Advances in Operator TheoryAdvances in Operator Theory
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Sun, 18 Nov 2018 14:23:02 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.The Bishop-Phelps-Bollobás modulus for functionals on classical Banach spaces
http://www.aot-math.org/article_58504_8968.html
‎In this manuscript we compute the Bishop-Phelps-Bollobás modulus for functionals in classical Banach spaces‎, ‎such as Hilbert spaces‎, ‎spaces of continuous functions‎, ‎$c_0$ and $ell_1$‎. Mon, 31 Dec 2018 20:30:00 +0100Approximation by Chlodowsky variant of Szasz operators involving Sheffer polynomials
http://www.aot-math.org/article_65207_0.html
‎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.Fri, 29 Jun 2018 19:30:00 +0100Semicircular-like and semicircular laws on Banach $*$-probability spaces induced by dynamical ...
http://www.aot-math.org/article_58535_8968.html
‎Starting from the finite Adele ring $A_{Bbb{Q}},$ we construct semigroup‎ ‎dynamical systems of $A_{Bbb{Q}},$ acting on certain $C^{*}$-probability‎ ‎spaces‎. ‎From such dynamical-systematic $C^{*}$-probability spaces‎, ‎we‎ ‎construct Banach-space operators acting on the $C^{*}$-probability spaces‎, ‎and corresponding Banach $*$-probability spaces‎. ‎In particular‎, ‎we are‎ ‎interested in Banach-space operators whose free distributions are the‎ ‎(weighted-)semicircular law(s)‎.Mon, 31 Dec 2018 20:30:00 +0100Lie centralizers on triangular rings and nest algebras
http://www.aot-math.org/article_66452_0.html
‎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.‎Tue, 24 Jul 2018 19:30:00 +0100Banach partial $*$-algebras: an overview
http://www.aot-math.org/article_59546_8968.html
A Banach partial $*$-algebra is a locally convex partial $*$-algebra whose total space is a Banach space. A Banach partial $*$-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely $L^p$-like function spaces and spaces of operators on Hilbert scales or lattices. Finally we analyze the important cases of Banach quasi $*$-algebras and $CQ^*$-algebras.Mon, 31 Dec 2018 20:30:00 +0100Partial isometries and a general spectral theorem
http://www.aot-math.org/article_68086_0.html
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, 26 Aug 2018 19:30:00 +0100Convolution dominated operators on compact extensions of abelian groups
http://www.aot-math.org/article_60135_8968.html
If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$, then an important question is: Is $mathbb{C} 1+CD(G)$ (or $CD(G)$ if $G$ is discrete) inverse-closed in the algebra of bounded operators on $L^2(G)$? In this note we answer this question in the affirmative, provided $G$ is such that one of the following properties is satisfied.(1) There is a discrete, rigidly symmetric, and amenable subgroup $Hsubset G$ and a(measurable) relatively compact neighbourhood of the identity $U$, invariant under conjugation by elements of $H$, such that ${hU;:;hin H}$ is a partition of $G$.(2) The commutator subgroup of $G$ is relatively compact. (If $G$ is connected, this just means that $G$ is an IN group.) All known examples where $CD(G)$ is inverse-closed in $B(L^2(G))$ are covered by this.Mon, 31 Dec 2018 20:30:00 +0100Some classes of Banach spaces and complemented subspaces of operators
http://www.aot-math.org/article_68608_0.html
‎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, 02 Sep 2018 19:30:00 +0100Norm estimates for resolvents of linear operators in a Banach space and spectral variations
http://www.aot-math.org/article_60382_8968.html
‎Let $P_t$ $(ale tle b)$ be a function whose values‎ ‎are projections in a Banach space‎. ‎The paper is devoted to bounded‎ ‎linear operators $A$ admitting the representation‎ $$‎‎A=int_a^b phi(t)dP_{t}+V‎,$$ where $phi(t)$ is a scalar function‎ ‎and $V$ is a compact quasi-nilpotent operator‎, ‎such that‎ ‎$P_tVP_t=VP_t$ $(ale tle b)$‎. ‎We obtain norm estimates‎ ‎for the resolvent of $A$ and a bound for the spectral‎ ‎variation of $A$‎. ‎In addition‎, ‎the representation for the resolvents of the considered operators is established via multiplicative operator integrals‎. ‎That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space‎. ‎It is also shown that the considered operators are‎ ‎Kreiss-bounded‎. ‎Applications to integral operators‎ ‎in $L^p$ are also discussed‎. ‎In particular‎, ‎bounds for the upper and lower spectral radius of integral operators are derived.Mon, 31 Dec 2018 20:30:00 +0100Compact embeddings on a subspace of weighted variable exponent Sobolev spaces
http://www.aot-math.org/article_69354_0.html
‎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 some‎Mon, 17 Sep 2018 19:30:00 +0100The structure of fractional spaces generated by a two-dimensional neutron transport operator ...
http://www.aot-math.org/article_60561_8968.html
‎In this study‎, ‎the structure of fractional spaces generated by the‎ ‎two-dimensional neutron transport operator~$A$~defined by formula $Au=omega‎‎_{1}frac{partial u}{partial x}+omega _{2}frac{partial u}{partial y}$‎ ‎is investigated‎. ‎The positivity of $A$~in $Cleft( mathbb{R}^{2}right)$‎ ‎and $L_{p}left( mathbb{R}^{2}right)‎ ,‎$ $1leq p<infty‎ ~‎$ is‎ ‎established. It is established that for any~$0<alpha <1~$and $1leq‎ ‎p<infty‎ ~‎$the norms in spaces $E_{alpha‎ ,‎p}left( L_{p}left( mathbb{R}‎^{2}right)‎ ,‎~Aright)‎ ~‎$and $E_{alpha }left( Cleft( mathbb{R}‎^{2}right)‎ ,‎~Aright)‎ ,‎~~W_{p}^{alpha } left( mathbb{R}^{2}right)‎ $ ‎and $C^{alpha }left( mathbb{R}^{2}right)‎ ~‎$are equivalent‎, ‎respectively‎. ‎The positivity of the neutron transport operator in H"{o}lder space $‎C^{alpha }left( mathbb{R}^{2}right)‎ ~‎$and Slobodeckij space $‎W_{p}^{alpha }left( mathbb{R}^{2}right) $ is proved‎. ‎In applications‎, ‎theorems on the stability of Cauchy problem for the neutron transport‎ ‎equation in H"{o}lder and Slobodeckij spaces are provided‎.Mon, 31 Dec 2018 20:30:00 +0100A descriptive definition of the It\^{o}-Henstock integral for the operator-valued stochastic process
http://www.aot-math.org/article_69612_0.html
‎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, 23 Sep 2018 20:30:00 +0100On some inequalities for the approximation numbers in Banach Algebras
http://www.aot-math.org/article_60610_8968.html
‎In this paper we generalize some inequalities for the approximation numbers of‎ ‎an element in a normed (Banach) algebra $X$ and‎, ‎as an application‎, ‎we present‎ ‎inequalities for the quasinorms of some ideals defined by means of the‎ ‎approximation numbers‎. ‎In particular‎, ‎if $X=L(E)$‎ - ‎the algebra of linear and bounded operators‎ ‎$T:ETo E$‎, ‎where $E$ is a Banach space‎, ‎we obtain inequalities for certain‎ ‎quasinorms of operators‎.Mon, 31 Dec 2018 20:30:00 +0100A note on irreducible representations of some vector-valued function algebras
http://www.aot-math.org/article_69702_0.html
‎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.Mon, 24 Sep 2018 20:30:00 +0100Quantum groups, from a functional analysis perspective
http://www.aot-math.org/article_61669_8968.html
‎‎It is well-known that any compact Lie group appears as closed subgroup of a unitary group‎, ‎$Gsubset U_N$‎. ‎The unitary group $U_N$ has a free analogue $U_N^+$‎, ‎and the study of the closed quantum subgroups $Gsubset U_N^+$ is a problem of general interest‎. ‎We review here the basic tools for dealing with such quantum groups‎, ‎with all the needed preliminaries included‎, ‎and we discuss as well a number of more advanced topics‎.Mon, 31 Dec 2018 20:30:00 +0100Monomial decomposition of homogeneous polynomials in vector lattices
http://www.aot-math.org/article_69739_0.html
‎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.Tue, 25 Sep 2018 20:30:00 +0100Numerical radius inequalities for operator matrices
http://www.aot-math.org/article_64261_8968.html
Several numerical radius inequalities for operator matrices areproved by generalizing earlier inequalities. In particular, thefollowing inequalities are obtained:if $n$ is even, \$$2w(T) leq max{| A_1 |, | A_2 |,ldots, | A_n |}+frac{1}{2}displaystylesum_{k=0}^{n-1} |~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} |$$\and if $n$ is odd, \$$2w(T) leq max{| A_1 |,| A_2 |,ldots,| A_n |}+ wbigg(widetilde{A}_{(frac{n+1}{2})t}bigg)+ frac{1}{2}displaystylesum_{k=0}^{n-1} |~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} |$$for all $tin [0, 1]$, $ A_i$'s are bounded linear operators on theHilbert space $mathcal{H}$, and $ T=begin{bmatrix}{bigzero} & & A_1\& hspace{4mm}A_2 & \& hspace{-8mm}reflectbox{$ddots$} & \A_n & & {!!bigzero}end{bmatrix}$.Mon, 31 Dec 2018 20:30:00 +0100Multicentric holomorphic calculus for $n-$tuples of commuting operators
http://www.aot-math.org/article_70587_0.html
‎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 $n=2$‎.Mon, 01 Oct 2018 20:30:00 +0100General exponential dichotomies: from finite to infinite time
http://www.aot-math.org/article_65079_8968.html
‎We consider exponential dichotomies on finite intervals and show that if the constants in the notion of an exponential dichotomy are chosen appropriately and uniformly on those intervals‎, ‎then there exists an exponential dichotomy on the whole line‎. ‎We consider the general case of a nonautonomous dynamics that need not be invertible‎. ‎Moreover‎, ‎we consider both cases of discrete and continuous time‎.Mon, 31 Dec 2018 20:30:00 +0100A variational inequality theory for constrained problems in reflexive Banach spaces
http://www.aot-math.org/article_73853_0.html
‎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.Sat, 13 Oct 2018 20:30:00 +0100Operators of Laplace transform type and a new class of hypergeometric coefficients
http://www.aot-math.org/article_65772_8968.html
‎A differential identity on the hypergeometric function ${}_2F_1(a,b;c;z)$ unifying and extending certain spectral results on the scale of‎ ‎Gegenbauer and Jacobi polynomials and leading to a new class of hypergeometric related scalars $mathsf{c}_j^m(a,b,c)$ and‎ ‎polynomials $mathscr{R}_m=mathscr{R}_m(X)$ is established‎. ‎The Laplace-Beltrami operator on a compact rank one symmetric‎ ‎space is considered next and for operators of the Laplace transform type by invoking an operator trace relation‎, ‎the Maclaurin spectral‎ ‎coefficients of their Schwartz kernel are fully described‎. ‎Other representations as well as extensions of the differential‎ ‎identity to the generalised hypergeometric function ${}_pF_q({bf a}; {bf b}; z)$ are formulated and proved‎.Mon, 31 Dec 2018 20:30:00 +0100$M$-operators on partially ordered Banach spaces
http://www.aot-math.org/article_75601_0.html
‎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.Fri, 26 Oct 2018 20:30:00 +0100Dominated orthogonally additive operators in lattice-normed spaces
http://www.aot-math.org/article_66676_8968.html
‎In this paper we introduce a new class of operators in‎ ‎lattice-normed spaces‎. ‎We say that an orthogonally additive operator‎ ‎$T$ from a lattice-normed space $(V,E)$ to a lattice-normed space‎ ‎$(W,F)$ is dominated if there exists a positive orthogonally‎ ‎additive operator $S$ from $E$ to $F$ such that $ls Tx rsleq Sls‎ ‎xrs$ for any element $x$ of $(V,E)$‎. ‎We show that under some mild‎ ‎conditions‎, ‎a dominated orthogonally additive operator has an exact‎ ‎dominant and obtain formulas for calculating the exact dominant of a‎ ‎dominated orthogonally additive operator‎. ‎In the last part of the‎ ‎paper we consider laterally-to-order continuous operators‎. ‎We prove‎ ‎that a dominated orthogonally additive operator is‎ ‎laterally-to-order continuous if and only if the same is its exact‎ ‎dominant‎.Mon, 31 Dec 2018 20:30:00 +0100Multiplicity of solutions for a class of Neumann elliptic systems in anisotropic Sobolev spaces ...
http://www.aot-math.org/article_76653_0.html
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.Thu, 01 Nov 2018 20:30:00 +0100Some approximation properties and nuclear operators in spaces of analytical functions
http://www.aot-math.org/article_67535_8968.html
‎We introduce and investigate a new notion of the approximation property $AP_{[c]}$‎, ‎where $c= (c_n)$ is an arbitrary positive real sequence‎, ‎tending to infinity‎. ‎Also‎, ‎we study the corresponding notion of $[c]$-nuclear operators in Banach spaces‎. ‎Some characterization of the $AP_{[c]}$ in terms of tensor products‎, ‎as well as‎ ‎sufficient conditions for a Banach space to have the $AP_{[c]},$ are given‎. ‎We give also sufficient conditions for a positive answer to the question‎: ‎when it follows from the $[c]$-nuclearity of an adjoint operator the nuclearity of‎ ‎the operator itself‎. ‎Obtained results are applied then to the study of properties‎ ‎of nuclear operators in some spaces of analytical functions‎. ‎Many examples are given‎.Mon, 31 Dec 2018 20:30:00 +0100Existence of weak solutions for an infinite system of second order differential equations
http://www.aot-math.org/article_77163_0.html
‎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.Mon, 12 Nov 2018 20:30:00 +0100$\ell_1$-summability and Lebesgue points of $d$-dimensional Fourier transforms
http://www.aot-math.org/article_68936_8968.html
‎The classical Lebesgue's theorem is generalized and it is proved that under some conditions on the summability function $theta$‎, ‎the $ell_1$-$theta$-means of a function $f$ from the Wiener amalgam space $W(L_1,ell_infty)(R^d)supset L_1(R^d)$ converge to $f$ at each modified strong Lebesgue point and thus almost everywhere‎. ‎The $theta$-summability contains the Weierstrass‎, ‎Abel‎, ‎Picard‎, ‎Bessel‎, ‎Fejér‎, ‎de La Vall'ee-Poussin‎, ‎Rogosinski and Riesz summations‎.Mon, 31 Dec 2018 20:30:00 +0100on Herz's extension theorem
http://www.aot-math.org/article_77387_0.html
{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)}Thu, 15 Nov 2018 20:30:00 +0100Topological properties of operations on spaces of differentiable functions
http://www.aot-math.org/article_69610_8968.html
‎In this paper‎, ‎we consider different notions of openness for the scalar multiplication on sequence spaces and spaces of continuous functions‎. ‎We apply existing techniques to derive weak openness of multiplication on spaces of differentiable functions‎, ‎endowed with a large collection of quasi-algebra norms‎. Mon, 31 Dec 2018 20:30:00 +0100