Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001On Zipf-Mandelbrot entropy and 3-convex functions7247378241210.15352/aot.1810-1426ENSadia KhalidCOMSATS University Islamabad, PakistanDilda PecaricCatholic University of Croatia, CroatiaJosip PecaricRudn University, RussiaJournal Article20181013In 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.http://www.aot-math.org/article_82412_e0a3b71e903f4fa41527f1f5ba8de8b6.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Analytic variable exponent Hardy spaces7387498307910.15352/aot.1901-1459ENGerardo A. ChaconUniversidad Antonio Narino, ColombiaGerardo R. ChaconGallaudet University, USAJournal Article20190111We 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.http://www.aot-math.org/article_83079_0520005e4978543e9ce85c508f3f20b2.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Riesz transform and fractional integral operators generated by non-degenerate elliptic differential operators7507668316810.15352/aot.1812-1443ENYoshihiro SawanoDenny Ivanal HakimDaniel SalimJournal Article20181206The 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.http://www.aot-math.org/article_83168_313ec1e8651e7162b31c414b0617e5c4.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001On $m$-convexity of set-valued functions7677838354410.15352/aot.1810-1429ENTeodoro LaraUniversity of los Andes, VenezuelaNelson MerentesUniversidad Central de Venezuela, VenezuelaRoy QuinteroUniversity of Iowa, USAEdgar RosalesUniversidad de Los Andes, Venezuela.Journal Article20181023We 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.http://www.aot-math.org/article_83544_0fd80bd66cdd95574c29be0edc5bbdf7.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001A trick for investigation of near-martingales in quantum probability spaces7847928566910.15352/aot.1903-1484ENGhadir SadeghiHakim Sabzevari University, IranAli TalebiFerdowsi University of Mashhad, IranJournal Article20190322In this paper, we introduce near-martingales in the setting of quantum probability spaces and present a trick for investigating some of their properties. For instance, we give a near-martingale analogous result of the fact that the space of all bounded $L^p$-martingales, equipped with the norm $|cdot|_p$, is isometric to $L^p(mathfrak{M})$ for $p>1$. We also present Doob and Riesz decompositions for the near-submartingale and provide Gundy's decomposition for $L^1$-bounded near-martingales. In addition, the interrelation between near-martingales and the instantly independence is studied.http://www.aot-math.org/article_85669_2a9953f25927e8118e28ed09f957b82a.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Invertibility of Toeplitz operators with polyanalytic symbols7938018569310.15352/aot.1812-1451ENAkaki TikaradzeUniversity of Toledo, USAJournal Article20181223For a class of continuous functions including complex polynomials in $z,bar{z},$ we show that the corresponding Toeplitz operator on the Bergman space of the unit disc can be expressed as a quotient of certain differential operators with holomorphic coefficients. This enables us to obtain several nontrivial operator theoretic results about such Toeplitz operators, including a new criterion for invertibility of a Toeplitz operator for a class of harmonic symbols.http://www.aot-math.org/article_85693_62a43f8d788bd3e616b8b2a17c894450.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Approximate and trajectory controllability of fractional neutral differential equation8028208581610.15352/aot.1812-1444ENRajesh DhayalIndian Institute of Technology Mandi, IndiaMuslim MalikIndian Institute of Technology Mandi, IndiaSyed AbbasIndian Institute of Technology Mandi, IndiaJournal Article20181212In this article, we study a new class of fractional neutral differential control system with non-instantaneous impulses and state-dependent delay. The resolvent family and Krasnoselskii's fixed point theorem are utilized to examine the approximate controllability outcomes for the proposed system. Further, we derive the trajectory controllability outcomes for the proposed fractional control system. Finally, the main results are validated with the aid of an example.http://www.aot-math.org/article_85816_99a8743b7f4764eea1d36417d3ea894c.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Three solutions for a Kirchhoff type problem involving nonlocal fractional $p$-Laplacian8218358615910.15352/aot.1901-1464ENElhoussine AzroulSidi Mohamed Ben Abdellah University, MoroccoAbdelmoujib BenkiraneSidi Mohamed Ben Abdellah University, MoroccoMohammed SratiSidi Mohamed Ben Abdellah University, Morocco0000-0003-0816-9743Journal Article20190201In this paper, using the three critical points theorem we obtain the existence of three weak solutions for a Kirchhoff type problem involving the nonlocal fractional $p$-Laplacian operator in a fractional Sobolev space, with homogeneous Dirichlet boundary conditions.http://www.aot-math.org/article_86159_57f5feb2837210bb6fac7e434e025846.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001Extension of the truncated bi-indexed weighted shifts, recursiveness and subnormality8368518693710.15352/aot.1812-1442ENRajae Ben TaherUniversity Moulay Ismail, MoroccoMustapha RachidiUniversidade Federal de Mato Grosso do Sul, BrazilJournal Article20181203In this paper, we build a process in order to extend the truncated weighted shift, using techniques of the bi-indexed recursive sequences. We apply this process to solve the subnormality of $2$-variable weighted shifts, whose associated moment sequence is a bi-indexed recursive sequence. Notably, we detail the case of the truncated $2$-variable weighted shift $Tequiv(T_1, T_2)$ of order $(2,2)$.http://www.aot-math.org/article_86937_e64b88d3cb1a4907c0e43aad5ba0f39a.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X4420191001An ultrapower construction of the multiplier algebra of a $C^*$-algebra and an application to boundary amenability of groups8528648736410.15352/aot.1904-1501ENFacundo PoggiUniversity of Buenos Aires, Argentina.Roman SasykUniversity of Buenos Aires, Argentina.Journal Article20190430Using ultrapowers of $C^*$-algebras we provide a new construction of the multiplier algebra of a $C^*$-algebra. This extends the work of Avsec and Goldbring [Houston J. Math., to appear, arXiv:1610.09276]. to the setting of noncommutative and non separable $C^*$-algebras. We also extend their work to give a new proof of the fact that groups that act transitively on locally finite trees with boundary amenable stabilizers are boundary amenable.http://www.aot-math.org/article_87364_4c5b190da76806524a8cb9f7ce70b745.pdf