Miek Messerschmidt

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• PhD
• Senior Lecturer at University of Pretoria

We give an overview of normality and conormality properties of pre-ordered Banach spaces. For pre-ordered Banach spaces $X$ and $Y$ with closed cones we investigate normality of $B(X,Y)$ in terms of normality and conormality of the underlying spaces $X$ and $Y$. Furthermore, we define a class of ordered Banach spaces called quasi-lattices which str...

This paper will generalize what may be termed the "geometric duality theory" of real pre-ordered Banach spaces which relates geometric properties of a closed cone in a real Banach space, to geometric properties of the dual cone in the dual Banach space. We show that geometric duality theory is not restricted to real pre-ordered Banach spaces, as is...

In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a c...

For $$d\in {\mathbb {N}}$$ d ∈ N , a compact sphere packing of Euclidean space $${\mathbb {R}}^{d}$$ R d is a set of spheres in $${\mathbb {R}}^{d}$$ R d with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial d -complex that covers all of $${\mathbb {R}}^{d}$$ R d . We are motivated by...

By a compact packing of the plane by discs, P, we mean a collection of closed discs in the plane with pairwise disjoint interior so that, for every disc \(C\in P\), there exists a sequence of discs \(D_{0},\ldots ,D_{m-1}\in P\) so that each \(D_i\) is tangent to both C and \(D_{i+1\,mod \,m}\). We prove, for every \(n\in \mathbb {N}\), that there...

Inspired by the theories of Kaplansky-Hilbert modules and probability theory in vector lattices, we generalise functional analysis by replacing the scalars R or C by a real or complex Dedekind complete unital f-algebra L; such an algebra can be represented as a suitable space of continuous functions. We set up the basic theory of L-normed and L-Ban...

We show that there exists a Banach space X X which contains closed subspaces Y Y and Z Z with Y + Z = X Y+Z=X such that the associated surjective summation operator Σ : Y × Z → X \Sigma \colon Y\times Z\to X defined by Σ ( y , z ) = y + z \Sigma (y,z)= y+z for y ∈ Y y\in Y and z ∈ Z z\in Z has no Lipschitz right inverse.

For $d\in\mathbb{N}$, a compact sphere packing of Euclidean space $\mathbb{R}^{d}$ is a set of spheres in $\mathbb{R}^{d}$ with disjoint interiors so that the contact hypergraph of the packing is the vertex scheme of a homogeneous simplicial $d$-complex that covers all of $\mathbb{R}^{d}$. We are motivated by the question: For $d,n\in\mathbb{N}$ wi...

By a compact packing of the plane by discs, $P$, we mean a collection of closed discs in the plane with pairwise disjoint interior so that, for every disc $C\in P$, there exists a sequence of discs $D_{0},\ldots,D_{m-1}\in P$ so that each $D_{i}$ is tangent to both $C$ and $D_{i+1\mod m}.$ We prove, for every $n\in\mathbb N$, that there exist only...

It was recently shown in Ter Horst et al. (Bull Lond Math Soc 51:1005–1014, 2019) that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredhol...

It was recently shown in [24] that the Banach space operator relations Equivalence After Extension (EAE) and Schur Coupling (SC) do not coincide by characterizing these relations for operators acting on essentially incomparable Banach spaces. The examples that prove the non-coincidence are Fredholm operators, which is a subclass of relatively regul...

In 1994, H. Bart and V. É. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC ⇒ SC open. Despite several affirmative results, in this paper we show that the answer in general is no. This follows f...

Previously only two examples of Banach space quotient maps which do not admit uniformly continuous right inverses were known: one due to Aharoni and Lindenstrauss and one due to Kalton ($\ell^\infty\to\ell^\infty/c_{0}$). We show through an application of Kalton's Monotone Transfinite Sequence Theorem that a quotient map of a subspace of $\ell^\inf...

Consider the following still-open problem: for any Banach space X, ordered by a closed generating cone C ⊆ X, do there always exist Lipschitz functions ⋅⁺ : X → C and ⋅⁻ : X → C satisfying x = x⁺ − x⁻ for every x ∈ X?

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical B...

In 1994 H. Bart and V.\'{E}. Tsekanovskii posed the question whether the Banach space operator relations matricial coupling (MC), equivalence after extension (EAE) and Schur coupling (SC) coincide, leaving only the implication EAE/MC $\Rightarrow$ SC open. Despite several affirmative results, in this paper we show that the answer in general is no....

A version of the classical Klee–Andô Theorem states the following: For every Banach space X, ordered by a closed generating cone C⊆X, there exists some α>0 so that, for every x∈X, there exist x±∈C so that x=x⁺−x⁻ and ‖x⁺‖+‖x⁻‖≤α‖x‖. The conclusion of the Klee–Andô Theorem is what is known as a conormality property. We prove stronger and somewhat mo...

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}$...

A compact circle-packing $P$ of the Euclidean plane is a set of circles which bound mutually disjoint open discs with the property that, for every circle $S\in P$, there exists a maximal indexed set $\{A_{0},\ldots,A_{n-1}\}\subseteq P$ so that, for every $i\in\{0,\ldots,n-1\}$, the circle $A_{i}$ is tangent to both circles $S$ and $A_{i+1\mod n}.$...

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators...

In recent years the coincidence of the operator relations equivalence after extension (EAE) and Schur coupling (SC) was settled for the Hilbert space case. For Banach space operators, it is known that SC implies EAE, but the converse implication is only known for special classes of operators, such as Fredholm operators with index zero and operators...

We prove that any correspondence (multi-function) mapping a metric space into a Banach space that satisfies a certain pointwise Lipschitz condition, always has a continuous selection that is pointwise Lipschitz on a dense set of its domain. We apply our selection theorem to demonstrate a slight improvement to a well-known version of the classical B...

A theorem of Wickstead from 1975 characterizes the ordered Banach spaces with order bounded precompact sets in terms of a geometric property, "coadditivity", relating the space's order with its topology. We strengthen Wickstead's Theorem by showing for an ordered Banach space to have all its precompact sets be order bounded, it is necessary and suf...

Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$ denotes the intrinsic metric on $S$.

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In this paper we investigate consequences of equivalence after extension for compact Banach space operators. We sh...

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous...

In earlier work a crossed product of a Banach algebra was con-structed from a Banach algebra dynamical system (A, G, α) and a class R of continuous covariant representations, and its representations were determined. In this paper we adapt the theory to the ordered context. We construct a pre-ordered crossed product of a Banach algebra from a pre-or...

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,\alpha)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper the theory is developed further. We consider the dependence of the crossed product on the class $\mathca...

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous...