2017
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Fixed point results for a new mapping related to mean nonexpansive mappings.
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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.
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1
16


Torrey
Gallagher
Bucknell University
Bucknell University
USA
torreymg@gmail.com
Mean nonexpansive
Fixed point
approximate fixed point sequence
nonexpansive
nonlinear operator
The AHSp is inherited by $E$summands
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2
In this short note we prove that the Approximate Hyperplane Series property (AHSp) is hereditary to $E$summands via characterizing the $E$projections.
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17
20


Francisco
GarciaPacheco
University of Cadiz
University of Cadiz
Spain
garcia.pacheco@uca.es
projection
complemented
normattaining
Lipschitz properties of convex functions
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2
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 equiLipschitz 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.
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21
49


Stefan
Cobzas
BabesBolyai University,
Department of Mathematics
BabesBolyai University,
Department of Mathematic
USA
scobzas@math.ubbcluj.ro
convex function
convex operator
Lipschitz property
normal cone
normed lattice
On the generalized free energy inequality
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2
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 maximumentropy inference principle and numerical range methods.
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50
58


Natalia
Bebiano
Universidade de Coimbra
Universidade de Coimbra
Portugal
bebiano@mat.uc.pt


Joao
da Providencia
Portugal
providencia@teor.fis.uc.pt
Maximumentropy inference
generalized free energy inequality
von Neumann entropy
Numerical Range
Various notions of best approximation property in spaces of Bochner integrable functions
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2
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), 7281]. 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 Mideal of a space with $3.2.I.P.$ are uniformly proximinal. A new class of examples are given having this property.
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59
77


Tanmoy
Paul
India
tanmoy@iith.ac.in
$L_p(I
X)$
proximinality
strong proximinality
ball proximinality
On the numerical radius of a quaternionic normal operator
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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.
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78
86


Ramesh
Golla
IIT Hyderabad
IIT Hyderabad
India
rameshg@iith.ac.in
Quaternionic Hilbert space
compact operator
right eigenvalue
norm attaining operator
Lindenstrauss theorem
Trigonometric polynomials over homogeneous spaces of compact groups
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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.
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87
97


Arash
Ghaani Farashahi
Austria
arash.ghaani.farashahi@univie.ac.at
Compact homogeneous space
compact group
dual space
unitary representation
irreducible representation
trigonometric polynomials