Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101Fixed point results for a new mapping related to mean nonexpansive mappings.1164104510.22034/aot.1610.1029ENTorrey MGallagherBucknell UniversityJournal Article20161012Mean 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.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101The AHSp is inherited by $E$-summands17204134110.22034/aot.1610.1033ENFrancisco Garcia-PachecoUniversity of Cadiz0000-0001-6208-6071Journal Article20161015In this short note we prove that the Approximate Hyperplane Series property (AHSp) is hereditary to $E$-summands via characterizing the $E$-projections.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101Lipschitz properties of convex functions21494145810.22034/aot.1610.1022ENStefan CobzasBabes-Bolyai University,
Department of MathematicsJournal Article20161003The 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.<br />One proves also equi-Lipschitz properties for pointwise bounded families of continuous convex<br />mappings, provided the source space $X$ is barrelled.<br /> Some results on Lipschitz properties of continuous convex functions defined on metrizable topological vector spaces are included as well.<br />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.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101On the generalized free energy inequality50584181510.22034/aot.1610.1041ENNatalia BebianoUniversidade de CoimbraJoao da ProvidenciaJournal Article20161028The 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.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101Various notions of best approximation property in spaces of Bochner integrable functions59774234710.22034/aot.1611-1052ENTanmoy PaulJournal Article20161106We 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 $1leq pleqinfty$.<br />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.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101On the numerical radius of a quaternionic normal operator78864234310.22034/aot.1611-1060ENRamesh GollaIIT HyderabadJournal Article20161122We 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.Tusi Mathematical Research Group (TMRG)Advances in Operator Theory2538-225X2120170101Trigonometric polynomials over homogeneous spaces of compact groups87974239710.22034/aot.1701-1090ENArash Ghaani FarashahiJournal Article20170109This paper presents a systematic study for trigonometric polynomials over homogeneous spaces of compact groups.<br />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 $G/H$.<br />As an application for harmonic analysis of trigonometric polynomials, we prove that the abstract dual space of any<br />homogeneous space of compact groups separates points of the homogeneous space in some sense.