2018-06-24T02:07:45Z
http://www.aot-math.org/?_action=export&rf=summon&issue=5213
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Homomorphic conditional expectations as noncommutative retractions
Robert
Pluta
Bernard
Russo
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
2017
10
01
396
408
http://www.aot-math.org/article_46633_0951f9fe6e9ebddc2336176992f3fa2a.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Variants of Weyl's theorem for direct sums of closed linear operators
Anuradha
Gupta
Karuna
Mamtani
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
2017
10
01
409
418
http://www.aot-math.org/article_46634_1d7bcdb1b169b981bcfa677780fe1a1b.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
On orthogonal decomposition of a Sobolev space
Dejenie
Lakew
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
2017
10
01
419
427
http://www.aot-math.org/article_46656_2f36d0c24bbc89d4e269167986e59a54.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
On symmetry of Birkhoff-James orthogonality of linear operators
Puja
Ghosh
Debmalya
Sain
Kallol
Paul
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
2017
10
01
428
434
http://www.aot-math.org/article_46810_689fe803bc0e591e2c5b81af0d53b265.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Traces for fractional Sobolev spaces with variable exponents
Leandro
Del Pezzo
Julio
Rossi
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
2017
10
01
435
446
http://www.aot-math.org/article_47208_88cb54f5c152f3cc09373bb798ea5a26.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Structures on the way from classical to quantum spaces and their tensor products
Alexander
Helemskii
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
2017
10
01
447
467
http://www.aot-math.org/article_48029_0d40409e1cfb1b74fe5c9c32cf76cdbe.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
On skew [m,C]-symmetric operators
Muneo
Cho
Biljana
Nacevska-Nastovska
Jun
Tomiyama
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
2017
10
01
468
474
http://www.aot-math.org/article_48114_7ad52fbbb1299adae9980e6759e8f244.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Pseudospectra of elements of reduced Banach algebras
Arundhathi
Krishnan
S. H.
Kulkarni
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
2017
10
01
475
493
http://www.aot-math.org/article_48321_79bf8c4df3d69120cd091ee229b6bc45.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
2-Local derivations on matrix algebras and algebras of measurable operators
Shavkat
Ayupov
Karimbergen
Kudaybergenov
Amir
Alauadinov
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
2017
10
01
494
505
http://www.aot-math.org/article_43482_5c7c7ad78c756b7cccec7aecf018feb2.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
A formulation of the Jacobi coefficients $c^l_j(alpha, beta)$ via Bell polynomials
Stuart
Day
Ali
Taheri
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
2017
10
01
506
515
http://www.aot-math.org/article_48949_48b57b4882775471cdf742aeac4354e6.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Besov-Dunkl spaces connected with generalized Taylor formula on the real line
Chokri
Abdelkefi
Faten
Rached
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
2017
10
01
516
530
http://www.aot-math.org/article_49228_6a57f1b28dc7c614b886ecd7150ded5c.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
Stability of the cosine-sine functional equation with involution
Jeongwook
Chang
Chang-Kwon
Choi
Jongjin
Kim
Prasanna. K
Sahoo
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
2017
10
01
531
546
http://www.aot-math.org/article_50055_de62150c980277715846bddca92ea10a.pdf
Advances in Operator Theory
Adv. Operator Theory (AOT)
2017
2
4
$L^p$ Fourier transformation on non-unimodular locally compact groups
Marianne
Terp
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
2017
10
01
547
583
http://www.aot-math.org/article_50287_73922bd1a3c72852c000ad5144599937.pdf