Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2412.10391

Hahn-Banach type extension results for linear operators on asymmetric normed spaces

S. Cobzaş
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present some results related to Hahn-Banach extension theorem for linear operators on asymmetric normed spaces. L. Nachbin, Trans. Amer. Math. Soc. 68 (1950), proved that a Banach space has the extension property for linear operators (a property also called injectivity) if and only if it has the Binary Intersection Property (BIP), meaning that every family of mutually intersecting closed balls has nonempty intersection. Its analog for quasi-metric spaces, called mixed BIP, was considered by Kemajou et al. Topology Appl. 159 (2012). The equivalence of mixed BIP to the injectivity of an asymmetric normed space was proved by Conradie et al., Topology Appl. 231 (2017), derived from some properties of the injective hull of a quasi-metric space. The aim of the present paper is to give a direct proof of this result by adapting Nachbin's ideas to the asymmetric case. Keywords: quasi-metric space, asymmetric normed space, injectivity, binary intersection property, hyperconvexity, Isbell-completeness, Isbell-convexity

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

S. Cobzaş. 2024-12-01. Hahn-Banach type extension results for linear operators on asymmetric normed spaces. https://arxiv.org/abs/2412.10391

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The Weyl-von Neumann theorem for antilinear skew-self-adjoint operators

In this article, we prove the Weyl--von Neumann theorem for bounded antilinear skew-self-adjoint operators. More specifically, we prove the following: Let $A$ be a bounded antilinear skew-self-adjoint operator on a separable Hilbert space $H$ whose kernel is either even dimensional or infinite dimensional. Let $1 0$ there exists an antilinear block skew-diagonal operator $D$ and an antilinear skew-self-adjoint Schatten $p$-class operator $K$ such that $A=K+D$ with $\|K\|_{p}<ε$. As a consequence, we prove the Weyl--von Neumann theorem for complex skew-symmetric operators: Let $τ$ be a conjugation on $H$ and let $T$ be a bounded linear operator $τ$-skew-symmetric with $\dim N(T)=\infty$ or $\dim N(T)$ is even. Let $1 0$, there exists a $τ$-skew-symmetric Schatten $p$-class operator $K$, a skew-symmetric block diagonal operator $D$ and a unitary operator $U$ such that $T=K+UDU^{tr}$ and $\|K\|_{p}<ε$, where $U^{tr}$ is the transpose of $U$ with respect to an orthonormal basis ${\{e_n:n\in \mathbb N}\}$ such that $τ(e_n)=e_n$ for each $n\in \mathbb N$. Furthermore, the above result holds even without any assumption on the dimension of $N(T)$, provided that $N(T)=N(T^*)$.

math.FA↗
arXiv2026

Schur bounded patterns, submajorisation and operator Lipschitz functions

A Schur bounded pattern is a subset $S\subset \mathbb{N}^2$ such that Schur multiplication by every bounded function on $\mathbb{N}^2$ supported on $S$ defines a bounded linear operator in the norm of $\mathcal{B}(\ell_2(\mathbb{N})).$ Schur bounded patterns were characterised by Davidson-Donsig as being unions of row-bounded and column-bounded sets. We study the analogous question for sets $S$ such that element-wise multiplication by every bounded function on $S$ is bounded in ideals of compact operators that are not closed under submajorisation, in particular the Schatten ideals $\mathcal{L}_p$ with $0<p<1$ and the weak Schatten ideal $\mathcal{L}_{1,\infty}.$ Conversely we characterise the ideals that are not closed under submajorisation by their Schur bounded patterns. This has implications for the functions which are Lipschitz in the norm of ideals that are not closed under submajorisation. In particular such functions must be differentiable and have derivative that is asymptotically constant at infinity.

math.FA↗
arXiv2026

Derivatives of Tensor Products and Applications to Spaces $C(K^n,X)$

In this paper, we develop an abstract theory of derivatives for Banach spaces based on objects that we call \emph{bidual assignments}. This framework encompasses both the Semadeni derivative and the recently introduced Semadeni--Pełczyński derivative. More generally, suitable ideals of subsets of dual spaces give rise to a broad family of derivatives within this setting. We establish direct-sum and tensor-product formulas for these derivatives, showing that they behave naturally with respect to direct sums and injective tensor products. We then obtain explicit descriptions of derivatives associated with compact trees and finite products of compact lines. In particular, we compute iterated derivatives and use them to derive isomorphic invariants for vector-valued spaces of continuous functions. As one consequence, if $K=\prod_{i=1}^n K_i$ and $L=\prod_{j=1}^m L_j$, where $n,m\geq1$ and all the factors are compact lines of uncountable character, then, for $1\leq p,q<\infty$, \[ C(K,\ell_p)\sim C(L,\ell_q) \] implies that $n=m$ and $p=q$. We also establish classification results for spaces of the form $C(K^n,X)$. In particular, for uncountable ordinals $α$ and $β$, an integer $n\geq1$, and Banach spaces $X$ satisfying suitable rigidity assumptions, we prove that \[ C([0,α]^n,X)\sim C([0,β]^n,X) \quad\text{if and only if}\quad C([0,α])\sim C([0,β]). \] This extends Kislyakov's classification of the spaces $C([0,α])$ and its vector-valued extension due to Galego to finite powers of ordinal intervals.

math.FA↗