Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Square inequality and strong order relation173844210.22034/aot.1610.1035ENTsuyoshi AndoJournal Article20161023It is well-known that for Hilbert space linear operators $0 leq A$ and $0 leq C$, inequality<br />$C leq A$ does not imply $C^2 leq A^2.$ We introduce a strong order relation $0 leq B lll A$, which guarantees that $C^2 leq B^{1/2}AB^{1/2} text{for all} 0 leq C leq B,$ and that $C^2 leq A^2$ when $B$ commutes with $A$. Connections of this approach with the arithmetic-geometric mean inequality of Bhatia--Kittaneh as well as the Kantorovich constant of $A$ are mentioned.http://www.aot-math.org/article_38442_d9989f3fd74949a9277c13928345bcef.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Operators reversing orthogonality in normed spaces8143847810.22034/aot.1610.1021ENJacek ChmielinskiPedagogical University of CracowJournal Article20161002We consider linear operators $Tcolon Xto X$ on a normed space $X$ which reverse orthogonality, i.e., satisfy the condition<br />$$<br />xbot yquad Longrightarrowquad Tybot Tx,qquad x,yin X,<br />$$<br />where $bot$ stands for Birkhoff orthogonality.http://www.aot-math.org/article_38478_7c15ca13cf82bd7c234123b9bb787e61.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Recent developments of Schwarz's type trace inequalities for operators in Hilbert spaces15913890610.22034/aot.1610.1032ENSever DragomirJournal Article20161013In this paper, we survey some recent trace inequalities for operators in<br />Hilbert spaces that are connected to Schwarz's, Buzano's and Kato's<br />inequalities and the reverses of Schwarz inequality known in the literature<br />as Cassels' inequality and Shisha--Mond's inequality. Applications for some<br />functionals that are naturally associated to some of these inequalities and<br />for functions of operators defined by power series are given. Examples for<br />fundamental functions such as the power, logarithmic, resolvent and<br />exponential functions are provided as well.http://www.aot-math.org/article_38906_2284ce53f9a52e67a0bd59db77882ece.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Fixed points of contractions and cyclic contractions on $C^{*}$-algebra-valued $b$-metric spaces921033895310.22034/aot.1610.1030ENZoran KadelburgAntonella NastasiDepartment of Mathematics and Computer Science,
University of PalermoStojan RadenovicFaculty of Mechanical Engineering, University of BelgradePasquale VetroDepartment of Mathematics and Computer Science,
University of PalermoJournal Article20161013In this paper, we discuss and improve some recent results about<br />contractive and cyclic mappings established in the framework of<br />$C^{*}$-algebra-valued $b$-metric spaces. Our proofs are much<br />shorter than the ones in existing literature. Also, we give two<br />examples that support our approach.http://www.aot-math.org/article_38953_c05393d482953043bf82592dbe9115d3.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Strengthened converses of the Jensen and Edmundson-Lah-Ribaric inequalities1041223960210.22034/aot.1610.1040ENMario KrnicRozarija MikicJosip PecaricJournal Article20161028In this paper, we give converses of the Jensen and Edmundson-Lah-Ribaric inequalities which are more accurate than the existing ones. These converses are given in a difference form and they rely on the recent refinement of the Jensen inequality obtained via linear interpolation of a convex function.<br /> As an application, we also derive improved converse relations for generalized means, for the Holder and Hermite-Hadamard inequalities as well as for the inequalities of Giaccardi and Petrovic.http://www.aot-math.org/article_39602_68b5a4686f6f70886b4597b3324fecf9.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201Positive definite kernels and boundary spaces1231334054710.22034/aot.1610.1044ENFeng TianPalle JorgensenJournal Article20161029We consider a kernel based harmonic analysis of "boundary,"<br />and boundary representations. Our setting is general: certain classes<br />of positive definite kernels. Our theorems extend (and are motivated<br />by) results and notions from classical harmonic analysis on the disk.<br />Our positive definite kernels include those defined on infinite discrete<br />sets, for example sets of vertices in electrical networks, or discrete<br />sets which arise from sampling operations performed on positive definite<br />kernels in a continuous setting. <br /><br />Below we give a summary of main conclusions in the paper: Starting<br />with a given positive definite kernel $K$ we make precise generalized<br />boundaries for $K$. They are measure theoretic "boundaries."<br />Using the theory of Gaussian processes, we show that there is always<br />such a generalized boundary for any positive definite kernel.http://www.aot-math.org/article_40547_eff3ba46ba5c59cdb0769db9b537f59e.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X1120161201(p,q)-type beta functions of second kind1341464054810.22034/aot.1609.1011ENAli AralVijay GuptaNoJournal Article20161017In the present article, we propose the (p,q)-variant of beta function of second kind and establish a relation between the generalized beta and gamma functions using some identities of the post-quantum calculus. As an application, we also propose the (p,q)-Baskakov-Durrmeyer operators, estimate moments and establish some direct results.http://www.aot-math.org/article_40548_62e3853082d62ca9f3f5adb1dcc194c2.pdf