-limited property ($1le p< infty$) 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. Sun, 31 Mar 2019 19:30:00 +0100 Compact embeddings on a subspace of weighted variable exponent Sobolev spaces ‎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 Sun, 31 Mar 2019 19:30:00 +0100 A descriptive definition of the It\^{o}-Henstock integral for the operator-valued stochastic process ‎In this paper‎, ‎we formulate a version of Fundamental Theorem for the It$hat{text{o}}$-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 It$hat{text{o}}$-Henstock integral for the operator-valued stochastic process.   Sun, 31 Mar 2019 19:30:00 +0100 A note on irreducible representations of some vector-valued function algebras ‎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. Sun, 31 Mar 2019 19:30:00 +0100 Monomial decomposition of homogeneous polynomials in vector lattices ‎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. Sun, 31 Mar 2019 19:30:00 +0100 Multicentric holomorphic calculus for $n-$tuples of commuting operators ‎In multicentric holomorphic calculus, one represents the function $varphi$ using a new polynomial variable $w=p(z),$ $zin mathbb{C},$ 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 $n=2$‎. Sun, 31 Mar 2019 19:30:00 +0100 A variational inequality theory for constrained problems in reflexive Banach spaces ‎Let $X$ be a real locally uniformly convex reflexive Banach space with locally uniformly convex dual space $X^*$ and $K$ be a nonempty‎, ‎closed and convex subset of $X$‎. ‎Let $T‎: ‎Xsupseteq D(T)to 2^{X^*}$ be maximal monotone‎, ‎$S‎: ‎Kto 2^{X^*}$ be bounded and of type $(S_+)$ and $C‎: ‎Xsupseteq D(C)to X^*$ with $D(T)cap D(partial phi)cap Ksubseteq D(C)$‎. ‎Let $phi‎ : ‎Xto (-infty‎, ‎infty]$ 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. Sun, 31 Mar 2019 19:30:00 +0100 $M$-operators on partially ordered Banach spaces ‎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. Sun, 31 Mar 2019 19:30:00 +0100 Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces ... 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. Sun, 31 Mar 2019 19:30:00 +0100 Existence of weak solutions for an infinite system of second order differential equations ‎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 Fr'echet spaces is established in conjunction with the technique of measures of weak noncompactness‎. Sun, 31 Mar 2019 19:30:00 +0100 on Herz's extension theorem {Large Abstract. 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 $p$hskip1pt-synthesis in $G$ if and only for every hbox{$uin A_p(H)cap C_{00}(H)$} and for every $varepsilon >0$ there is hbox{$vin A_p(G)cap C_{00}(G),$} an extension of $u,$ such that$$|v|_{A_p(G)} Sun, 31 Mar 2019 19:30:00 +0100 Eigenvalue problems involving the fractional $p(x)$-Laplacian operator Sun, 31 Mar 2019 19:30:00 +0100 Generalized almost convergence of double sequences in modular function spaces ‎This paper deals with almost convergence‎ ‎of double sequences using a new generalization of fractional-order difference operator in modular spaces and application to Korovkin-type approximation in the context of modular spaces for positive linear operators‎. ‎We then obtain several inclusion relations and present some examples‎, ‎include proper non-trivial extensions of the corresponding classical ones‎. ‎Further‎, ‎we extend our study to new modular forms of Korovkin-type approximation theorems‎. ‎Finally‎, ‎we give an example using bivariate Chlodowsky-Szsaz-Kantorovich-Charlier-type operators and outline possible further extensions and improvements‎, ‎in order to illustrate the effectiveness of the proposed methods. Sat, 17 Nov 2018 20:30:00 +0100 A universal Banach space with a $K$-unconditional basis ‎For a constant $Kgeq 1$ let $mathfrak{B}_K$ be the class of pairs $(X,(mathbf e_n)_{ninomega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(mathbf e_n)_{ninomega}$ for $X$‎, ‎having the unconditional basic constant $K_ule K$‎. ‎Such pairs are called $K$-based Banach spaces‎. ‎A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$‎. ‎Using the technique of Fra"iss'e theory‎, ‎we construct a rational $K$-based Banach space $big(mathbb U_K,(mathbf e_n)_{ninomega}big)$ which is $mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:Lambdatomathbb U_K$ defined on a based subspace $Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $bar f:Atomathbb U_K$ of the based Banach space $A$‎. ‎We also prove that the $K$-based Banach space $mathbb U_K$ is almost $mathfrak{FI}_1$-universal in the sense that any base preserving‎ ‎$varepsilon$-isometry $f:Lambdatomathbb U_K$ defined on a based subspace $Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $varepsilon$-isometry $bar f:Atomathbb U_K$ of the based Banach space $A$‎. ‎On the other hand‎, ‎we show that no almost $mathfrak{FI}_K$-universal based Banach space exists for $K>1$‎. ‎The Banach space $mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis‎, ‎constructed by Pel czy'nski in 1969‎. Sun, 09 Dec 2018 20:30:00 +0100 Characterization of K-frame vectors and K-frame generator multipliers ‎Let $mathcal{U}$ be a unitary system and let $mathcal{B(U)}$ be the Bessel vector space for $mathcal{U}$‎. ‎In this paper‎, ‎we give a characterization of the Bessel vector space and the local commutant space at different complete frame vectors‎. ‎The relation between local commutant spaces at different complete frame vectors is investigated‎. ‎Moreover‎, ‎by introducing multiplication and adjoint on the Bessel vector space for a unital semigroup of unitary operators‎, ‎we give a $C^*$-algebra structure to $mathcal{B(U)}$‎. ‎Then‎, ‎we construct some subsets of $K$-frame vectors that have Banach space or Banach algebra structure‎. ‎Also‎, ‎as a consequence‎, ‎the set of complete frame vectors for different unitary systems contains Banach spaces or Banach algebras‎. ‎In the end‎, ‎we give several characterizations of $K$-frame generator multipliers and Parseval $K$-frame generator multipliers‎. Mon, 17 Dec 2018 20:30:00 +0100 Atomic characterizations of Hardy spaces ‎‎associated to Schr\"{o}dinger type operators ‎In this article‎, ‎the authors consider the Schr"{o}dinger type‎ ‎operator $L:=-{rm div}(Anabla)+V$ on $mathbb{R}^n$ with $ngeq 3$‎, ‎where the matrix $A$ is symmetric and satisfies‎ ‎uniformly elliptic condition and the nonnegative potential‎ ‎$V$ belongs to the reverse H"{o}lder class $RH_q(mathbb{R}^n)$‎ ‎with $qin(n/2,,infty)$‎. ‎Let $p(cdot): mathbb{R}^nto(0,,1]$ be a variable exponent function‎ ‎satisfying the globally $log$-H"{o}lder continuous condition‎. ‎The authors introduce the variable Hardy space $H_L^{p(cdot)}(mathbb{R}^n)$ associated to $L$‎ ‎and establish its atomic characterization‎. ‎The atoms here are closer to the atoms of‎ ‎variable Hardy space $H^{p(cdot)}(mathbb{R}^n)$ in spirit‎, ‎which further implies that $H^{p(cdot)}(mathbb{R}^n)$ is continuously embedded in‎ ‎$H_L^{p(cdot)}(mathbb{R}^n)$‎. Tue, 18 Dec 2018 20:30:00 +0100 A Riemann-type definition of the It\^{o}'s integral for the operator-valued dtochastic process ‎In this paper‎, ‎we introduce the It^{o}-McShane integral and show that the classical It^{o} integral of an operator-valued stochastic process with respect to a Hilbert space-valued $Q$-Wiener process can be defined using the It^{o}-McShane integral. Tue, 08 Jan 2019 20:30:00 +0100 Special factors of invertible elements in simple unital purely infinite $C^*$-algebras In simple unital purely infinite $C^*$-algebra $A$‎, ‎M‎. ‎Leen proved that any element in the identity component of the invertible group is‎ ‎a finite product of symmetries of $A$‎. ‎Revising Leen's factorization‎, ‎we show that a multiple of eight of such factors are $*$-symmetries of the form $1-2P_{i,j}(u)$‎, ‎where $P_{i,j}(u)$ are certain projections of the $C^*$-matrix algebra‎, ‎defined by H‎. ‎Dye as‎ ‎begin{equation*}‎ ‎P_{i,j}(u) = frac{1}{2}(e_{i,i}+e_{j,j}‎ ‎+e_{i,1}ue_{1,j}+e_{j,1}u^*e_{1,i}),‎ ‎end{equation*}‎ ‎for a given system of matrix units ${e_{i,j}}_{i,j=1}^n$ of $A$ and a unitary $uin mathcal{U}(A)$. Tue, 08 Jan 2019 20:30:00 +0100 New coupled order Hadamard operators and some applications ‎In this paper‎, ‎we introduce new Hadamard type operators ‎``‎‎with respect to‎ ‎another function''‎. ‎Some properties of the introduced operators are proved‎ ‎and some applications are discussed‎. ‎For our results‎, ‎some recent results‎ ‎related to Hadamard operators are deduced as some special cases. Tue, 08 Jan 2019 20:30:00 +0100