Advances in Operator TheoryAdvances in Operator Theory
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Sat, 25 Mar 2017 04:24:17 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Complex interpolation and non-commutative integration
http://www.aot-math.org/article_42356_0.html
We will show that under suitable conditions interpolation between a Banach space and its dual yields a Hilbert space at θ = 1/2. By application of this result to the special case of the non-commutative Lp-spaces of Leinert and Terp we conclude that L2 is a Hilbert space and that Lp is isometrically isomorphic to the dual of Lq without using the isomorphisms of these spaces to Hilsum’s and Haagerup’s Lp-spaces. U. Haagerup, G. Pisier, and F. Watbled give conditions under which interpolation between a Banach space and its conjugate dual yields a Hilbert space at 1/2. The result mentioned above when put in “conjugate form” extends their results.Thu, 26 Jan 2017 20:30:00 +0100Some lower bounds for the numerical radius of Hilbert space operators
http://www.aot-math.org/article_42504_4671.html
We show that if $T$ is a bounded linear operator on a complex Hilbert space, thenbegin{equation*}frac{1}{2}Vert TVertleq sqrt{frac{w^2(T)}{2} + frac{w(T)}{2}sqrt{w^2(T) - c^2(T)}} leq w(T),end{equation*}where $w(cdot)$ and $c(cdot)$ are the numerical radius and the Crawford number, respectively.We then apply it to prove that for each $tin[0, frac{1}{2})$ and natural number $k$,begin{equation*}frac{(1 + 2t)^{frac{1}{2k}}}{{2}^{frac{1}{k}}}m(T)leq w(T),end{equation*}where $m(T)$ denotes the minimum modulus of $T$. Some other related results are also presented.Fri, 31 Mar 2017 19:30:00 +0100On maps compressing the numerical range between $C^*$-algebras
http://www.aot-math.org/article_43297_4671.html
In this paper, we deal with the problem of characterizing linear maps compressing the numerical range. Acounterexample is given to show that such a map need not be a Jordan *-homomorphism in general even if the C*-algebras are commutative. Next, under an auxiliary condition we show that such a map is a Jordan *-homomorphism.Fri, 31 Mar 2017 19:30:00 +0100Normalized tight vs. general frames in sampling problems
http://www.aot-math.org/article_43335_4671.html
We present a new approach to sampling theory using the operator theory framework. We use a replacement operator and replace general frames of the sampling and reconstruction subspaces by normalized tight frames. The replacement can be done in a numerically stable and efficient way. The approach enables us to unify the standard consistent reconstruction results with the results for quasiconsistent reconstruction. Our approach naturally generalizes to consistent and quasiconsistent reconstructions from several samples. Not only we can handle sampling problems in a more efficient way, we also answer questions that seem to be open so far.Fri, 31 Mar 2017 19:30:00 +0100Reproducing pairs of measurable functions and partial inner product spaces
http://www.aot-math.org/article_43461_4671.html
We continue the analysis of reproducing pairs of weakly measurable functions, which generalize continuous frames. More precisely, we examine the case where the defining measurable functions take their values in a partial inner product space (PIP spaces). Several examples, both discrete and continuous, are presented.Fri, 31 Mar 2017 19:30:00 +0100Some results about fixed points in the complete metric space of zero at infinity varieties and ...
http://www.aot-math.org/article_43478_4671.html
‎This paper aims to study fixed points in the complete metric space ofvarieties which are zero at infinity as a subspace of the complete metric space of allvarieties. Also, the convex structure of the complete metric space of all varietieswill be introduced.Fri, 31 Mar 2017 19:30:00 +01002-Local derivations on matrix algebras and algebras of measurable operators
http://www.aot-math.org/article_43482_0.html
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.Thu, 23 Feb 2017 20:30:00 +0100Direct estimates of certain Mihesan-Durrmeyer type operators
http://www.aot-math.org/article_43785_4671.html
In this note we consider a Durrmeyer type operator having the basis functions in summation and integration due to Mihec{s}an [Creative Math. Inf. 17 (2008), 466--472.] and Pv{a}ltv{a}nea [Carpathian J. Math. 24 (2008), no. 3, 378--385.] that preserve the linear functions. We present a Voronovskaja type theorem, local approximation theorem by means of second order modulus of continuity and weighted approximation for these operators. In the last section of the paper, we obtain the rate of approximation for absolutely continuous functions having a derivative equivalent with a function of bounded variation.Fri, 31 Mar 2017 19:30:00 +0100Semicontinuity and closed faces of C*-algebras
http://www.aot-math.org/article_43918_0.html
C. Akemann and G.K. Pedersen [Duke Math. J. 40 (1973), 785--795.] defined three concepts of semicontinuity for self-adjoint elements of $A^{**}$, the enveloping von Neumann algebra of a $C^*$-algebra $A$. We give the basic properties of the analogous concepts for elements of $pA^{**}p$, where $p$ is a closed projection in $A^{**}$. In other words, in place of affine functionals on $Q$, the quasi--state space of $A$, we consider functionals on $F(p)$, the closed face of $Q$ suppported by $p$. We prove an interpolation theorem: If $hgeq k$, where $h$ is lower semicontinuous on $F(p)$ and $k$ upper semicontinuous, then there is a continuous affine functional $x$ on $F(p)$ such that $kleq xleq h$. We also prove an interpolation--extension theorem: Now $h$ and $k$ are given on $Q$, $x$ is given on $F(p)$ between $h_{|F(p)}$ and $k_{|F(p)}$, and we seek to extend $x$ to $widetilde x$ on $Q$ so that $kleqwidetilde xleq h$. We give a characterization of $pM(A)_{{text{sa}}}p$ in terms of semicontinuity. And we give new characterizations of operator convexity and strong operator convexity in terms of semicontinuity.Fri, 03 Mar 2017 20:30:00 +0100The closure of ideals of $\ell^1(\Sigma)$ in its enveloping $\mathrm{C}^*$-algebra
http://www.aot-math.org/article_44047_0.html
If~$X$ is a compact Hausdorff space and~$sigma$ is a homeomorphism of~$X$, then an involutive Banach algebra $ell^1(Sigma)$ of crossed product type is naturally associated with the topological dynamical system $Sigma=(X,sigma)$. We initiate the study of the relation between two-sided ideals of $ell^1(Sigma)$ and ${mathrm C}^ast(Sigma)$, the enveloping $mathrm{C}^ast$-algebra ${mathrm C}(X)rtimes_sigmamathbb Z$ of $ell^1(Sigma)$. Among others, we prove that the closure of a proper two-sided ideal of $ell^1(Sigma)$ in ${mathrm C}^ast(Sigma)$ is again a proper two-sided ideal of ${mathrm C}^ast(Sigma)$.Tue, 07 Mar 2017 20:30:00 +0100On spectral synthesis in several variables
http://www.aot-math.org/article_44065_4671.html
In a recent paper we proposed a possible generalization of L. Schwartz's classical spectral synthesis result for continuous functions in several variables. The idea is based on Gelfand pairs and spherical functions while "translation invariance" is replaced by invariance with respect to the action of affine groups. In this paper we describe the function classes which play the role of the exponential monomials in this setting.Fri, 31 Mar 2017 19:30:00 +0100Positive map as difference of two completely positive or super-positive maps
http://www.aot-math.org/article_44116_0.html
For a linear map from ${mathbb M}_m$ to ${mathbb M}_n$, besides the usual positivity, there are two stronger notions, complete positivity and super positivity. Given a positive linear map $varphi$ we study a decomposition $varphi = varphi^{(1)} - varphi^{(2)}$ with completely positive linear maps $varphi^{(j)} (j = 1,2)$. Here $varphi^{(1)} + varphi^{(2)}$ is of simple form with norm small as possible. The same problem is discussed with super-positivity in place of complete positivity.Fri, 10 Mar 2017 20:30:00 +0100On the weak compactness of Weak* Dunford--Pettis operators on Banach lattices
http://www.aot-math.org/article_44450_0.html
We characterize Banach lattices on which each positive weak* Dunford--Pettis operator is weakly (resp., M-weakly, resp., order weakly) compact. More precisely, we prove that if $F$ is a Banach lattice with order continuous norm, then each positive weak* Dunford--Pettis operator $T : Elongrightarrow F$ is weakly compact if, and only if, the norm of $E^{prime}$ is order continuous or $F$ is reflexive. On the other hand, when the Banach lattice $F$ is Dedekind $sigma$-complete, we show that every positive weak* Dunford--Pettis operator $T: Elongrightarrow F$ is M-weakly compact if, and only if, the norms of $E^{prime}$ and $F$ are order continuous or $E$ is finite-dimensional.Thu, 16 Mar 2017 20:30:00 +0100Two-weight norm inequalities for the higher-order commutators of fractional integral operator
http://www.aot-math.org/article_44490_0.html
In this paper, we obtain several sufficient conditions such that the higher-order commutators $I_{alpha,b}^m$ generated by $I_alpha$ and $bin textrm{BMO}(mathbb{R}^n)$ is bounded from $L^p(v)$ to $L^q(u)$, where $frac{1}{q}=frac{1}{p}-frac{alpha}{n}$ and $0<alpha<n$.Mon, 20 Mar 2017 20:30:00 +0100Properties of $J$-fusion frames in Krein spaces
http://www.aot-math.org/article_44491_0.html
In this article we introduce the notion of $J$-Parseval fusion frames in a Krein space $mathbb{K}$ and characterize 1-uniform $J$-Parseval fusion frames with $zeta=sqrt{2}$. We provide some results regarding construction of new $J$-tight fusion frame from given $J$-tight fusion frames. We also characterize an uniformly $J$-definite subspace of a Krein space $mathbb{K}$ in terms of $J$-fusion frame. Finally we generalize the fundamental identity of Hilbert space frames in the setting of Krein spaces.Mon, 20 Mar 2017 20:30:00 +0100