arXiv subjects

Sainik Karak

Publications and source records attributed to Sainik Karak.

4 recordsLinked to original sources

arXiv2026

Unique Hahn-Banach extensions in Bochner $L^p$ spaces

Let $Y$ be a closed subspace of a real Banach space $X$. Let $(Ω,Σ,μ)$ be a complete finite measure space and let $1<p<\infty$. We prove that $Y$ has property-$(U)$, property-$(SU)$, property-$(wU)$ in $X$ if and only if $L^p(μ,Y)$ has the corresponding property in $L^p(μ,X)$. No Radon-Nikodým assumption on $X^*$ is required. The proofs use $w^*$-measurable representations of dual functionals and a simultaneous scalar-lifting construction. A norm-minimal representation for an element in $L^p(μ,X)^*$ is found suitable in this case. We also study the point of continuity of the identity map under different topologies on the unit sphere of $L^p(μ,X)^*$. For Bochner-integrable functionals, the pointwise $w^*-w$-continuity yields the uniqueness of the extension to $L^p(μ,X^{**})$. Examples show why arbitrary $w^*$-measurable representatives cannot replace the norm-minimal representatives in this context.

math.FA↗

arXiv2026

On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces

This article aims to examine the Hahn-Banach smoothness of Banach spaces and its connections to various geometrical aspects. We examine the circumstances that allow linear functionals to have unique norm-preserving extensions, with particular attention to the behavior of these properties in $L_1$-preduals and in spaces of affine continuous functions. Banach spaces which are $L_1$-preduals and also Hahn-Banach smooth are completely characterized. It is demonstrated that if $X$ is an $M$-embedded space then $X^*$ admits a predual which is not weakly Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ and also represents a split face, no subspace of $A(S)$ retains the property-$(wU)$ in $A(S)^{**}$. Furthermore, when $X=C_0(L)$, in the context of a locally compact Hausdorff space $L$, the continuity of the identity mapping $I:(B_{X^*},w^*)\to (B_{X^*},w)$ in $ext (B_{X^*})$ significantly influences the subspaces of $X$ that have unique extension property in $X^{**}$. Collectively, this study provides structural characterizations of specialized geometric property, so called Hahn-Banach smoothness, and offers solutions to some natural problems enlisted at the beginning that involve spaces that are $L_1$-preduals and also spaces that are $M$-embedded.

math.FA↗

arXiv2026

Extension of $p$-compact operators in Banach spaces

We analyze various consequences in relation to the extension of operators $T:X\to Y$ that are $p$-compact, as well as the extension of operators $T:X\to Y$ whose adjoints $T^*:Y^*\to X^*$ are $p$-compact. In most cases, we discuss these extension properties when the underlying spaces, either domain or codomain, are $P_λ$ spaces. We also answer if these extensions are almost norm-preserving in such circumstances where the extension $\widetilde{T}$ of a $T$ exists. It is observed that an operator can often be extended to a larger domain when the codomain is appropriately extended as well. Specific assumptions might enable us to obtain an extension of an operator that maintains the same range. Necessary and sufficient conditions are derived for a Banach space to be $L_1$-predual.

math.FA↗

arXiv2025

Uniqueness of Hahn-Banach extensions in locally convex spaces

We intend to study the uniqueness of the Hahn-Banach extensions of linear functionals on a subspace in locally convex spaces. Various characterizations are derived when a subspace $Y$ has an analogous version of property-U (introduced by Phelps) in a locally convex space, referred to as the property-SNP. We characterize spaces where every subspace has this property. It is demonstrated that a subspace $M$ of a Banach space $E$ has property-U if and only if the subspace $M$ of the locally convex space $E$ endowed with the weak topology has the property-SNP, mentioned above. This investigation circles around exploring the potential connections between the family of seminorms and the unique extension of functionals previously mentioned. We extensively studied this property on the spaces of continuous functions on Tychonoff spaces endowed with the topology of pointwise convergence.

math.FA↗