arXiv subjects

Daniel Akech Thiong

Publications and source records attributed to Daniel Akech Thiong.

6 recordsLinked to original sources

arXiv2026

$H$-Operator Approximation Spaces via Delayed Riesz Means and Quasi-Banach Moduli

We establish a constructive and quasi-Banach framework for approximation spaces $A_μ^ρ$ of compact $H$-operators between Banach and quasi-Banach spaces. By leveraging delayed Riesz means $V_{2^{bn}}$ generated by self-adjoint differential operators $P(D)$, we replace abstract best approximants with explicit linear operator decompositions. Furthermore, we extend the theory of $H$-operator approximation spaces to quasi-Banach settings ($0 < p < 1$), bypassing the collapse of Peetre's $K$-functional by employing localized moduli of smoothness $ω_φ^r(f,t)_p$. Integrating seminal spectral bounds due to Markus \cite{M1966} and factorization techniques for operator ideals, we prove that operators belonging to $A_μ^ρ$ possess complete systems of root vectors whose expansions are Abel summable.

math.FA↗

arXiv2026

Local Reflexivity and Duality for Subspace Approximation Schemes

We address a structural gap in the quantitative theory of operator approximation by investigating how subspace approximation schemes interact with local reflexivity. Recent literature establishing the duality of scheme-relative approximation numbers, $a_n(T,Q) = a_n(T^{**}, Q^{\perp\perp})$, has relied on imposing an Extended Local Reflexivity Property (ELRP) as an independent axiom. In this note, we establish a dichotomy. For infinite-dimensional admissible schemes, we prove they naturally generate complete nests, perfectly satisfying the topological hypotheses of the Oja-Veidenberg Nest Principle of Local Reflexivity. Conversely, we demonstrate that for approximation schemes with finite-dimensional components, the bidual geometry collapses. Consequently, the ELRP is entirely superfluous, and the duality of approximation numbers holds unconditionally for all bounded linear operators via an elementary isometric restriction.

math.FA↗

arXiv2026

Measuring the Infinite: Interpolation Theory, Lorentz Spaces, and Dispersive PDEs

This article explores the vital role of interpolation theory and Lorentz spaces in the rigorous analysis of linear operators. While classical Lebesgue spaces ($L_{p}$) successfully measure the magnitude of functions, they frequently fail to bound evolution operators at critical endpoints of $p=1$ or $p = \infty$ because they conflate a function's amplitude with its spatial spread. To resolve this analytic bottleneck, we introduce distribution functions and decreasing rearrangements, culminating in the construction of Lorentz spaces ($L_{p, q}$). By utilizing the Complex (Riesz-Thorin), Real (Peetre's K-functional), and Marcinkiewicz methods of interpolation, these highly sensitive intermediate spaces act as geometric bridges between endpoint extremes. We conclude by applying this abstract framework to two distinct illustrative models: deriving the continuous smoothing decay of the parabolic Heat equation, and establishing the foundational dispersive Strichartz estimates for the dispersive free Schrödinger equation.

math.AP↗

arXiv2023

Equality in Degrees of Compactness: Schauder's Theorem and s-numbers

We investigate an extension of Schauder's theorem by studying the relationship between various $s$-numbers of an operator $T$ and its adjoint $T^*$. We have three main results. First, we present a new proof that the approximation number of $T$ and $T^*$ are equal for compact operators. Second, for non-compact, bounded linear operators from $X$ to $Y$, we obtain a relationship between certain $s$-numbers of $T$ and $T^*$ under natural conditions on $X$ and $Y$. Lastly, for non-compact operators that are compact with respect to certain approximation schemes, we prove results by comparing the degree of compactness of $T$ with that of its adjoint $T^*$.

math.FA↗

arXiv2023

Approximation spaces for H-operators

This paper defines and establishes relations among approximation spaces of certain operators called \textit{H-operators}, which generalize the notion of self-adjoint to Banach spaces.

math.FA↗

arXiv2022

Schauder's Theorem and s-Numbers

Motivated by the well known theorem of Schauder, we study the relationship between various s-numbers of an operator T and its adjoint T* between Banach spaces.

math.FA↗