Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Homomorphic conditional expectations as noncommutative retractions3964084663310.22034/aot.1705-1161ENRobert PlutaDepartment of Mathematics, University of California, IrvineBernard RussoDepartment of Mathematics, University of
California, IrvineJournal Article20170512Let $A$ 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Variants of Weyl's theorem for direct sums of closed linear operators4094184663410.22034/aot.1701-1087ENAnuradha GuptaUniversity of Delhi, Delhi.Karuna MamtaniJournal Article20170103If $T$ 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001On orthogonal decomposition of a Sobolev space4194274665610.22034/aot.1703-1135ENDejenie LakewBryant & Stratton CollegeJournal Article20170311The 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001On symmetry of Birkhoff-James orthogonality of linear operators4284344681010.22034/aot.1703-1137ENPuja GhoshJadavpur UNiversityDebmalya SainKallol PaulJadavpur UNiversityJournal Article20170315A 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Traces for fractional Sobolev spaces with variable exponents4354464720810.22034/aot.1704-1152ENLeandro Del PezzoU Buenos AiresJulio D.RossiUniversidad de Buenos Aires
Facultad de Ciencias Exactas y Naturales
Depto Matematica
Ciudad Universitaria, pab 1,
Buenos Aires, ArgentinaJournal Article20170410In 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)}}dxdyhttp://www.aot-math.org/article_47208_88cb54f5c152f3cc09373bb798ea5a26.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Structures on the way from classical to quantum spaces and their tensor products4474674802910.22034/aot.1706-1189ENAlexander HelemskiiJournal Article20170620We 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001On skew [m,C]-symmetric operators4684744811410.22034/aot.1703-1147ENMuneo ChoKanagawa UniversityBiljana Nacevska-NastovskaDepartment of Mathematics and Physics
Faculty of Electrical Engineering and Information Technology
Ss. Cyril and Methodius University in SkopjeJun TomiyamaJournal Article20170331In 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Pseudospectra of elements of reduced Banach algebras4754934832110.22034/aot.1702-1112ENArundhathi KrishnanS. H. KulkarniIndian Institute of Technology MadrasJournal Article20170203Let $A$ 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X24201710012-Local derivations on matrix algebras and algebras of measurable operators4945054348210.22034/aot.1612-1074ENShavkat AyupovKarimbergen KudaybergenovAmir AlauadinovJournal Article20161208Let $mathcal{A}$ 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001A formulation of the Jacobi coefficients $c^l_j(alpha, beta)$ via Bell polynomials5065154894910.22034/aot.1705-1163ENStuart DayDepartment of Mathematics, University of SussexAli TaheriDepartment of Mathematics, University of SussexJournal Article20170513The 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Besov-Dunkl spaces connected with generalized Taylor formula on the real line5165304922810.22034/aot.1704-1154ENChokri AbdelkefiDepartment of Mathematics, University of TunisFaten RachedJournal Article20170424In 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001Stability of the cosine-sine functional equation with involution5315465005510.22034/aot.1706-1190ENJeongwook ChangChang-Kwon ChoiJongjin KimPrasanna. K SahooJournal Article20170627 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.pdfTusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2420171001$L^p$ Fourier transformation on non-unimodular locally compact groups5475835028710.22034/AOT.1709-1231ENMarianne TerpJournal Article20170819Let $G$ 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