Bishop-Phelps-Bolloba's theorem on bounded closed convex sets

This paper deals with the \emph{Bishop-Phelps-Bollob\'as property} (\emph{BPBp} for short) on bounded closed convex subsets of a Banach space $X$, not just on its closed unit ball $B_X$. We firstly prove that the \emph{BPBp} holds for bounded linear functionals on arbitrary bounded closed convex subsets of a real Banach space. We show that for all finite dimensional Banach spaces $X$ and $Y$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed convex subset $D$ of $X$, and also that for a Banach space $Y$ with property $(\beta)$ the pair $(X,Y)$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of an arbitrary Banach space $X$. For a bounded closed absorbing convex subset $D$ of $X$ with positive modulus convexity we get that the pair $(X,Y)$ has the \emph{BPBp} on $D$ for every Banach space $Y$. We further obtain that for an Asplund space $X$ and for a locally compact Hausdorff $L$, the pair $(X, C_0(L))$ has the \emph{BPBp} on every bounded closed absolutely convex subset $D$ of $X$. Finally we study the stability of the \emph{BPBp} on a bounded closed convex set for the $\ell_1$-sum or $\ell_{\infty}$-sum of a family of Banach spaces

Isometries on spaces of absolutely continuous functions in a noncompact framework

In this paper we deal with surjective linear isometries between spaces of scalar-valued absolutely continuous functions on arbitrary (not necessarily closed or bounded) subsets of the real line (with at least two points). As a corollary, it is shown that when the underlying spaces are connected, each surjective linear isometry of these function spaces is a weighted composition operator, a result which generalizes all the previous known results concerning such isometries

Mean Li-Yorke chaos in Banach spaces

We investigate the notion of mean Li-Yorke chaos for operators on Banach spaces. We show that it differs from the notion of distributional chaos of type 2, contrary to what happens in the context of topological dynamics on compact metric spaces. We prove that an operator is mean Li-Yorke chaotic if and only if it has an absolutely mean irregular vector. As a consequence, absolutely Ces\`aro bounded operators are never mean Li-Yorke chaotic. Dense mean Li-Yorke chaos is shown to be equivalent to the existence of a dense (or residual) set of absolutely mean irregular vectors. As a consequence, every mean Li-Yorke chaotic operator is densely mean Li-Yorke chaotic on some infinite-dimensional closed invariant subspace. A (Dense) Mean Li-Yorke Chaos Criterion and a sufficient condition for the existence of a dense absolutely mean irregular manifold are also obtained. Moreover, we construct an example of an invertible hypercyclic operator $T$ such that every nonzero vector is absolutely mean irregular for both $T$ and $T^{-1}$. Several other examples are also presented. Finally, mean Li-Yorke chaos is also investigated for $C_0$-semigroups of operators on Banach spaces.Comment: 26 page

Normality of spaces of operators and quasi-lattices

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which strictly contains the Banach lattices, and we prove that every strictly convex reflexive ordered Banach space with a closed proper generating cone is a quasi-lattice. These spaces provide a large class of examples $X$ and $Y$ that are not Banach lattices, but for which $B(X,Y)$ is normal. In particular, we show that a Hilbert space $\mathcal{H}$ endowed with a Lorentz cone is a quasi-lattice (that is not a Banach lattice if $\dim\mathcal{H}\geq3$), and satisfies an identity analogous to the elementary Banach lattice identity $\||x|\|=\|x\|$ which holds for all elements $x$ of a Banach lattice. This is used to show that spaces of operators between such ordered Hilbert spaces are always absolutely monotone and that the operator norm is positively attained, as is also always the case for spaces of operators between Banach lattices.Comment: Minor typos fixed. Exact solution now provided in Example 5.10. To appear in Positivit

A class of spaces containing all generalized absolutely closed (almost compact) spaces

[EN] The class of θ-compact spaces is introduced which properly contains the class of almost compact (generalized absolutely closed) spaces and is strictly contained in the class of quasicompact spaces. In the realm of almost regular spaces, the class of θ-compact spaces coincides with the class of nearly compact spaces. Moreover, an almost regular θ-compact space is mildly normal (= K-normal). A θ-closed, θ-embedded subset of a θ-compact space is θ-compact and the product of two θ-compact space is θ-compact if one of them is compact. A (strongly) θ-continuous image of a θ-compact space is θ-compact (compact). A space is compact if and only if it is θ-compact and θ-point paracompact.Kohli, J.; Das, A. (2006). A class of spaces containing all generalized absolutely closed (almost compact) spaces. Applied General Topology. 7(2):233-244. doi:10.4995/agt.2006.1926.SWORD2332447

Internal Neighbourhood Structures II: Closure and closed morphisms

Internal preneighbourhood spaces were initiated in \cite{2020}{.} The present paper introduces a closure operator on an internal preneighbourhood space of a finitely complete category with finite coproducts and a proper $(\mathsf{E}, \mathsf{M})$ system. The closure operator is shown to be grounded, idempotent, additive (if every filter of admissible subobjects is contained in a prime filter), hereditary, transitive, and satisfy \emph{finite structure preservation property} whenever product projections are $\mathsf{E}$-morphisms. The closure operator agrees with the usual closure operators for topological spaces and locales. The paper discuss closed morphisms, dense morphisms, proper morphisms, separated morphisms and perfect morphisms. Alongwith the paper introduces special classes of internal preneighbourhood spaces, namely the compact spaces, Hausdorff spaces, compact Hausdorff spaces, Tychonoff spaces and absolutely closed spaces