Hahn-Banach Extension Research Articles

ON SOME PROPERTIES OF BANACH SPACE-VALUED FIBONOCCI SEQUENCE SPACES

In this work, we give some results about the basic properties of the vector-valued Fibanocci sequence spaces. In general, sequence spaces with Banach space-valued cannot have a Schauder Basis unless the terms of the sequences are complex or real terms. Instead, we de ned the concept of relative basis in [11] by generalizing the de nition of a basis in Banach spaces. Using this de nition, we can characterize certain important properties of vector- term Fibanocci sequence spaces, such as separability, Dunford-Pettis Property, approximation property, Radon-Riesz Property and Hahn-Banach extension property.

On Certain Spaces of Ideal Operators

We determine some important spaces of ideal operators and ideal characteristics. Special considerationis given to Frechet spaces, Spaces of finite rank operators and spaces of Hahn-Banach extension operators.The characteristics of ideals and related properties in these spaces as well as in some of their dual spacesare obtained.

On Some Sequence Spaces via q-Pascal Matrix and Its Geometric Properties

We develop some new sequence spaces 𝓁p(P(q)) and 𝓁∞(P(q)) by using q-Pascal matrix P(q). We discuss some topological properties of the newly defined spaces, obtain the Schauder basis for the space 𝓁p(P(q)) and determine the Alpha-(α-), Beta-(β-) and Gamma-(γ-) duals of the newly defined spaces. We characterize a certain class (𝓁p(P(q)),X) of infinite matrices, where X∈{𝓁∞,c,c0}. Furthermore, utilizing the proposed results, we characterize certain other classes of infinite matrices. We also examine some geometric properties, like the approximation property, Dunford–Pettis property, Hahn–Banach extension property, and Banach–Saks-type p property of the spaces 𝓁p(P(q)) and 𝓁∞(P(q)).

On a norming property of subspaces of a Banach space and some applications

We introduce and study m-heavy subspaces of a Banach space and obtain a complete analytic characterization of the same. The importance of m-heavy subspaces in studying the geometry of a Banach space is illustrated by establishing its various applications, including a strengthening of the classical Birkhoff–James orthogonality, duality theory and linear Hahn–Banach extension operators.

On Some Topological and Geometric Properties of Some q-Cesáro Sequence Spaces

Mathematical concepts are aesthetic tools that are useful to create methods or solutions to real-world problems in theory and practice, and that sometimes contain symmetrical and asymmetrical structures due to the nature of the problems. In this study, we investigate whether the sequence spaces Xpq, 0≤p<∞, and X∞, which are constructed by q-Cesáro matrix, satisfy some of the further properties described with respect to the bounded linear operators on them. More specifically, we answer to the question: “Which of these spaces have the Approximation, Dunford-Pettis, Radon–Riesz and Hahn–Banach extension properties?”. Furthermore, we try to investigate some geometric properties such as rotundity and smootness of these spaces.

Uniqueness of Hahn–Banach extensions and some of its variants

In this paper, we analyze the various strengthening and weakening of the uniqueness of the Hahn–Banach extension. In addition, we consider the case in which Y is an ideal of X. In this context, we study the property (U)/(SU)/(HB) and property (wU)/(k-U) for a subspace Y of a Banach space X. We obtain various new characterizations of these properties. We study different kinds of stabilities resulting from these properties in the tensor product spaces, spaces of Bochner integrable functions, and the higher duals of Banach spaces. We discuss various examples in the classical Banach spaces, where the aforementioned properties are satisfied and where they fail. It is established that a hyperplane in \(c_0\) has property (HB) if and only if it is an M-summand. It is observed that a finite-dimensional subspace Y has property (k-U) in \(c_0\), in addition to that if Y is an ideal, then \(Y^*\) is a k-strictly convex subspace of \(\ell _1\) for some natural k.

On Hahn-Banach theorem and some of its applications

Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Von Neumann algebra conditional expectations with applications to generalized representing measures for noncommutative function algebras

We establish several deep existence criteria for conditional expectations on von Neumann algebras, and then apply this theory to develop a noncommutative theory of representing measures of characters of a function algebra. Our main cycle of results describes what may be understood as a ‘noncommutative Hoffman-Rossi theorem’ giving the existence of weak* continuous ‘noncommutative representing measures’ for so-called D-characters. These results may also be viewed as ‘module’ Hahn-Banach extension theorems for weak* continuous ‘characters’ into possibly noninjective von Neumann algebras. In closing we introduce the notion of ‘noncommutative Jensen measures’, and show that as in the classical case representing measures of logmodular algebras are Jensen measures. The proofs of the two main cycles of results rely on the delicate interplay of Tomita-Takesaki theory, noncommutative Radon-Nikodym derivatives, Connes cocycles, Haagerup noncommutative Lp-spaces, Haagerup's reduction theorem, etc.

Representation of uniform boundedness principle and Hahn–Banach theorem in linear n-normed space

The concept of b-linear functional and its different types of continuity in linear n-normed space are presented and some of their properties are being established. We derive the Uniform Boundedness Principle and Hahn-Banach extension Theorem with the help of bounded b-linear functionals in the case of linear n-normed spaces and discuss some examples and applications. Finally, we present the concept of weak*convergence for the sequence of bounded b-linear functionals in linear n-normed space.

Stability of unique Hahn–Banach extensions and associated linear projections

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in and when this association forms a linear operator of norm-1 from to . It is proved that, under certain geometric assumptions on X, Y, Z, these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if has property-U (SU) in . We prove that when has the Radon–Nikodým Property for 1 < p < ∞, L p (μ, Y) has property-U (property-SU) in L p (μ, X) if and only if Y is so in X. We show that if Z⊆ Y⊆ X and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and Z (⊆ Y) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in c 0 which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of c 0 which have property-U (SU).

From the Hahn–Banach extension theorem to the isotonicity of convex functions and the majorization theory

The property of isotonicity of a continuous convex function on the positive cone is characterized via subdifferentials. This is used to illustrate a new generalization of the Hardy–Littlewood–Polya inequality of majorization to the case of functions of a vector variable.

Amenability of representations and invariant Hahn–Banach theorems

Invariant Hahn–Banach theorems are proved in the context of right amenable, weakly almost periodic representations of semigroups on locally convex spaces. Applications are made to invariant means, invariant measures, and to Banach limits on weakly almost periodic flows. Additionally, invariant linear Hahn–Banach extension operators are shown to exist on suitable invariant subspaces of Banach spaces. The key construct on which these results depend is the lifting of a representation on a semigroup S to its coefficient compactification, the minimal compactification of S for which the lift is injective.

Amenability and Hahn-Banach extension property for set-valued mappings

Amenability is an important notion in harmonic analysis on groups and semigroups, and their associated Banach algebras. In this paper, we present some characterization of a semitopological semigroup $S$ on the existence of a left invariant mean on $\text{\rm LUC}(S)$, $\text{\rm AP}(S)$ and $\text{\rm WAP}(S)$ in terms of Hahn-Banach extension theorem, which extend the first author's early results in 1970s. Moreover, we refine and extend the well known Day's result and Mitchell's results on fixed point properties for set-valued mappings. As an application, we give an application of our result to a class of Banach algebras related to amenability of groups and semigroups.

Linear Hahn-Banach type extension operators in Banach algebras of operators

Linear Hahn-Banach type extension operators in Banach algebras of operators

Hahn-Banach extension theorem for interval-valued functions and existence of quasilinear functionals

Hahn-Banach extension theorem for interval-valued functions and existence of quasilinear functionals

Amenability and Fan–Glicksberg theorem for set-valued mappings

In this paper, we begin by discussion of some well known results on the existence of left invariant means in the spaces: LUC(S), AP(S) and W AP(S) with Hahn-Banach extension theorem. We then give a new and precise proof of the well known Fan–Glicksberg fixed point theorem. This is then followed by a discussion on some related open problems.

Uniqueness of the weights in Harsanyi-type theorems

A necessary and sufficient condition for the uniqueness of the weights in Harsanyi-type aggregation theorems is obtained as an application of Phelps’ theorem on the uniqueness of Hahn–Banach extensions.

Global solutions in the critical Besov space for the non-cutoff Boltzmann equation

The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin–Lerner space. In order to analyze the collisional term of the equation, a Chemin–Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn–Banach extension theorem, the desired solution is obtained.

Subdifferential calculus for invariant linear ordered vector space-valued operators and applications

We give a direct proof of sandwich-type theorems for linear invariant partially ordered vector space operators in the setting of convexity. As consequences, we deduce equivalence results between sandwich, Hahn-Banach, separation and Krein-type extension theorems, Fenchel duality, Farkas and Kuhn-Tucker-type minimization results and subdifferential formulas in the context of invariance. As applications, we give Tarski-type extension theorems and related examples for vector lattice-valued invariant probabilities, defined on suitable kinds of events.

The Hahn–Banach theorem almost everywhere

The aim of this work is to present an almost everywhere version of the Hahn–Banach extension theorem.