Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Square inequality and strong order relation
1
7
EN
Tsuyoshi
Ando
ando@es.hokudai.ac.jp
10.22034/aot.1610.1035
It is well-known that for Hilbert space linear operators $0 leq A$ and $0 leq C$, inequality$C leq A$ 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.
Square inequality,strong order relation,operator arithmetic-geometric mean inequality,Kantorovich constant
http://www.aot-math.org/article_38442.html
http://www.aot-math.org/article_38442_d9989f3fd74949a9277c13928345bcef.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Operators reversing orthogonality in normed spaces
8
14
EN
Jacek
Chmielinski
Pedagogical University of Cracow
jacek@up.krakow.pl
10.22034/aot.1610.1021
We consider linear operators $Tcolon Xto X$ on a normed space $X$ which reverse orthogonality, i.e., satisfy the condition$$xbot yquad Longrightarrowquad Tybot Tx,qquad x,yin X,$$where $bot$ stands for Birkhoff orthogonality.
Birkhoff orthogonality,orthogonality preserving mappings,orthogonality reversing map-pings,linear similarities,characterizations of inner product spaces
http://www.aot-math.org/article_38478.html
http://www.aot-math.org/article_38478_7c15ca13cf82bd7c234123b9bb787e61.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Recent developments of Schwarz's type trace inequalities for operators in Hilbert spaces
15
91
EN
Sever
Dragomir
sever.dragomir@vu.edu.au
10.22034/aot.1610.1032
In this paper, we survey some recent trace inequalities for operators inHilbert spaces that are connected to Schwarz's, Buzano's and Kato'sinequalities and the reverses of Schwarz inequality known in the literatureas Cassels' inequality and Shisha--Mond's inequality. Applications for somefunctionals that are naturally associated to some of these inequalities andfor functions of operators defined by power series are given. Examples forfundamental functions such as the power, logarithmic, resolvent andexponential functions are provided as well.
Trace class operators,Hilbert-Schmidt operators,Trace,Schwarz inequality,Kato inequality,Cassels inequality,Shisha--Mond inequality,Trace inequalities for matrices,Power series of operators
http://www.aot-math.org/article_38906.html
http://www.aot-math.org/article_38906_2284ce53f9a52e67a0bd59db77882ece.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Fixed points of contractions and cyclic contractions on $C^{*}$-algebra-valued $b$-metric spaces
92
103
EN
Zoran
Kadelburg
kadelbur@matf.bg.ac.rs
Antonella
Nastasi
Department of Mathematics and Computer Science,
University of Palermo
ella.nastasi.93@gmail.com
Stojan
Radenovic
Faculty of Mechanical Engineering, University of Belgrade
radens@beotel.rs
Pasquale
Vetro
Department of Mathematics and Computer Science,
University of Palermo
pasquale.vetro@unipa.it
10.22034/aot.1610.1030
In this paper, we discuss and improve some recent results aboutcontractive and cyclic mappings established in the framework of$C^{*}$-algebra-valued $b$-metric spaces. Our proofs are muchshorter than the ones in existing literature. Also, we give twoexamples that support our approach.
$C^{*}$-algebra-valued $b$-metric space,$b$-metric space,cyclic type mapping,expansive mapping
http://www.aot-math.org/article_38953.html
http://www.aot-math.org/article_38953_c05393d482953043bf82592dbe9115d3.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Strengthened converses of the Jensen and Edmundson-Lah-Ribaric inequalities
104
122
EN
Mario
Krnic
mario.krnic@fer.hr
Rozarija
Mikic
jaksic.rozarija@gmail.com
Josip
Pecaric
pecaric@element.hr
10.22034/aot.1610.1040
In 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. 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.
positive linear functional,convex function,converse,Jensen inequality,Edmundson-Lah-Ribaric inequality,Holder inequality,Hermite-Hadamard inequality
http://www.aot-math.org/article_39602.html
http://www.aot-math.org/article_39602_68b5a4686f6f70886b4597b3324fecf9.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
Positive definite kernels and boundary spaces
123
133
EN
Feng
Tian
james.ftian@gmail.com
Palle
Jorgensen
palle-jorgensen@uiowa.edu
10.22034/aot.1610.1044
We consider a kernel based harmonic analysis of "boundary,"and boundary representations. Our setting is general: certain classesof positive definite kernels. Our theorems extend (and are motivatedby) results and notions from classical harmonic analysis on the disk.Our positive definite kernels include those defined on infinite discretesets, for example sets of vertices in electrical networks, or discretesets which arise from sampling operations performed on positive definitekernels in a continuous setting. Below we give a summary of main conclusions in the paper: Startingwith a given positive definite kernel $K$ we make precise generalizedboundaries for $K$. They are measure theoretic "boundaries."Using the theory of Gaussian processes, we show that there is alwayssuch a generalized boundary for any positive definite kernel.
Gaussian free fields,reproducing kernel Hilbert space,discrete analysis,Green's function,non-uniform sampling
http://www.aot-math.org/article_40547.html
http://www.aot-math.org/article_40547_eff3ba46ba5c59cdb0769db9b537f59e.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
1
1
2016
12
01
(p,q)-type beta functions of second kind
134
146
EN
Ali
Aral
aliaral73@yahoo.com
Vijay
Gupta
No
vijaygupta2001@hotmail.com
10.22034/aot.1609.1011
In 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.
(p,q)-beta function of second kind, (p,q)-gamma function, Baskakov operator, Durrmeyer variant
http://www.aot-math.org/article_40548.html
http://www.aot-math.org/article_40548_62e3853082d62ca9f3f5adb1dcc194c2.pdf