Advances in Operator TheoryAdvances in Operator Theory
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Sat, 21 Jul 2018 15:41:36 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Linear preservers of two-sided right matrix majorization on $\mathbb{R}_{n}$
http://www.aot-math.org/article_53654_6058.html
A nonnegative real matrix $Rin textbf{M}_{n,m}$ with the property that all its row sums are one is said to be row stochastic. For $x, y in mathbb{R}_{n}$, we say $x$ is right matrix majorized by $y$ (denoted by $xprec_{r} y$) if there exists an $n$-by-$n$ row stochastic matrix $R$ such that $x=yR$. The relation $sim_{r}$ on $mathbb{R}_{n}$ is defined as follows. $xsim_{r}y$ if and only if $ xprec_{r} yprec_{r} x$. In the present paper, we characterize the linear preservers of $sim_{r}$ on $mathbb{R}_{n}$, and answer the question raised by F. Khalooei [Wavelet Linear Algebra textbf{1} (2014), no. 1, 43--50].Sat, 30 Jun 2018 19:30:00 +0100Operators of Laplace transform type and a new class of hypergeometric coefficients
http://www.aot-math.org/article_65772_0.html
‎A differential identity on the hypergeometric function ${}_2F_1(a,b;c;z)$ unifying and extending certain spectral results on the scale of‎ ‎Gegenbauer and Jacobi polynomials and leading to a new class of hypergeometric related scalars $mathsf{c}_j^m(a,b,c)$ and‎ ‎polynomials $mathscr{R}_m=mathscr{R}_m(X)$ is established‎. ‎The Laplace-Beltrami operator on a compact rank one symmetric‎ ‎space is considered next and for operators of the Laplace transform type by invoking an operator trace relation‎, ‎the Maclaurin spectral‎ ‎coefficients of their Schwartz kernel are fully described‎. ‎Other representations as well as extensions of the differential‎ ‎identity to the generalised hypergeometric function ${}_pF_q({bf a}; {bf b}; z)$ are formulated and proved‎.Sat, 14 Jul 2018 19:30:00 +0100Pompeiu-Čebyšev type inequalities for selfadjoint operators in Hilbert spaces
http://www.aot-math.org/article_54087_6058.html
In this work, generalization of some inequalities for continuous h-synchronous (h-asynchronous) functions of selfadjoint linear operators in Hilbert spaces are proved. .Sat, 30 Jun 2018 19:30:00 +0100Projections and isolated points of parts of the spectrum
http://www.aot-math.org/article_65648_0.html
In this paper, we relate the existence of certain projections‎, ‎commuting with a bounded linear operator $Tin L(X)$ acting on Banach space $X$‎, ‎with the generalized Kato decomposition GKD of $T$‎. ‎We also relate the existence of these projections with some properties of the quasi-nilpotent part $H_0(T)$ and the analytic core $K(T)$‎. ‎Further results are given for the isolated points of some parts of the spectrum‎.Thu, 12 Jul 2018 19:30:00 +0100Perturbation of minimum attaining operators
http://www.aot-math.org/article_54270_6058.html
We prove that the minimum attaining property of a bounded linear operator on a Hilbert space $H$ whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact it is a porous set in the ideal of all compact operators on $H$. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results.Sat, 30 Jun 2018 19:30:00 +0100The polar decomposition for adjointable operators on Hilbert $C^*$-modules and centered operators
http://www.aot-math.org/article_65609_0.html
‎Let $T$ be an adjointable operator between two Hilbert $C^*$-modules and $T^*$ be the adjoint operator of $T$‎. ‎The polar decomposition of $T$ is characterized as $T=U(T^*T)^frac12$ and $mathcal{R}(U^*)=overline{mathcal{R}(T^*)}$‎, ‎where $U$ is a partial isometry‎, ‎$mathcal{R}(U^*)$ and $overline{mathcal{R}(T^*)}$ denote the range of $U^*$ and the norm closure of the range of $T^*$‎, ‎respectively‎. ‎Based on this new characterization of the polar decomposition‎, ‎an application to the study of centered operators is carried out.Wed, 11 Jul 2018 19:30:00 +0100Besicovitch almost automorphic solutions of nonautonomous differential equations of first order
http://www.aot-math.org/article_54492_6058.html
The main purpose of this paper is to analyze the existence and uniqueness of Besicovitch almost automorphic solutions and weighted Besicovitch pseudo-almost automorphic solutions of nonautonomous differential equations of first order. We provide an interesting application of our abstract theoretical results.Sat, 30 Jun 2018 19:30:00 +0100Regular spectrum of elements in topological algebras
http://www.aot-math.org/article_65400_0.html
‎Main properties of the regular (or extended) spectrum of elements in topological algebras (introduced by L‎. ‎Waelbroeck and G‎. ‎R‎. ‎Allan for unital locally convex algebras) are presented‎. ‎Descriptions of the relationship between the usual spectrum and the regular spectrum of elements in topological algebras with jointly continuous multiplication are given‎. ‎It is shown that the usual spectrum and the regular spectrum of elements coincide for Hausdorff locally convex Waelbroeck algebras‎. ‎Main properties of the disolvent map of elements in topological algebras are studied‎.Fri, 06 Jul 2018 19:30:00 +0100A Kakutani-Mackey-like theorem
http://www.aot-math.org/article_56029_6058.html
We give a partial extension of a Kakutani-Mackey theorem for quasi-complemented vector spaces. This can be applied in the representation theory of certain complemented (non-normed) topological algebras. The existence of continuous linear maps, in the context of quasi-complemented vector spaces, is a very important issue in their study. Relative to this, we prove that every Hausdorff quasi-complemented locally convex space has continuous linear maps, under which a certain quasi-complemented locally convex space, turns to be pre-Hilbert.Sat, 30 Jun 2018 19:30:00 +0100Approximation by Chlodowsky variant of Szasz operators involving Sheffer polynomials
http://www.aot-math.org/article_65207_0.html
‎In this article‎, ‎we present a Chlodowsky type variation of Sz'{a}sz operators defined by means of the Sheffer type‎ ‎polynomials‎. ‎We established convergence properties and the order of‎ ‎convergence through a classical approach‎, ‎the second order modulus of‎ ‎continuity‎, ‎Peetre's $K$-functional and a new type of weighted modulus of‎ ‎continuity‎. ‎Furthermore‎, ‎$A$-statistical approximation of Korokin type for the operators is also shown and the rate of convergence of operators for‎ ‎functions having derivatives of bounded variation is also‎ ‎obtained‎. ‎Moreover‎, ‎some numerical and graphical examples are also given to support our results‎.Fri, 29 Jun 2018 19:30:00 +0100$T1$ theorem for inhomogeneous Triebel--Lizorkin and Besov spaces on RD-spaces and its application
http://www.aot-math.org/article_57072_6058.html
Using Calder'{o}n's reproducing formulas and almost orthogonal estimates, the $T1$ theorem for the inhomogeneous Triebel--Lizorkin and Besov spaces on RD-spaces is obtained. As an application, new characterizations for these spaces with ``half" the usual conditions of the approximate to the identity are presented.Sat, 30 Jun 2018 19:30:00 +0100Compact and ``compact'' operators on standard Hilbert modules over ...
http://www.aot-math.org/article_65174_0.html
‎We construct a topology on the standard Hilbert module $H_{mathcal{A}}$ over a unital $C^*$-algebra and topology on $H_{mathcal{A}}^{#}$ (the extension of the module $H_{mathcal{A}}$ by the algebra $mathcal{A}^{**}$) such that any‎ ‎"compact"‎ ‎operator‎, ‎(i.e‎. ‎any operator in the norm closure of the linear span of the operators of the form $zmapsto xleft<y,zright>$‎, ‎$x,yin H_{mathcal{A}}$ (or $x,yin H_{mathcal{A}}^{#}$)) maps bounded sets into totally bounded sets.Thu, 28 Jun 2018 19:30:00 +0100Fixed points of a class of unitary operators
http://www.aot-math.org/article_57403_6058.html
In this paper, we consider a class of unitary operators defined on the Bergman space of the right half plane and characterize the fixed points of these unitary operators. We also discuss certain intertwining properties of these operators. Applications of these results are also obtained.Sat, 30 Jun 2018 19:30:00 +0100General exponential dichotomies: from finite to infinite time
http://www.aot-math.org/article_65079_0.html
‎We consider exponential dichotomies on finite intervals and show that if the constants in the notion of an exponential dichotomy are chosen appropriately and uniformly on those intervals‎, ‎then there exists an exponential dichotomy on the whole line‎. ‎We consider the general case of a nonautonomous dynamics that need not be invertible‎. ‎Moreover‎, ‎we consider both cases of discrete and continuous time‎.Tue, 26 Jun 2018 19:30:00 +0100Well-posedness issues for a class of coupled nonlinear Schr"odinger equations with ...
http://www.aot-math.org/article_57444_6058.html
The initial value problem for some coupled nonlinear Schrodinger equations in two space dimensions with exponential growth is investigated. In the defocusing case, global well-posedness and scattering are obtained. In the focusing sign, global and non global existence of solutions are discussed via potential well- method.Sat, 30 Jun 2018 19:30:00 +0100Numerical radius inequalities for operator matrices
http://www.aot-math.org/article_64261_0.html
Several numerical radius inequalities for operator matrices areproved by generalizing earlier inequalities. In particular, thefollowing inequalities are obtained:if $n$ is even, \$$2w(T) leq max{| A_1 |, | A_2 |,ldots, | A_n |}+frac{1}{2}displaystylesum_{k=0}^{n-1} |~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} |$$\and if $n$ is odd, \$$2w(T) leq max{| A_1 |,| A_2 |,ldots,| A_n |}+ wbigg(widetilde{A}_{(frac{n+1}{2})t}bigg)+ frac{1}{2}displaystylesum_{k=0}^{n-1} |~ |A_{n-k}|^{t}|A_{k+1}^{*}|^{1-t} |$$for all $tin [0, 1]$, $ A_i$'s are bounded linear operators on theHilbert space $mathcal{H}$, and $ T=begin{bmatrix}{bigzero} & & A_1\& hspace{4mm}A_2 & \& hspace{-8mm}reflectbox{$ddots$} & \A_n & & {!!bigzero}end{bmatrix}$.Mon, 18 Jun 2018 19:30:00 +0100Closedness and invertibility for the sum of two closed operators
http://www.aot-math.org/article_57481_6058.html
We show a Kalton--Weis type theorem for the general case of non-commuting operators. More precisely, we consider sums of two possibly non-commuting linear operators defined in a Banach space such that one of the operators admits a bounded $H^infty$-calculus, the resolvent of the other one satisfies some weaker boundedness condition and the commutator of their resolvents has certain decay behavior with respect to the spectral parameters. Under this consideration, we show that the sum is closed and that after a sufficiently large positive shift it becomes invertible, and moreover sectorial. As an application we recover a classical result on the existence, uniqueness and maximal $L^{p}$-regularity for solutions of the abstract linear non-autonomous parabolic problem.Sat, 30 Jun 2018 19:30:00 +0100On an elasto-acoustic transmission problem in anisotropic, inhomogeneous media
http://www.aot-math.org/article_63618_0.html
We consider a coupled system describing the interaction between acoustic and elastic regions‎, ‎where the coupling occurs not via material properties but through an interaction on an interface separating the two regimes‎. ‎Evolutionary well-posedness in the sense of Hadamard well-posedness supplemented by causal dependence is shown for a natural choice of generalized interface conditions‎. ‎The results are obtained in a real Hilbert space setting incurring no regularity constraints on the boundary and almost none on the interface of the underlying regions‎. ‎Mon, 11 Jun 2018 19:30:00 +0100Parallel iterative methods for solving the common null point problem in Banach spaces
http://www.aot-math.org/article_57735_6058.html
We consider the common null point problem in Banach spaces. Then, using the hybrid projection method and the $varepsilon $- enlargement of maximal monotone operators, we prove two strong convergence theorems for finding a solution of this problem.Sat, 30 Jun 2018 19:30:00 +0100On tensors of factorizable quantum channels with the completely depolarizing channel
http://www.aot-math.org/article_63014_0.html
‎In this paper‎, ‎we obtain results for factorizability of quantum channels‎. ‎Firstly‎, ‎we prove that if a tensor $Totimes S_k$ of a quantum channel $T$ on $M_n(mathbb{C})$ with the completely depolarizing channel $S_k$ is written as a convex combination of automorphisms on the matrix algebra $M_n(mathbb{C})otimes M_k(mathbb{C})$ with rational coefficients‎, ‎then the quantum channel $T$ has an exact factorization through some matrix algebra with the normalized trace‎. ‎Next‎, ‎we prove that if a quantum channel has an exact factorization through a finite dimensional von Neumann algebra with a convex combination of normal faithful tracial states with rational coefficients‎, ‎then it also has an exact factorization through some matrix algebra with the normalized trace‎.Wed, 23 May 2018 19:30:00 +0100Complex isosymmetric operators
http://www.aot-math.org/article_57759_6058.html
‎In this paper‎, ‎we introduce complex isosymmetric and $(m,n,C)$-isosymmetric operators on a Hilbert space $mathcal H$ and study properties of such operators‎. ‎In particular‎, ‎we prove that if $T in {mathcal B}(mathcal H)$ is an $(m,n,C)$-isosymmetric operator and $N$ is a $k$-nilpotent operator such that $T$ and $N$ are $C$-doubly commuting‎, ‎then $T‎ + ‎N$ is an $(m+2k-2‎, ‎n+2k-1,C)$-isosymmetric operator‎. ‎Moreover‎, ‎we show that if $T$ is $(m,n,C)$-isosymmetric and if $S$ is $(m',D)$-isometric and $n'$-complex symmetric with a conjugation $D$‎, ‎then $T otimes S$ is $(m+m'-1,n+n'-1,C otimes D)$-isosymmetric‎.Sat, 30 Jun 2018 19:30:00 +0100A Banach algebra with its applications over paths of bounded variation
http://www.aot-math.org/article_61910_0.html
‎Let $C[0,T]$ denote the space of continuous‎, ‎real-valued functions on $[0,T]$‎. ‎In this paper we introduce two Banach algebras‎: ‎one of them is defined on $C[0,T]$ and the other is a space of equivalence classes of measures over paths of bounded variation on $[0,T]$‎. ‎We establish an isometric isomorphism between them‎, ‎and evaluate analytic Feynman integrals of the functions in the Banach algebras which play significant roles in the Feynman integration theories and quantum mechanics‎.Sun, 13 May 2018 19:30:00 +0100Variant versions of the Lewent type determinantal inequality
http://www.aot-math.org/article_58027_6058.html
‎In this paper‎, ‎we present a refinement of the Lewent determinantal inequality and then‎, ‎we show that the following inequality holds‎ ‎begin{align*}‎ ‎ &detfrac{I_{mathcal{H}}+A_1}{I_{mathcal{H}}-A_1}+detfrac{I_{mathcal{H}}+A_n}{I_{mathcal{H}}-A_n}-sum_{j=1}^nlambda_j detleft(frac{I_{mathcal{H}}+A_j}{I_{mathcal{H}}-A_j}right)\‎ ‎ & ge detleft[left(frac{I_{mathcal{H}}+A_1}{I_{mathcal{H}}-A_1}right)left(frac{I_{mathcal{H}}+A_n}{I_{mathcal{H}}-A_n}right)prod_{j=1}^n left(frac{I_{mathcal{H}}+A_j}{I_{mathcal{H}}-A_j}right)^{-lambda_j}right],‎, ‎end{align*}‎ ‎where $A_jinmathbb{B}(mathcal{H})$‎, ‎$0le A_j < I_mathcal{H}$‎, ‎$A_j's$ are trace class operators and $A_1 le A_j le A_n~(j=1,cdots,n)$ and $sum_{j=1}^nlambda_j=1,‎~ ‎lambda_j ge 0‎~ ‎(j=1,cdots,n)$‎. In addition‎, ‎we present some new versions of the Lewent type determinantal inequality‎.Sat, 30 Jun 2018 19:30:00 +0100On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system
http://www.aot-math.org/article_61855_0.html
‎Suppose that $hat{b}_mdownarrow 0, {hat{b}_m}_{m=1}^inftynotin l^2,$‎ ‎and $b_n=2^{-frac{m}{2}}hat{b}_m$ for all $ nin(2^m,2^{m+1}].$‎ ‎In this paper‎, ‎it is proved that any measurable and almost everywhere finite‎ ‎function $f(x)$ on $[0,1]$ can be corrected on a set of arbitrarily small measure‎ ‎to a bounded measurable function $widetilde{f}(x)$; so that the nonzero Fourier--Haar‎ ‎coefficients of the corrected function present some subsequence of ${b_n}$‎, ‎and its‎ ‎Fourier--Haar series converges uniformly on $[0,1]$‎.Fri, 11 May 2018 19:30:00 +0100wUR modulus and normal structure in Banach spaces
http://www.aot-math.org/article_58068_6058.html
‎Let $X$ be a Banach space‎. ‎In this paper‎, ‎we study the properties of wUR modulus of $X$‎, ‎$delta_X(varepsilon‎, ‎f),$ where $0 le varepsilon le 2$ and $f in S(X^*),$ and the relationship between the values of wUR modulus and reflexivity‎, ‎uniform non-squareness and normal structure respectively‎. ‎Among other results‎, ‎we proved that if $ delta_X(1‎, ‎f)> 0$ for any $fin S(X^*),$ then $X$ has weak normal structure‎.Sat, 30 Jun 2018 19:30:00 +0100Quantum groups, from a functional analysis perspective
http://www.aot-math.org/article_61669_0.html
‎‎It is well-known that any compact Lie group appears as closed subgroup of a unitary group‎, ‎$Gsubset U_N$‎. ‎The unitary group $U_N$ has a free analogue $U_N^+$‎, ‎and the study of the closed quantum subgroups $Gsubset U_N^+$ is a problem of general interest‎. ‎We review here the basic tools for dealing with such quantum groups‎, ‎with all the needed preliminaries included‎, ‎and we discuss as well a number of more advanced topics‎.Mon, 07 May 2018 19:30:00 +0100The matrix power means and interpolations
http://www.aot-math.org/article_58111_6058.html
‎It is well-known that the Heron mean is a linear interpolation between the arithmetic and the geometric means while the matrix power mean $P_t(A,B):= A^{1/2}left(frac{I+(A^{-1/2}BA^{-1/2})^t}{2}right)^{1/t}A^{1/2}$ interpolates between the harmonic‎, ‎the geometric‎, ‎and the arithmetic means‎. ‎In this article‎, ‎we establish several comparisons between the matrix power mean‎, ‎the Heron mean and the Heinz mean‎. ‎Therefore‎, ‎we have a deeper understanding about the distribution of these matrix means‎.Sat, 30 Jun 2018 19:30:00 +0100Estimations of the Lehmer mean by the Heron mean and their generalizations involving refined ...
http://www.aot-math.org/article_60974_0.html
As generalizations of the arithmetic and the geometric means, for positive real numbers $a$ and $b$,the power difference mean $J_{q}(a,b)=frac{q}{q+1}frac{a^{q+1}-b^{q+1}}{a^{q}-b^{q}}$,the Lehmer mean $L_{q}(a,b)=frac{a^{q+1}+b^{q+1}}{a^{q}+b^{q}}$ and the Heron mean $K_{q}(a,b)=(1-q)sqrt{ab}+qfrac{a+b}{2}$ are well known.In this paper, concerning our recent results on estimations of the power difference mean, we obtain the greatest value $alpha=alpha(q)$ and the least value $beta=beta(q)$ such that the double inequalityfor the Lehmer mean [K_{alpha}(a,b)<L_{q}(a,b)<K_{beta}(a,b)] holds for any $q in mathbb{R}$.We also obtain an operator version of this estimation. Moreover, we discuss generalizations of the results on estimations of the power difference and the Lehmer means. This argument involves refined Heinz operator inequalities by Liang and Shi.Mon, 23 Apr 2018 19:30:00 +0100$C^*$-algebra distance filters
http://www.aot-math.org/article_58258_6058.html
‎We use non-symmetric distances to give a self-contained account of $C^*$-algebra filters and their corresponding compact projections‎, ‎simultaneously simplifying and extending their general theory‎. Sat, 30 Jun 2018 19:30:00 +0100On some inequalities for the approximation numbers in Banach Algebras
http://www.aot-math.org/article_60610_0.html
‎In this paper we generalize some inequalities for the approximation numbers of‎ ‎an element in a normed (Banach) algebra $X$ and‎, ‎as an application‎, ‎we present‎ ‎inequalities for the quasinorms of some ideals defined by means of the‎ ‎approximation numbers‎. ‎In particular‎, ‎if $X=L(E)$‎ - ‎the algebra of linear and bounded operators‎ ‎$T:ETo E$‎, ‎where $E$ is a Banach space‎, ‎we obtain inequalities for certain‎ ‎quasinorms of operators‎.Wed, 18 Apr 2018 19:30:00 +0100On Neugebauer's covering theorem
http://www.aot-math.org/article_58259_6058.html
We present a new proof of a covering theorem of C‎. ‎J‎. ‎Neugebauer‎, ‎stated‎ ‎in a slightly more general form than the original version; we also give an application to restricted weak‎ ‎type (1,1) inequalities for the uncentered maximal operator‎.Sat, 30 Jun 2018 19:30:00 +0100The structure of fractional spaces generated by a two-dimensional neutron transport operator ...
http://www.aot-math.org/article_60561_0.html
‎In this study‎, ‎the structure of fractional spaces generated by the‎ ‎two-dimensional neutron transport operator~$A$~defined by formula $Au=omega‎‎_{1}frac{partial u}{partial x}+omega _{2}frac{partial u}{partial y}$‎ ‎is investigated‎. ‎The positivity of $A$~in $Cleft( mathbb{R}^{2}right)$‎ ‎and $L_{p}left( mathbb{R}^{2}right)‎ ,‎$ $1leq p<infty‎ ~‎$ is‎ ‎established. It is established that for any~$0<alpha <1~$and $1leq‎ ‎p<infty‎ ~‎$the norms in spaces $E_{alpha‎ ,‎p}left( L_{p}left( mathbb{R}‎^{2}right)‎ ,‎~Aright)‎ ~‎$and $E_{alpha }left( Cleft( mathbb{R}‎^{2}right)‎ ,‎~Aright)‎ ,‎~~W_{p}^{alpha } left( mathbb{R}^{2}right)‎ $ ‎and $C^{alpha }left( mathbb{R}^{2}right)‎ ~‎$are equivalent‎, ‎respectively‎. ‎The positivity of the neutron transport operator in H"{o}lder space $‎C^{alpha }left( mathbb{R}^{2}right)‎ ~‎$and Slobodeckij space $‎W_{p}^{alpha }left( mathbb{R}^{2}right) $ is proved‎. ‎In applications‎, ‎theorems on the stability of Cauchy problem for the neutron transport‎ ‎equation in H"{o}lder and Slobodeckij spaces are provided‎.Tue, 17 Apr 2018 19:30:00 +0100The existence of hyper-invariant subspaces for weighted shift operators
http://www.aot-math.org/article_58113_6058.html
‎We introduce some classes of Banach spaces for which the hyperinvariant subspace problem for the‎ ‎shift operator has positive answer‎. ‎Moreover‎, ‎we provide sufficient conditions on weights which ensure that certain subspaces of $ell^2_{{beta}}(mathbb{Z})$ are closed under convolution‎. ‎Finally we consider some cases of weighted spaces for which the problem remains open‎.Sat, 30 Jun 2018 19:30:00 +0100$L^p$ Hardy -Rellich and uncertainty principle inequalities on the sphere
http://www.aot-math.org/article_60419_0.html
‎In this paper, we study the Hardy-Rellich type inequalities and uncertainty principle on the geodesic sphere‎. ‎Firstly‎, ‎we derive $L^p$-Hardy inequalities via divergence theorem‎, ‎which are in turn used to establish the $L^p$ Rellich inequalities‎. ‎We also establish Heisenberg uncertainty principle on the sphere via the Hardy-Rellich type inequalities‎. ‎The best constants appearing in the inequalities are shown to be sharp‎.Sat, 14 Apr 2018 19:30:00 +0100Orthogonality of bounded linear operators on complex Banach spaces
http://www.aot-math.org/article_58482_6058.html
‎We study Birkhoff-James orthogonality of compact linear operators on complex reflexive Banach spaces and obtain its characterization‎. ‎By means of introducing new definitions‎, ‎we illustrate that it is possible in the complex case‎, ‎to develop a study of orthogonality of compact linear operators‎, ‎analogous to the real case‎. ‎Furthermore‎, ‎earlier operator theoretic characterizations of Birkhoff-James orthogonality in the real case‎, ‎can be obtained as simple corollaries to our present study‎. ‎In fact‎, ‎we obtain more than one equivalent characterizations of Birkhoff-James orthogonality of compact linear operators in the complex case‎, ‎in order to distinguish the complex case from the real case‎.Sat, 30 Jun 2018 19:30:00 +0100Norm estimates for resolvents of linear operators in a Banach space and spectral variations
http://www.aot-math.org/article_60382_0.html
‎Let $P_t$ $(ale tle b)$ be a function whose values‎ ‎are projections in a Banach space‎. ‎The paper is devoted to bounded‎ ‎linear operators $A$ admitting the representation‎ $$‎‎A=int_a^b phi(t)dP_{t}+V‎,$$ where $phi(t)$ is a scalar function‎ ‎and $V$ is a compact quasi-nilpotent operator‎, ‎such that‎ ‎$P_tVP_t=VP_t$ $(ale tle b)$‎. ‎We obtain norm estimates‎ ‎for the resolvent of $A$ and a bound for the spectral‎ ‎variation of $A$‎. ‎In addition‎, ‎the representation for the resolvents of the considered operators is established via multiplicative operator integrals‎. ‎That representation can be considered as a generalization of the representation for the resolvent of a normal operator in a Hilbert space‎. ‎It is also shown that the considered operators are‎ ‎Kreiss-bounded‎. ‎Applications to integral operators‎ ‎in $L^p$ are also discussed‎. ‎In particular‎, ‎bounds for the upper and lower spectral radius of integral operators are derived.Thu, 12 Apr 2018 19:30:00 +0100Convolution dominated operators on compact extensions of abelian groups
http://www.aot-math.org/article_60135_0.html
If $G$ is a locally compact group, $CD(G)$ the algebra of convolution dominated operators on $L^2(G)$, then an important question is: Is $mathbb{C} 1+CD(G)$ (or $CD(G)$ if $G$ is discrete) inverse-closed in the algebra of bounded operators on $L^2(G)$? In this note we answer this question in the affirmative, provided $G$ is such that one of the following properties is satisfied.(1) There is a discrete, rigidly symmetric, and amenable subgroup $Hsubset G$ and a(measurable) relatively compact neighbourhood of the identity $U$, invariant under conjugation by elements of $H$, such that ${hU;:;hin H}$ is a partition of $G$.(2) The commutator subgroup of $G$ is relatively compact. (If $G$ is connected, this just means that $G$ is an IN group.) All known examples where $CD(G)$ is inverse-closed in $B(L^2(G))$ are covered by this.Mon, 02 Apr 2018 19:30:00 +0100Banach partial $*$-algebras: an overview
http://www.aot-math.org/article_59546_0.html
A Banach partial $*$-algebra is a locally convex partial $*$-algebra whose total space is a Banach space. A Banach partial $*$-algebra is said to be of type (B) if it possesses a generating family of multiplier spaces that are also Banach spaces. We describe the basic properties of these objects and display a number of examples, namely $L^p$-like function spaces and spaces of operators on Hilbert scales or lattices. Finally we analyze the important cases of Banach quasi $*$-algebras and $CQ^*$-algebras.Tue, 13 Mar 2018 20:30:00 +0100Semicircular-like and semicircular laws on Banach $*$-probability spaces induced by dynamical ...
http://www.aot-math.org/article_58535_0.html
‎Starting from the finite Adele ring $A_{Bbb{Q}},$ we construct semigroup‎ ‎dynamical systems of $A_{Bbb{Q}},$ acting on certain $C^{*}$-probability‎ ‎spaces‎. ‎From such dynamical-systematic $C^{*}$-probability spaces‎, ‎we‎ ‎construct Banach-space operators acting on the $C^{*}$-probability spaces‎, ‎and corresponding Banach $*$-probability spaces‎. ‎In particular‎, ‎we are‎ ‎interested in Banach-space operators whose free distributions are the‎ ‎(weighted-)semicircular law(s)‎.Sun, 11 Mar 2018 20:30:00 +0100The Bishop-Phelps-Bollobás modulus for functionals on classical Banach spaces
http://www.aot-math.org/article_58504_0.html
‎In this manuscript we compute the Bishop-Phelps-Bollobás modulus for functionals in classical Banach spaces‎, ‎such as Hilbert spaces‎, ‎spaces of continuous functions‎, ‎$c_0$ and $ell_1$‎. Sat, 10 Mar 2018 20:30:00 +0100Affine actions and the Yang-Baxter equation
http://www.aot-math.org/article_60104_6058.html
‎In this paper‎, ‎the relations between the Yang-Baxter equation and affine actions are explored in detail‎. ‎In particular‎, ‎we classify the injective set-theoretic solutions of the Yang-Baxter equation in two ways‎: ‎(i) by their associated affine actions of‎ ‎their structure groups on their derived structure groups‎, ‎and (ii) by the $C^*$-dynamical systems obtained from their associated‎ ‎affine actions‎. ‎On the way to our main results‎, ‎several‎ ‎other useful results are also obtained‎.Sat, 30 Jun 2018 19:30:00 +0100Characterizing projections among positive operators in the unit sphere
http://www.aot-math.org/article_60341_6058.html
Let $E$ and $P$ be subsets of a Banach space $X$‎, ‎and let us define the unit sphere around $E$ in $P$ as the set $$Sph(E;P)‎ :‎=left{ xin P‎ : ‎|x-b|=1 hbox{ for all } bin E right}.$$ Given a $C^*$-algebra $A$ and a subset $Esubset A,$ we shall write $Sph^‎+ ‎(E)$ or $Sph_A^‎+ ‎(E)$ for the set $Sph(E;S(A^+)),$ where $S(A^+)$ denotes the unit sphere of $A^+$‎. ‎We prove that‎, ‎for every complex Hilbert space $H$‎, ‎the following statements are equivalent for every positive element $a$ in the unit sphere of $B(H)$‎: (a) $a$ is a projection (b) $Sph^+_{B(H)} left( Sph^+_{B(H)}({a}) right) ={a}$. ‎We also prove that the equivalence remains true when $B(H)$ is replaced with an atomic von Neumann algebra or with $K(H_2)$‎, ‎where $H_2$ is an infinite-dimensional and separable complex Hilbert space‎. ‎In the setting of compact operators we establish a stronger conclusion by showing that the identity $$Sph^+_{K(H_2)} left( Sph^+_{K(H_2)}(a) right) =left{ bin S(K(H_2)^+)‎ : ‎!! begin{array}{c}‎s_{_{K(H_2)}} (a) leq s_{_{K(H_2)}} (b)‎, ‎hbox{ and }\‎ ‎textbf{1}-r_{_{B(H_2)}}(a)leq textbf{1}-r_{_{B(H_2)}}(b)‎ ‎end{array}‎right},$$ holds for every $a$ in the unit sphere of $K(H_2)^+$‎, ‎where $r_{_{B(H_2)}}(a)$ and $s_{_{K(H_2)}} (a)$ stand for the range and support projections of $a$ in $B(H_2)$ and $K(H_2)$‎, ‎respectively‎.Sat, 30 Jun 2018 19:30:00 +0100