arXiv subjects

Richard J. Smith

Publications and source records attributed to Richard J. Smith.

28 records · Page 2Linked to original sources

arXiv2018

Topology, isomorphic smoothness and polyhedrality in Banach spaces

In recent decades, topology has come to play an increasing role in some geometric aspects of Banach space theory. The class of so-called $w^*$-locally relatively compact sets was introduced recently by Fonf, Pallares, Troyanski and the author, and were found to be a useful topological tool in the theory of isomorphic smoothness and polyhedrality in Banach spaces. We develop the topological theory of these sets and present some Banach space applications.

math.FA↗

arXiv2017

Enhanced sensing and conversion of ultrasonic Rayleigh waves by elastic metasurfaces

Recent years have heralded the introduction of metasurfaces that advantageously combine the vision of sub- wavelength wave manipulation, with the design, fabrication and size advantages associated with surface excitation. An important topic within metasurfaces is the tailored rainbow trapping and selective spatial frequency separation of electromagnetic and acoustic waves using graded metasurfaces. This frequency dependent trapping and spatial frequency segregation has implications for energy concentrators and associated energy harvesting, sensing and wave filtering techniques. Different demonstrations of acoustic and electromagnetic rainbow devices have been performed, however not for deep elastic substrates that support both shear and compressional waves, together with surface Rayleigh waves; these allow not only for Rayleigh wave rainbow effects to exist but also for mode conversion from surface into bulk waves. We design, and test experimentally, a device creating not only elastic Rayleigh wave rainbow trapping, but also selective mode conversion of surface Rayleigh waves to shear waves.

physics.class-ph↗

arXiv2015

Smooth and polyhedral approximation in Banach spaces

We show that norms on certain Banach spaces $X$ can be approximated uniformly, and with arbitrary precision, on bounded subsets of $X$ by $C^{\infty}$ smooth norms and polyhedral norms. In particular, we show that this holds for any equivalent norm on $c_0(Γ)$, where $Γ$ is an arbitrary set. We also give a necessary condition for the existence of a polyhedral norm on a weakly compactly generated Banach space, which extends a well-known result of Fonf.

math.FA↗

arXiv2015

Chains of functions in $C(K)$-spaces

The Bishop property ($\symbishop$), introduced recently by K.P. Hart, T. Kochanek and the first-named author, was motivated by Pełczy{ń}ski's classical work on weakly compact operators on $C(K)$-spaces. This property asserts that certain chains of functions in said spaces, with respect to a particular partial ordering, must be countable. There are two versions of ($\symbishop$): one applies to linear operators on $C(K)$-spaces and the other to the compact Hausdorff spaces themselves. We answer two questions that arose after ($\symbishop$) was first introduced. We show that if $\mathscr{D}$ is a class of compact spaces that is preserved when taking closed subspaces and Hausdorff quotients, and which contains no non-metrizable linearly ordered space, then every member of $\mathscr{D}$ has ($\symbishop$). Examples of such classes include all $K$ for which $C(K)$ is Lindelöf in the topology of pointwise convergence (for instance, all Corson compact spaces) and the class of Gruenhage compact spaces. We also show that the set of operators on a $C(K)$-space satisfying ($\symbishop$) does not form a right ideal in $\mathscr{B}(C(K))$. Some results regarding local connectedness are also presented.

math.GN↗

arXiv2015

The Metric Approximation Property and Lipschitz-Free Spaces over Subsets of $\mathbb{R}^N$

We prove that for certain subsets $M \subseteq \mathbb{R}^N$, $N \geqslant 1$, the Lipschitz-free space $\mathcal{F}(M)$ has the metric approximation property (MAP), with respect to any norm on $\mathbb{R}^N$. In particular, $\mathcal{F}(M)$ has the MAP whenever $M$ is a finite-dimensional compact convex set. This should be compared with a recent result of Godefroy and Ozawa, who showed that there exists a compact convex subset $M$ of a separable Banach space, for which $\mathcal{F}(M)$ fails the approximation property.

math.FA↗

arXiv2011

Tree duplicates, $G_δ$-diagonals and Gruenhage spaces

We present an example in ZFC of a locally compact, scattered Hausdorff non-Gruenhage space $D$ having a $\G_delta$-diagonal. This answers a question posed by Orihuela, Troyanski and the author in a study of strictly convex norms on Banach spaces. In addition, we show that the Banach space of continuous functions $C_0(D)$ admits a $C^\infty$-smooth bump function.

math.FA↗

arXiv2010

Strictly convex norms and topology

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_δ$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit equivalent strictly convex norms, and give an internal topological characterisation of those scattered compact spaces $K$, for which the dual Banach space $C(K)^*$ admits an equivalent strictly convex dual norm. We establish some relationships between (*) and other topological concepts, and the position of several well-known examples in this context. For instance, we show that $C(\mathcal{K})^*$ admits an equivalent strictly convex dual norm, where $\mathcal{K}$ is Kunen's compact space. Also, under the continuum hypothesis CH, we give an example of a compact scattered non-Gruenhage space having (*).

math.FA↗

arXiv2009

Operator machines on directed graphs

We show that if an infinite-dimensional Banach space X has a symmetric basis then there exists a bounded, linear operator R : X --> X such that the set A = {x in X : ||R^n(x)|| --> infinity} is non-empty and nowhere dense in X. Moreover, if x in X\A then some subsequence of (R^n(x)) converges weakly to x. This answers in the negative a recent conjecture of Prajitura. The result can be extended to any Banach space containing an infinite-dimensional, complemented subspace with a symmetric basis; in particular, all 'classical' Banach spaces admit such an operator.

math.FA↗

arXiv2007

Gruenhage compacta and strictly convex dual norms

We prove that if K is a Gruenhage compact space then C(K)* admits an equivalent, strictly convex dual norm. As a corollary, we show that if X is a Banach space and X* is the |.|-closed linear span of K, where K is a Gruenhage compact in the w*-topology and |.| is equivalent to a coarser, w*-lower semicontinuous norm on X*, then X* admits an equivalent, strictly convex dual norm. We give a partial converse to the first result by showing that if T is a tree, then C(T)* admits an equivalent, strictly convex dual norm if and only if T is a Gruenhage space. Finally, we present some stability properties satisfied by Gruenhage spaces; in particular, Gruenhage spaces are stable under perfect images.

math.FA↗

arXiv2007

Trees, linear orders and Gâteaux smooth norms

We introduce a linearly ordered set Z and use it to prove a necessity condition for the existence of a Gâteaux smooth norm on C(T), where T is a tree. This criterion is directly analogous to the corresponding equivalent condition for Fréchet smooth norms. In addition, we prove that if C(T) admits a Gâteaux smooth lattice norm then it also admits a lattice norm with strictly convex dual norm.

math.FA↗