Advances in Operator TheoryAdvances in Operator Theory
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Tue, 28 Feb 2017 06:51:40 +0100FeedCreatorAdvances in Operator Theory
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Feed provided by Advances in Operator Theory. Click to visit.Fixed point results for a new mapping related to mean nonexpansive mappings.
http://www.aot-math.org/article_41045_4671.html
Mean nonexpansive mappings were first introduced in 2007 by Goebel and Jap'on Pineda and advances have been made by several authors toward understanding their fixed point properties in various contexts. For any given mean nonexpansive mapping of a Banach space, many of the positive results have been derived from knowing that a certain average of some iterates of the mapping is nonexpansive. However, nothing is known about the properties of a mean nonexpansive mapping which has been averaged with the identity. In this paper we prove some fixed point results for a mean nonexpansive mapping which has been composed with a certain average of itself and the identity and we use this study to draw connections to the original mapping.Tue, 28 Feb 2017 20:30:00 +0100The AHSp is inherited by $E$-summands
http://www.aot-math.org/article_41341_4671.html
In this short note we prove that the Approximate Hyperplane Series property (AHSp) is hereditary to $E$-summands via characterizing the $E$-projections.Tue, 28 Feb 2017 20:30:00 +0100Lipschitz properties of convex functions
http://www.aot-math.org/article_41458_4671.html
The present paper is concerned with Lipschitz properties of convex mappings. One considers the general context of mappings defined on an open convex subset $Omega$ of a locally convex space $X$ and taking values in a locally convex space $Y$ ordered by a normal cone.One proves also equi-Lipschitz properties for pointwise bounded families of continuous convexmappings, provided the source space $X$ is barrelled. Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well.The paper has a methodological character - its aim is to show that some geometric properties (monotonicity of the slope, the normality of the seminorms) allow to extend the proofs from the scalar case to the vector one. In this way the proofs become more transparent and natural.Tue, 28 Feb 2017 20:30:00 +0100On the generalized free energy inequality
http://www.aot-math.org/article_41815_4671.html
The generalized free energy inequality known from statistical mechanics is stated in the finite dimension setting and the maximizing matrix is restored. Our approach uses the maximum-entropy inference principle and numerical range methods.Tue, 28 Feb 2017 20:30:00 +0100Various notions of best approximation property in spaces of Bochner integrable functions
http://www.aot-math.org/article_42347_4671.html
We show that a separable proximinal subspace of $X$, say $Y$ is strongly proximinal (strongly ball proximinal) if and only if $L_p(I,Y)$ is strongly proximinal (strongly ball proximinal) in $L_p(I,X)$, for $1leq p<infty$. The $p=infty$ case requires a stronger assumption, that of 'uniform proximinality'. Further, we show that a separable subspace $Y$ is ball proximinal in $X$ if and only if $L_p(I,Y)$ is ball proximinal in $L_p(I,X)$ for $1leq pleqinfty$.We develop the notion of 'uniform proximinality' of a closed convex set in a Banach space, rectifying one that was defined in a recent paper by P.-K Lin et al. [J. Approx. Theory 183 (2014), 72--81]. We also provide several examples having this property; viz. any $U$-subspace of a Banach space has this property. Recall the notion of $3.2.I.P.$ by Joram Lindenstrauss, a Banach space $X$ is said to have $3.2.I.P.$ if any three closed balls which are pairwise intersecting actually intersect in $X$. It is proved the closed unit ball $B_X$ of a space with $3.2.I.P$ and closed unit ball of any M-ideal of a space with $3.2.I.P.$ are uniformly proximinal. A new class of examples are given having this property.Tue, 28 Feb 2017 20:30:00 +0100Complex 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 +0100On the numerical radius of a quaternionic normal operator
http://www.aot-math.org/article_42343_4671.html
We prove that for a right linear bounded normal operator on a quaternionic Hilbert space (quaternionic bounded normal operator) the norm and the numerical radius are equal. As a consequence of this result we give a new proof of the known fact that a non zero quaternionic compact normal operator has a non zero right eigenvalue. Using this we give a new proof of the spectral theorem for quaternionic compact normal operators. Finally, we show that every quaternionic compact operator is norm attaining and prove the Lindenstrauss theorem on norm attaining operators, namely, the set of all norm attaining quaternionic operators is norm dense in the space of all bounded quaternionic operators defined between two quaternionic Hilbert spaces.Tue, 28 Feb 2017 20:30:00 +0100Some lower bounds for the numerical radius of Hilbert space operators
http://www.aot-math.org/article_42504_0.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.Sun, 29 Jan 2017 20:30:00 +0100Trigonometric polynomials over homogeneous spaces of compact groups
http://www.aot-math.org/article_42397_4671.html
This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups.Let $H$ be a closed subgroup of a compact group $G$. Using the abstract notion of dual space $widehat{G/H}$, we introduce the space of trigonometric polynomials $mathrm{Trig}(G/H)$ over the compact homogeneous space $G/H$.As an application for harmonic analysis of trigonometric polynomials, we prove that the abstract dual space of anyhomogeneous space of compact groups separates points of the homogeneous space in some sense.Tue, 28 Feb 2017 20:30:00 +0100On maps compressing the numerical range between $C^*$-algebras
http://www.aot-math.org/article_43297_0.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.Wed, 15 Feb 2017 20:30:00 +0100Normalized tight vs. general frames in sampling problems
http://www.aot-math.org/article_43335_0.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, 17 Feb 2017 20:30:00 +0100Reproducing pairs of measurable functions and partial inner product spaces
http://www.aot-math.org/article_43461_0.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.Tue, 21 Feb 2017 20:30:00 +0100Some results about fixed points in the complete metric space of zero at infinity varieties and ...
http://www.aot-math.org/article_43478_0.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.Wed, 22 Feb 2017 20:30:00 +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 +0100