20 research outputs found
Strong Klee-And\^o Theorems through an Open Mapping Theorem for cone-valued multi-functions
A version of the classical Klee-And\^o Theorem states the following: For
every Banach space , ordered by a closed generating cone ,
there exists some so that, for every , there exist
so that and
.
The conclusion of the Klee-And\^o Theorem is what is known as a conormality
property.
We prove stronger and somewhat more general versions of the Klee-And\^o
Theorem for both conormality and coadditivity (a property that is intimately
related to conormality). A corollary to our result shows that the functions
, as above, may be chosen to be bounded, continuous, and
positively homogeneous, with a similar conclusion yielded for coadditivity.
Furthermore, we show that the Klee-And\^o Theorem generalizes beyond ordered
Banach spaces to Banach spaces endowed with arbitrary collections of cones.
Proofs of our Klee-And\^o Theorems are achieved through an Open Mapping Theorem
for cone-valued multi-functions/correspondences.
We very briefly discuss a potential further strengthening of The Klee-And\^o
Theorem beyond what is proven in this paper, and motivate a conjecture that
there exists a Banach space , ordered by a closed generating cone
, for which there exist no Lipschitz functions
satisfying for all .Comment: Major rewrite. Large parts were removed which a referee pointed out
can be proven through much easier method
Normality of spaces of operators and quasi-lattices
We give an overview of normality and conormality properties of pre-ordered
Banach spaces. For pre-ordered Banach spaces and with closed cones we
investigate normality of in terms of normality and conormality of the
underlying spaces and .
Furthermore, we define a class of ordered Banach spaces called quasi-lattices
which strictly contains the Banach lattices, and we prove that every strictly
convex reflexive ordered Banach space with a closed proper generating cone is a
quasi-lattice. These spaces provide a large class of examples and that
are not Banach lattices, but for which is normal. In particular, we
show that a Hilbert space endowed with a Lorentz cone is a
quasi-lattice (that is not a Banach lattice if ), and
satisfies an identity analogous to the elementary Banach lattice identity
which holds for all elements of a Banach lattice. This is
used to show that spaces of operators between such ordered Hilbert spaces are
always absolutely monotone and that the operator norm is positively attained,
as is also always the case for spaces of operators between Banach lattices.Comment: Minor typos fixed. Exact solution now provided in Example 5.10. To
appear in Positivit
On compact packings of the plane with circles of three radii
A compact circle-packing of the Euclidean plane is a set of circles which
bound mutually disjoint open discs with the property that, for every circle
, there exists a maximal indexed set so that, for every , the circle is tangent to
both circles and
We show that there exist at most pairs with for
which there exist a compact circle-packing of the plane consisting of circles
with radii , and .
We discuss computing the exact values of such as roots of
polynomials and exhibit a selection of compact circle-packings consisting of
circles of three radii. We also discuss the apparent infeasibility of computing
\emph{all} these values on contemporary consumer hardware with the methods
employed in this paper.Comment: Dataset referred to in the text can be obtained at
http://dx.doi.org/10.17632/t66sfkn5tn.
Geometric duality theory of cones in dual pairs of vector spaces
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 done classically, but can be extended
to real Banach spaces endowed with arbitrary collections of closed cones.
We define geometric notions of normality, conormality, additivity and
coadditivity for members of dual pairs of real vector spaces as certain
possible interactions between two cones and two convex convex sets containing
zero. We show that, thus defined, these notions are dual to each other under
certain conditions, i.e., for a dual pair of real vector spaces , the
space is normal (additive) if and only if its dual is conormal
(coadditive) and vice versa. These results are set up in a manner so as to
provide a framework to prove results in the geometric duality theory of cones
in real Banach spaces. As an example of using this framework, we generalize
classical duality results for real Banach spaces pre-ordered by a single closed
cone, to real Banach spaces endowed with an arbitrary collections of closed
cones.
As an application, we analyze some of the geometric properties of naturally
occurring cones in C*-algebras and their duals
Equivalence after extension for compact operators on Banach spaces
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 show that generating the same operator ideal
is necessary but not sufficient for two compact operators to be equivalent
after extension. In analogy with the necessary and sufficient conditions on the
singular values for compact Hilbert space operators that are equivalent after
extension, we prove the necessity of similar relationships between the
-numbers of two compact Banach space operators that are equivalent after
extension, for arbitrary -functions.
We investigate equivalence after extension for operators on
-spaces. We show that two operators that act on different
-spaces cannot be equivalent after one-sided extension. Such
operators can still be equivalent after extension, for instance all invertible
operators are equivalent after extension, however, if one of the two operators
is compact, then they cannot be equivalent after extension. This contrasts the
Hilbert space case where equivalence after one-sided extension and equivalence
after extension are, in fact, identical relations.
Finally, for general Banach spaces and , we investigate consequences
of an operator on being equivalent after extension to a compact operator on
. We show that, in this case, a closed finite codimensional subspace of
must embed into , and that certain general Banach space properties must
transfer from to . We also show that no operator on can be
equivalent after extension to an operator on , if and are
essentially incomparable Banach spaces