Advances in Operator TheoryAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Generalized almost convergence of double sequences in modular function spaces
http://www.aot-math.org/article_78133_8968.html
‎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.Sun, 30 Jun 2019 19:30:00 +0100A universal Banach space with a $K$-unconditional basis
http://www.aot-math.org/article_80091_8968.html
‎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, 30 Jun 2019 19:30:00 +0100Characterization of K-frame vectors and K-frame generator multipliers
http://www.aot-math.org/article_80493_8968.html
‎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.Sun, 30 Jun 2019 19:30:00 +0100Atomic characterizations of Hardy spaces associated to Schr\"{o}dinger type operators
http://www.aot-math.org/article_80523_8968.html
‎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 ‎$H_L^{p(cdot)}(mathbb{R}^n)$‎.Sun, 30 Jun 2019 19:30:00 +0100A Riemann-type definition of the Itô's integral for the operator-valued dtochastic process
http://www.aot-math.org/article_81514_8968.html
‎In this paper‎, ‎we introduce the Itô-McShane integral and show that the classical Itô integral of an operator-valued stochastic process with respect to a Hilbert space-valued $Q$-Wiener process can be defined using theItô-McShane integral.Sun, 30 Jun 2019 19:30:00 +0100Special factors of invertible elements in simple unital purely infinite $C^*$-algebras
http://www.aot-math.org/article_81517_8968.html
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 $uin mathcal{U}(A)$.Sun, 30 Jun 2019 19:30:00 +0100New coupled order Hadamard operators and some applications
http://www.aot-math.org/article_81519_8968.html
‎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.Sun, 30 Jun 2019 19:30:00 +0100Class of operators with superiorly closed numerical ranges
http://www.aot-math.org/article_82132_8968.html
‎The aim of this paper is to introduce a class of operators acting on a complex Hilbert space‎. ‎This class will contain‎, ‎among others‎, ‎non zero compact operators‎. ‎We will give a characterization of this class in term of generalized numerical ranges‎. ‎We will deduce that if $A$ is a compact operator‎, ‎then $ w(A)=vert lambda vert $ with $ lambda in wa $‎, ‎where $ wa $ and $ w(A) $ are the numerical range and the numerical radius of $ A $‎, ‎respectively‎. ‎We will give some new necessary conditions for an operator to be compact‎. ‎We will also show some light on the generalized numerical ranges of the elementary operators $dd$ and $m$‎.Sun, 30 Jun 2019 19:30:00 +0100Hardy-Littlewood inequalities for multipolynomials
http://www.aot-math.org/article_82586_8968.html
‎The notion of multipolynomials was recently introduced and explored by T‎. ‎Velanga [‎Linear Multilinear Algebra‎. 66 (2018)‎, ‎no‎. ‎11‎, ‎2328--2348‎] as an attempt to encompass the theories of polynomials and multilinear operators‎. ‎In the present paper we push this subject further‎, ‎by proving Hardy--Littlewood inequalities for multipolynomials and‎, ‎textit{en passant}‎, ‎a variant of the Kahane--Salem--Zygmund inequality in this framework.Sun, 30 Jun 2019 19:30:00 +0100Algebraic properties of Toeplitz operators with symbols from the range of the heat transform on ...
http://www.aot-math.org/article_82571_8968.html
‎We develop new methods to study the zero product problem and the commutator of Toeplitz operators on the Fock space with harmonic symbols‎. ‎Our method gives us new results on the zero product problem and the commutator of Toeplitz operators on the Fock space‎. ‎We also extend some known result on the Bergman space setting to the Fock space‎.Sun, 30 Jun 2019 19:30:00 +0100On Zipf-Mandelbrot entropy and 3-convex functions
http://www.aot-math.org/article_82412_0.html
‎In this paper‎, ‎we present some interesting results related to the bounds of Zipf-Mandelbrot entropy and the $3$-convexity of the function‎. ‎Further‎, ‎we define linear functionals as the non-negative differences of the obtained inequalities and we present mean value theorems for the linear functionals‎. ‎Finally‎, ‎we discuss the n-exponential convexity and the log-convexity of the functions‎ ‎associated with the linear functionals‎.Thu, 31 Jan 2019 20:30:00 +0100Analytic variable exponent Hardy spaces
http://www.aot-math.org/article_83079_0.html
We introduce a variable exponent version of the Hardy space of analytic functions on the unit disk, we show some properties of the space, and give an example of a variable exponent $p(cdot)$ that satisfies the $log$-H"older condition, and $H^{p(cdot)}neq H^q$ for every constant exponent $1<q<infty$. We also consider the variable exponent version of the Hardy space on the upper-half plane.Tue, 26 Feb 2019 20:30:00 +0100Riesz transform and fractional integral operators generated by non-degenerate elliptic ...
http://www.aot-math.org/article_83168_0.html
‎The Morrey boundedness‎ ‎is proved‎ ‎for the Riesz transform and the inverse operator‎ ‎of the non-degenerate elliptic differential operator of divergence form‎ ‎generated by a vector-function in $(L^infty)^{n^2}$‎, ‎and for the inverse operator of the Schr"{o}dinger operators whose non-negative potentials satisfy a certain integrability condition‎. ‎In this note‎, ‎our result is not obtained directly from the estimates of integral formula‎, ‎which reflects the fact that the solution of the Kato conjecture‎ ‎did not use any integral expression of the operators‎. ‎One of the important tools in the proof‎ ‎is the decomposition of the functions in Morrey spaces‎ ‎based on the elliptic differential operators in question‎. ‎In some special cases‎ ‎where the integral kernel comes into play‎, ‎the boundedness property of the Littlewood--Paley operator‎ ‎was already obtained by Gong‎. ‎So‎, ‎the main novelties of this paper‎ ‎are the decomposition results associated with elliptic differential operators‎ ‎and the result in the case where the explicit formula of the integral kernel of the heat semigroup‎ ‎is unavailable.Wed, 27 Feb 2019 20:30:00 +0100On m-convexity of set-valued functions
http://www.aot-math.org/article_83544_0.html
‎‎We introduce the notion of an $m$-convex set-valued‎ ‎function and study some properties of this class of functions‎. ‎Several characterizations are given as well as certain algebraic‎ ‎properties and examples‎. ‎Finally‎, ‎an inclusion of Jensen type is‎ ‎presented jointly with a sandwich type theorem‎.Sun, 03 Mar 2019 20:30:00 +0100