Advances in Operator TheoryAdvances in Operator Theory
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Mon, 23 Jan 2017 03:34:40 +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_42172_0.html
We derive that a separable proximinal subspace $Y$ of $X$, $Y$ is strongly proximinal (strongly ball proximinal) if and only if $1leq p<infty$, $L_p(I,Y)$ is strongly proximinal (strongly ball proximinal) in $L_p(I,X)$. Case for $p=infty$ follows from stronger assumption on $Y$ in $X$ (uniform proximinality). It is observed that a separable proximinal subspace $Y$ in $X$, $Y$ is ball proximinal in $X$ if and only if $L_p(I,Y)$ is ball proximinal in $L_p(I,X)$ for $1leq pleqinfty$; this observation also extends to that for any (strongly) proximinal subspace $Y$ of $X$, if every separable subspace of $Y$ is ball (strongly) proximinal in $X$ then $L_p(I,Y)$ is ball (strongly) proximinal in $L_p(I,X)$ for $1leq p<infty$.We introduce the notion of uniform proximinality of a closed convex set in a Banach space, which is wrongly defined in a recent paper by P.-K Lin et al[J. Approx. Theory 183 (2014), 72--81]. Several examples are given 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.Fri, 20 Jan 2017 20:30:00 +0100