Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Homomorphic conditional expectations as noncommutative retractions
396
408
EN
Robert
Pluta
Department of Mathematics, University of California, Irvine
plutar@tcd.ie
Bernard
Russo
Department of Mathematics, University of
California, Irvine
brusso@uci.edu
10.22034/aot.1705-1161
Let $A$ be a $C^*$-algebra and $mathcal{E}colon A to A$ a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, $$mathcal{E}(x)^* mathcal{E}(x) leq mathcal{E}(x^* x),$$implies that$$leftVertmathcal{E}(x)rightVert^2 leq leftVertmathcal{E}(x^* x)rightVert.$$In this note we show that $mathcal{E}$ is homomorphic (in the sense that $mathcal{E}(xy) = mathcal{E}(x)mathcal{E}(y)$ for every $x, y$ in $A$) if and only if$$leftVertmathcal{E}(x)rightVert^2 = leftVertmathcal{E}(x^*x)rightVert,$$for every $x$ in $A$. We also prove that a homomorphic conditional expectation on a commutative $C^*$-algebra $C_0(X)$ is given by composition with a continuous retraction of $X$. One may therefore consider homomorphic conditional expectations as noncommutative retractions.
conditional expectation,Kadison inequality,Retraction,triple homomorphism,JC*-triple
http://www.aot-math.org/article_46633.html
http://www.aot-math.org/article_46633_0951f9fe6e9ebddc2336176992f3fa2a.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Variants of Weyl's theorem for direct sums of closed linear operators
409
418
EN
Anuradha
Gupta
University of Delhi, Delhi.
dishna2@yahoo.in
Karuna
Mamtani
karunamamtani@gmail.com
10.22034/aot.1701-1087
If $T$ is an operator with compact resolvent and $S$ is any densely defined closed linear operator, then the orthogonal direct sum of $T$ and $S$ satisfies various Weyl type theorems if some necessary conditions are imposed on the operator $S$. It is shown that if $S$ is isoloid and satisfies Weyl's theorem, then $T oplus S$ satisfies Weyl's theorem. Analogous result is proved for a-Weyl's theorem. Further, it is shown that Browder's theorem is directly transmitted from $S$ to $T oplus S$. The converse of these results have also been studied.
operators with compact resolvent,Direct sums,Weyl's Theorem,a-Weyl's Theorem,Browder's Theorem
http://www.aot-math.org/article_46634.html
http://www.aot-math.org/article_46634_1d7bcdb1b169b981bcfa677780fe1a1b.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
On orthogonal decomposition of a Sobolev space
419
427
EN
Dejenie
Lakew
Bryant & Stratton College
dejenieal@gmail.com
10.22034/aot.1703-1135
The theme of this short article is to investigate an orthogonal decomposition of the Sobolev space $W^{1,2}left( Omega right) $ as $ W^{1,2}left( Omega right) =A^{2,2}left( Omega right) oplus D^{2}left( W_{0}^{3,2}left( Omega right) right)$ and look at some of the properties of the inner product therein and the distance defined from the inner product. We also determine the dimension of the orthogonal difference space $W^{1,2}left( Omega right) ominus left(W_{0}^{1,2}left( Omega right) right) ^{perp }$ and show the expansion of Sobolev spaces as their regularity increases.
Sobolev space,orthogonal decomposition,,inner product,Distance
http://www.aot-math.org/article_46656.html
http://www.aot-math.org/article_46656_2f36d0c24bbc89d4e269167986e59a54.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
On symmetry of Birkhoff-James orthogonality of linear operators
428
434
EN
Puja
Ghosh
Jadavpur UNiversity
ghosh.puja1988@gmail.com
Debmalya
Sain
saindebmalya@gmail.com
Kallol
Paul
Jadavpur UNiversity
kalloldada@gmail.com
10.22034/aot.1703-1137
A bounded linear operator $T$ on a normed linear space $mathbb{X}$ is said to be right symmetric (left symmetric) if $Aperp_{B} T Rightarrow T perp_B A $ ($T perp_{B} A Rightarrow A perp_B T $) for all $ A in B(mathbb{X}),$ the space of all bounded linear operators on $mathbb{X}$. Turnsek [Linear Algebra Appl., 407 (2005), 189-195] proved that if $mathbb{X}$ is a Hilbert space then $T$ is right symmetric if and only if $T$ is a scalar multiple of an isometry or coisometry. This result fails in general if the Hilbert space is replaced by a Banach space. The characterization of right and left symmetric operators on a Banach space is still open. In this paper we study the orthogonality in the sense of Birkhoff-James of bounded linear operators on $ (mathbb{R}^n, |cdot|_{infty}) $ and characterize the right symmetric and left symmetric operators on $(mathbb{R}^n,|cdot|_{infty}).$
Birkhoff-James orthogonality,left symmetric operator,right symmetric operator
http://www.aot-math.org/article_46810.html
http://www.aot-math.org/article_46810_689fe803bc0e591e2c5b81af0d53b265.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Traces for fractional Sobolev spaces with variable exponents
435
446
EN
Leandro
Del Pezzo
U Buenos Aires
ldpezzo@dm.uba.ar
Julio
D.
Rossi
Universidad de Buenos Aires
Facultad de Ciencias Exactas y Naturales
Depto Matematica
Ciudad Universitaria, pab 1,
Buenos Aires, Argentina
jrossi@dm.uba.ar
10.22034/aot.1704-1152
In this note we prove a trace theorem in fractional spaces with variable exponents. To be more precise, we show that if $pcolonoverline{Omega }times overline{Omega } rightarrow (1,infty )$ and $qcolonpartial Omegarightarrow (1,infty )$ are continuous functions such that[ frac{(n-1)p(x,x)}{n-sp(x,x)}>q(x) qquad mbox{ in } partial Omega cap {xinoverline{Omega}colon n-sp(x,x) >0}, ]then the inequality $$ Vert fVert _{scriptstyle L^{q(cdot)}(partial Omega )} leq C left{ Vert fVert _{scriptstyle L^{bar{p}(cdot)}(Omega )}+ [f]_{s,p(cdot,cdot)} right} $$ holds. Here $bar{p}(x)=p(x,x)$ and $lbrack frbrack_{s,p(cdot,cdot)} $ denotes the fractional seminorm with variable exponent, that is given by begin{equation*} lbrack frbrack_{s,p(cdot,cdot)} := inf left{ lambda >0colon int_{Omega}int_{Omega }frac{|f(x)-f(y)|^{p(x,y)}}{lambda ^{p(x,y)} |x-y|^{n+sp(x,y)}}dxdy
$p-$Laplacian,fractional operators,variable exponents
http://www.aot-math.org/article_47208.html
http://www.aot-math.org/article_47208_88cb54f5c152f3cc09373bb798ea5a26.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Structures on the way from classical to quantum spaces and their tensor products
447
467
EN
Alexander
Helemskii
helemskii@rambler.ru
10.22034/aot.1706-1189
We study tensor products of two structures situated, in a sense, between normed spaces and (abstract) operator spaces. We call them Lambert and proto-Lambert spaces and pay more attention to the latter ones. The considered two tensor products lead to essentially different norms in the respective spaces. Moreover, the proto-Lambert tensor product is especially nice for spaces with the maximal proto-Lambert norm and in particular, for $L_1$-spaces. At the same time the Lambert tensor product is nice for Hilbert spaces with the minimal Lambert norm.
proto-Lambert space,L-bounded operator,proto-Lambert tensor product,Lambert space,Lambert tensor product
http://www.aot-math.org/article_48029.html
http://www.aot-math.org/article_48029_0d40409e1cfb1b74fe5c9c32cf76cdbe.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
On skew [m,C]-symmetric operators
468
474
EN
Muneo
Cho
Kanagawa University
chiyom01@kanagawa-u.ac.jp
Biljana
Nacevska-Nastovska
Department of Mathematics and Physics
Faculty of Electrical Engineering and Information Technology
Ss. Cyril and Methodius University in Skopje
bibanmath@gmail.com
Jun
Tomiyama
jtomiyama@fc.jwu.ac.jp
10.22034/aot.1703-1147
In this paper, first we characterize the spectra of skew $[m,C]$-symmetric operators and we also prove that if operators $T$ and $S$ are $C$-doubly commuting operators, $T$ is a skew $[m,C]$-symmetric operator and $Q$ is an $n$-nilpotent operator, then $T+Q$ is a skew $[m+2n-2,C]$-symmetric operator. Finally, we show that if $T$ is skew $[m,C]$-symmetric and $S$ is $[n,D]$-symmetric, then $Totimes S$ is skew $[m+n-1, C otimes D]$-symmetric.
Hilbert space,linear operator,conjugation,m-isometric operator,m-symetric
http://www.aot-math.org/article_48114.html
http://www.aot-math.org/article_48114_7ad52fbbb1299adae9980e6759e8f244.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Pseudospectra of elements of reduced Banach algebras
475
493
EN
Arundhathi
Krishnan
arundhathi.krishnan@gmail.com
S. H.
Kulkarni
Indian Institute of Technology Madras
shk@iitm.ac.in
10.22034/aot.1702-1112
Let $A$ be a Banach algebra with identity $1$ and $pin A$ be a non-trivial idempotent. Then $q=1-p$ is also an idempotent. The subalgebras $pAp$ and $qAq$ are Banach algebras, called reduced Banach algebras, with identities $p$ and $q$ respectively. For $ain A$ and $varepsilon>0$, we examine the relationship between the $varepsilon$-pseudospectrum $Lambda_{varepsilon}(A,a)$ of $ain A$, and $varepsilon$-pseudospectra of $papin pAp$ and $qaqin qAq$. We also extend this study by considering a finite number of idempotents $p_{1},cdots,p_{n}$, as well as an arbitrary family of idempotents satisfying certain conditions.
Banach algebra,Direct Sum,Reduced Banach algebra,idempotent,pseudospectrum,spectrum
http://www.aot-math.org/article_48321.html
http://www.aot-math.org/article_48321_79bf8c4df3d69120cd091ee229b6bc45.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
2-Local derivations on matrix algebras and algebras of measurable operators
494
505
EN
Shavkat
Ayupov
sh_ayupov@mail.ru
Karimbergen
Kudaybergenov
karim2006@mail.ru
Amir
Alauadinov
amir_85@mail.ru
10.22034/aot.1612-1074
Let $mathcal{A}$ be a unital Banach algebra such that any Jordan derivation from $mathcal{A}$ into any $mathcal{A}$-bimodule $mathcal{M}$ is a derivation. We prove that any 2-local derivation from the algebra $M_n(mathcal{A})$ into $M_n(mathcal{M}),,(ngeq 3)$ is a derivation. We apply this result to show that any 2-local derivation on the algebra of locally measurable operators affiliated with a von Neumann algebra without direct abelian summands is a derivation.
Matrix algebra,derivation,inner derivation,$2$-local derivation,measurable operator
http://www.aot-math.org/article_43482.html
http://www.aot-math.org/article_43482_5c7c7ad78c756b7cccec7aecf018feb2.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
A formulation of the Jacobi coefficients $c^l_j(alpha, beta)$ via Bell polynomials
506
515
EN
Stuart
Day
Department of Mathematics, University of Sussex
s.day@sussex.ac.uk
Ali
Taheri
Department of Mathematics, University of Sussex
a.taheri@sussex.ac.uk
10.22034/aot.1705-1163
The Jacobi polynomials $(mathscr{P}^{(alpha, beta)}_k: kge0, alpha, beta>-1)$ are deeply intertwined with the Laplacian on compact rank one symmetric spaces. They represent the {it spherical} or zonal functions and as such constitute the main ingredients in describing the spectral measures and spectral projections associated with the Laplacian on these spaces. In this note we strengthen this connection by showing that a set of spectral and geometric quantities associated with Jacobi operator fully describe the Maclaurin coefficients associated with the heat and other related Schwartzian kernels and present an explicit formulation of these quantities using the Bell polynomials.
Jacobi polynomials,Laplace-Beltrame operators,Heat kernel,Bell polynomials,Rank one symmetric spaces
http://www.aot-math.org/article_48949.html
http://www.aot-math.org/article_48949_48b57b4882775471cdf742aeac4354e6.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Besov-Dunkl spaces connected with generalized Taylor formula on the real line
516
530
EN
Chokri
Abdelkefi
Department of Mathematics, University of Tunis
chokri.abdelkefi@yahoo.fr
Faten
Rached
rached@math.jussieu.fr
10.22034/aot.1704-1154
In the present paper, we define for the Dunkl tranlation operators on the real line, the Besov-Dunkl space of functions for which the remainder in the generalized Taylor's formula has a given order. We provide characterization of these spaces by the Dunkl convolution.
Dunkl operator,Dunkl transform,Dunkl translation operators,Dunkl convolution,Generalized Taylor formula,Besov-Dunkl spaces
http://www.aot-math.org/article_49228.html
http://www.aot-math.org/article_49228_6a57f1b28dc7c614b886ecd7150ded5c.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
Stability of the cosine-sine functional equation with involution
531
546
EN
Jeongwook
Chang
jchang@dankook.ac.kr
Chang-Kwon
Choi
ck38@kunsan.ac.kr
Jongjin
Kim
jjkim@jbnu.ac.kr
Prasanna. K
Sahoo
sahoo@louisville.edu
10.22034/aot.1706-1190
Let $S$ and $G$ be a commutative semigroup and a commutative group respectively, $Bbb C$ and $Bbb R^+$ the sets of complex numbers and nonnegative real numbers respectively, $sigma : S to S$ or $sigma : G to G$ an involution and $psi : Gto Bbb R^+$ be fixed. In this paper, we first investigate general solutions of the equation begin{align}g(x+sigma y)=g(x)g(y)+f(x)f(y)nonumberend{align}for all $ x,y in S$, where $f, g : S to Bbb C$ are unknown functions to be determined. Secondly, we consider the Hyers-Ulam stability of the equation,i.e., we study the functional inequalitybegin{align}|g(x+sigma y)-g(x)g(y)-f(x)f(y)|le psi(y)nonumberend{align}for all $ x,y in G$, where $f, g : Gto Bbb C$.
additive function,cosine-sine functional equation,exponential function,Involution,Stability
http://www.aot-math.org/article_50055.html
http://www.aot-math.org/article_50055_de62150c980277715846bddca92ea10a.pdf
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2
4
2017
10
01
$L^p$ Fourier transformation on non-unimodular locally compact groups
547
583
EN
Marianne
Terp
mtgmohr@gmail.com
10.22034/AOT.1709-1231
Let $G$ be a locally compact group with modular function $Delta$ and left regular representation $lambda$. We define the $L^p$ Fourier transform of a function $f in L^p(G)$, $1le p le 2$, to be essentially the operator $lambda(f)Delta^{frac{1}{q}}$ on $L^2(G)$ (where $frac{1}{p}+frac{1}{q}=1$) and show that a generalized Hausdorff--Young theorem holds. To do this, we first treat in detail the spatial $L^p$ spaces $L^p(psi_0)$, $1 le p le infty$, associated with the von Neumann algebra $M=lambda(G)^{primeprime}$ on $L^2(G)$ and the canonical weight $psi_0$ on its commutant. In particular, we discuss isometric isomorphisms of $L^2(psi_0)$ onto $L^2(G)$ and of $L^1(psi_0)$ onto the Fourier algebra $A(G)$. Also, we give a characterization of positive definite functions belonging to $A(G)$ among all continuous positive definite functions.
$L^p$ Fourier transformation,locally compact group,Fourier algebra,positive definite function
http://www.aot-math.org/article_50287.html
http://www.aot-math.org/article_50287_73922bd1a3c72852c000ad5144599937.pdf