arXiv subjects

Francisco L. Hernández

Publications and source records attributed to Francisco L. Hernández.

5 recordsLinked to original sources

arXiv2024

Disjointly strictly singular inclusions between variable Lebesgue spaces

Disjointly strictly singular inclusions between variable Lebesgue spaces $L^{p(\cdot)}(μ)$ on finite measure are characterized. Suitable criteria in terms of the (bounded or unbounded) exponents are given. It is proved the equivalence of $L$-weak compactness (also called almost compactness) and disjoint strict singularity for variable Lebesgue space inclusions. For infinite measure any inclusion $L^{p(\cdot)}(μ) \hookrightarrow L^{q(\cdot)}(μ)$ is not disjointly strictly singular. No restrictions on the exponent are imposed.

math.FA↗

arXiv2020

Strictly singular non-compact operators between $L_p$ spaces

We study the structure of strictly singular non-compact operators between $L_p$ spaces. Answering a question raised in [Adv. Math. 316 (2017), 667-690], it is shown that there exist operators $T$, for which the set of points $(\frac1p,\frac1q)\in(0,1)\times (0,1)$ such that $T:L_p\rightarrow L_q$ is strictly singular but not compact contains a line segment in the triangle $\{(\frac1p,\frac1q):1<p<q<\infty\}$ of any positive slope. This will be achieved by means of Riesz potential operators between metric measure spaces with different Hausdorff dimension. The relation between compactness and strict singularity of regular operators defined on subspaces of $L_p$ is also explored.

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arXiv2019

On the structure of variable exponent spaces

The first part of this paper surveys several results on the lattice structure of variable exponent Lebesgue function spaces (or Nakano spaces) $\lpv$. In the second part strictly singular and disjointly strictly singular operators between spaces $\lpv$ are studied. New results on the disjoint strict singularity of the inclusions $ L^{p(\cdot)}(Ω) \hookrightarrow L^{q(\cdot)}(Ω)$ are given.

math.FA↗

arXiv2017

Interpolation and extrapolation of strictly singular operators between $L_p$ spaces

We study the interpolation and extrapolation properties of strictly singular operators between different $L_p$ spaces. To this end, the structure of strictly singular non-compact operators between $L_p-L_q$ spaces is analyzed. Among other things, we clarify the relation between strict singularity and the $L$-characteristic set of an operator. In particular, Krasnoselskii's interpolation theorem for compact operators is extended to the class of strictly singular operators.

math.FA↗

arXiv2015

Disjointly homogeneous Banach lattices and applications

This is a survey on disjointly homogeneous Banach lattices and their applicactions. Several structural properties of this class are analyzed. In addition we show how these spaces provide a natural framework for studying the compactness of powers of operators allowing for a unified treatment of well-known results.

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