eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
396
408
10.22034/aot.1705-1161
46633
Homomorphic conditional expectations as noncommutative retractions
Robert Pluta
plutar@tcd.ie
1
Bernard Russo
brusso@uci.edu
2
Department of Mathematics, University of California, Irvine
Department of Mathematics, University of
California, Irvine
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.
http://www.aot-math.org/article_46633_0951f9fe6e9ebddc2336176992f3fa2a.pdf
conditional expectation
Kadison inequality
Retraction
triple homomorphism
JC*-triple
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
409
418
10.22034/aot.1701-1087
46634
Variants of Weyl's theorem for direct sums of closed linear operators
Anuradha Gupta
dishna2@yahoo.in
1
Karuna Mamtani
karunamamtani@gmail.com
2
University of Delhi, Delhi.
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.
http://www.aot-math.org/article_46634_1d7bcdb1b169b981bcfa677780fe1a1b.pdf
operators with compact resolvent
Direct sums
Weyl's Theorem
a-Weyl's Theorem
Browder's Theorem
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
419
427
10.22034/aot.1703-1135
46656
On orthogonal decomposition of a Sobolev space
Dejenie Lakew
dejenieal@gmail.com
1
Bryant & Stratton College
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.
http://www.aot-math.org/article_46656_2f36d0c24bbc89d4e269167986e59a54.pdf
Sobolev space
orthogonal decomposition,
inner product
distance
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
428
434
10.22034/aot.1703-1137
46810
On symmetry of Birkhoff-James orthogonality of linear operators
Puja Ghosh
ghosh.puja1988@gmail.com
1
Debmalya Sain
saindebmalya@gmail.com
2
Kallol Paul
kalloldada@gmail.com
3
Jadavpur UNiversity
Jadavpur UNiversity
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 $(mathbb{R}^n,|cdot|_{infty}).$
http://www.aot-math.org/article_46810_689fe803bc0e591e2c5b81af0d53b265.pdf
Birkhoff-James orthogonality
left symmetric operator
right symmetric operator
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
435
446
10.22034/aot.1704-1152
47208
Traces for fractional Sobolev spaces with variable exponents
Leandro Del Pezzo
ldpezzo@dm.uba.ar
1
Julio Rossi
jrossi@dm.uba.ar
2
U Buenos Aires
Universidad de Buenos Aires Facultad de Ciencias Exactas y Naturales Depto Matematica Ciudad Universitaria, pab 1, Buenos Aires, Argentina
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
http://www.aot-math.org/article_47208_88cb54f5c152f3cc09373bb798ea5a26.pdf
$p-$Laplacian
fractional operators
variable exponents
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
447
467
10.22034/aot.1706-1189
48029
Structures on the way from classical to quantum spaces and their tensor products
Alexander Helemskii
helemskii@rambler.ru
1
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.
http://www.aot-math.org/article_48029_0d40409e1cfb1b74fe5c9c32cf76cdbe.pdf
proto-Lambert space
L-bounded operator
proto-Lambert tensor product
Lambert space
Lambert tensor product
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
468
474
10.22034/aot.1703-1147
48114
On skew [m,C]-symmetric operators
Muneo Cho
chiyom01@kanagawa-u.ac.jp
1
Biljana Nacevska-Nastovska
bibanmath@gmail.com
2
Jun Tomiyama
jtomiyama@fc.jwu.ac.jp
3
Kanagawa University
Department of Mathematics and Physics
Faculty of Electrical Engineering and Information Technology
Ss. Cyril and Methodius University in Skopje
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.
http://www.aot-math.org/article_48114_7ad52fbbb1299adae9980e6759e8f244.pdf
Hilbert space
linear operator
conjugation
m-isometric operator
m-symetric
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
475
493
10.22034/aot.1702-1112
48321
Pseudospectra of elements of reduced Banach algebras
Arundhathi Krishnan
arundhathi.krishnan@gmail.com
1
S. H. Kulkarni
shk@iitm.ac.in
2
Indian Institute of Technology Madras
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.
http://www.aot-math.org/article_48321_79bf8c4df3d69120cd091ee229b6bc45.pdf
Banach algebra
Direct Sum
Reduced Banach algebra
idempotent
pseudospectrum
spectrum
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
494
505
10.22034/aot.1612-1074
43482
2-Local derivations on matrix algebras and algebras of measurable operators
Shavkat Ayupov
sh_ayupov@mail.ru
1
Karimbergen Kudaybergenov
karim2006@mail.ru
2
Amir Alauadinov
amir_85@mail.ru
3
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.
http://www.aot-math.org/article_43482_5c7c7ad78c756b7cccec7aecf018feb2.pdf
Matrix algebra
derivation
inner derivation
$2$-local derivation
measurable operator
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
506
515
10.22034/aot.1705-1163
48949
A formulation of the Jacobi coefficients $c^l_j(alpha, beta)$ via Bell polynomials
Stuart Day
s.day@sussex.ac.uk
1
Ali Taheri
a.taheri@sussex.ac.uk
2
Department of Mathematics, University of Sussex
Department of Mathematics, University of Sussex
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.
http://www.aot-math.org/article_48949_48b57b4882775471cdf742aeac4354e6.pdf
Jacobi polynomials
Laplace-Beltrame operators
Heat kernel
Bell polynomials
Rank one symmetric spaces
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
516
530
10.22034/aot.1704-1154
49228
Besov-Dunkl spaces connected with generalized Taylor formula on the real line
Chokri Abdelkefi
chokri.abdelkefi@yahoo.fr
1
Faten Rached
rached@math.jussieu.fr
2
Department of Mathematics, University of Tunis
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.
http://www.aot-math.org/article_49228_6a57f1b28dc7c614b886ecd7150ded5c.pdf
Dunkl operator
Dunkl transform
Dunkl translation operators
Dunkl convolution
Generalized Taylor formula
Besov-Dunkl spaces
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
531
546
10.22034/aot.1706-1190
50055
Stability of the cosine-sine functional equation with involution
Jeongwook Chang
jchang@dankook.ac.kr
1
Chang-Kwon Choi
ck38@kunsan.ac.kr
2
Jongjin Kim
jjkim@jbnu.ac.kr
3
Prasanna. K Sahoo
sahoo@louisville.edu
4
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 $f, g : Gto Bbb C$.
http://www.aot-math.org/article_50055_de62150c980277715846bddca92ea10a.pdf
additive function
cosine-sine functional equation
exponential function
Involution
Stability
eng
Tusi Mathematical Research Group (TMRG)
Advances in Operator Theory
2538-225X
2538-225X
2017-10-01
2
4
547
583
10.22034/AOT.1709-1231
50287
$L^p$ Fourier transformation on non-unimodular locally compact groups
Marianne Terp
mtgmohr@gmail.com
1
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.
http://www.aot-math.org/article_50287_73922bd1a3c72852c000ad5144599937.pdf
$L^p$ Fourier transformation
locally compact group
Fourier algebra
positive definite function