


Positivity in Functional Analysis and Applications
Org: Charalambos Aliprantis (Purdue) and Vladimir Troitsky (Alberta) [PDF]
 SAFAK ALPAY, Middle East Technical University, Turkiye
On bweakly compact operators
[PDF] 
A subset A of a vector lattice E is called bbounded if A is
bounded in E^{ ~ ~ }, the order bidual of E. An operator T: E ® X, where E is a Banach lattice and X is a
Banach space, is called bweakly compact if for each bbounded subset
A in E, T(A) is relatively weakly compact in X. We will give
some recent results on bweakly compact operators. In particular, we
will give a dual characterization of bweakly compact operators and
investigate factorization properties of such operators.
 RAZVAN ANISCA, Lakehead University
Unconditional decompositions in Banach spaces
[PDF] 
We discuss properties related to unconditionality in Banach spaces
which admit a UFDD (unconditional finitedimensional decomposition).
As a consequence we obtain that if a Banach space X contains an
unconditional basic sequence then we have one of the following
"regularirregular" alternatives: either X contains a subspace
isomorphic to l_{2} or X contains a subspace which has a UFDD but
does not admit a UFDD with a uniform bound for the dimensions of the
decomposition. This result can be also viewed in the context of
Gowers' dichotomy theorem.
 EVGENIOS AVGERINOS, University of the Aegean, Rhodes, Greece
On the solutions of multivalued problems governed by vector
differential inclusions
[PDF] 
In this work we consider second order vector differential inclusions
with periodic boundary conditions and a general multivalued term. So
let T=[0,b]. The multivalued boundary value problem under
consideration is the following:

ì ï í
ï î

x^{¢¢}(t) Î F 
æ è

t,x(t),x^{¢}(t) 
ö ø

a.e. on T 

x(0)=x(b), x^{¢}(0) = x^{¢}(b) 


ü ï ý
ï þ

\tag1 
 (1) 
Second order differential inclusions have been studied recently
primarily with Dirichlet boundary conditions and with a multivalued
term which is compact, convex valued and satisfies a growth condition.
There have been some works where the convexity hypothesis on the
values of the multifunction has been dropped and/or the Dirichlet
boundary condition has been replaced by a more general nonlinear one.
This development of the research on second order multivalued boundary
value problems, can be traced in recent works. In the paper we
combine the method of HuPapageorgiou with techniques from the theory
of nonlinear operators and the multivalued LeraySchauder principle,
to establish the existence of a solution for problem (1) under very
general hypotheses on F(t,x,y).
 GERARD BUSKES, University of Mississippi
Geometric mean in vector lattices
[PDF] 
We compare the geometric mean and the square mean in vector lattices
and show that the geometric mean enables one to define the square of
an Archimedean vector lattice.
 ROMAN DRNOVSEK, Institute of Mathematics, Physics and Mechanics; University
of Ljubljana, Jadranska 19, SI1000 Ljubljana, Slovenia
On positive unipotent operators on Banach lattices
[PDF] 
An operator on a Banach space is said to be unipotent whenever its
spectrum contains only the number 1. Let T be a positive
unipotent operator on a complex Banach lattice. Huijsmans and de
Pagter posed a question whether T is necessarily greater than or
equal to the identity operator I. We give a partial answer to the
question by proving that this is true if lim_{n®¥} n(TI)^{n}^{1/n} = 0.
 EDUARD EMELYANOV, Middle East Technical University, Ankara, Turkey
Some recent results on constrictive operators
[PDF] 
Let X be a Banach space. An operator T Î L(X) is
called constrictive if there exists a compact A Í X
such that

lim
n®¥

dist(T^{n} x,A) = 0 ("x Î X, x £ 1). 

T is called quasiconstrictive if its stable space
X_{0} (T) = {x Î X : 
lim
n®¥

T^{n} x = 0} 

is closed and of finite codimension. We discuss some conditions on
T under which T is constrictive or quasiconstrictive.
 JULIO FLORES, Universidad Rey Juan Carlos, Madrid (Spain)
Domination by positive BanachSaks operators
[PDF] 
Given a positive BanachSaks operator T between two Banach lattices
E and F, we give sufficient conditions on E and F in order to
ensure that every positive operator dominated by T is BanachSaks.
A counterexample is also given when these conditions are dropped.
Moreover, we deduce a characterization of the BanachSaks property in
Banach lattices in terms of disjointness.
 VALENTINA GALVANI, University of Alberta, Department of Economics
Spanning with Options over a Borel Space
[PDF] 
The research question is whether portfolios of finitely many ordinary
call options (or put options) allow to hedge any financial claim. The
situation considered is a twodate incomplete securities market
defined over a metrizable statespace W containing uncountably
many states of nature. The investigation is limited to securities
markets for which the space of contingent claims is identified with an
L_{p}space, with 1 £ p < ¥, on the Borel space (W,B,P) where B is the Borel salgebra of
W and P is a nonatomic Borel regular probability.
A claim is an element of the space of contingent claims. The constant
function 1 represents the riskfree asset payoff. The
expression (sk1)^{+} describes the payoff of a call option
written on an underlying asset s with strike price k. The payoff
of a portfolio of the riskfree asset and of finitely many call options
is an element of the space O_{s} defined by
O_{s} = Span{1, (sk1)^{+} : k Î Â}. 

Options are said to span the market if the space O_{s} is dense in the
space of contingent claims. This paper proves that there exist
infinitely many underlying assets for which options span a separable
L_{p}space. In particular, if W is also Polish, this
collection of underlying assets is dense in the positive cone of
L_{p}(P).
 HAILEGEBRIEL GESSESSE, University of Alberta, Edmonton, AB
Minimal Vectors of Positive Operators On Ordered Banach
Spaces
[PDF] 
We extend the technique of minimal vectors to latticeordered Banach
Spaces which are not Banach Lattices, but with norm that satisfy
 x  = x for every x. We prove the existence of
invariant subspaces for some positive operators on these spaces. As a
special case, we consider operators on Sobolev Spaces.
 NIGEL KALTON, University of Missouri
Symmetric functionals and traces
[PDF] 
We will discuss some problems concerning the existence of symmetric or
rearrangementinvariant linear functionals on sequence and function
spaces, with applications to nonstandard traces on ideals of
operators.
This work includes joint work with Fyodor Sukochev and with Ken Dykema.
 ARKADY KITOVER, Community College of Philadelphia, 1700 Spring Garden Street,
Philadelphia, PA 19130, USA
Universally order bounded operators on Hilbert Spaces
[PDF] 
It is shown that linear bounded operators on a Hilbert space H which
are regular for any representation of H as a L^{2} are exactly the
sums of HilbertSchmidt operators and multiples of the identity
operator.
 MEHMET ORHON, University of New Hampshire, Mathematics Department,
Kingsbury Hall, Durham, NH 03824
On the ideal center of the dual of a Banach lattice
[PDF] 
Let X be a Banach lattice. Its ideal center Z(X) is embedded
naturally in the ideal center Z(X¢) of its dual. The embedding may
be extended to a contractive algebra and lattice homomorphism of
Z(X)" into Z(X¢). We show that the extension is onto Z(X¢) if
and only if X has a topologically full center (that is, the closure
of Z(X)x is the closed ideal generated by each x Î X). The result
can be generalized to the ideal center of the order dual of an
Archemedian Riesz space and in a modified form to the orthomorphisms
on the order dual of an Archemedian Riesz space.
 HEYDAR RADJAVI, University of Waterloo, Waterloo, Ontario N2L 3G1
Invariant Sublattices for Semigroups of Operators
[PDF] 
Let X be a Banach lattice and S a set of operators on X. This
is a report on joint work in progress with Vladimir Troitsky
concerning those sublattices of X which are invariant under every
member of S. We study conditions under which nontrivial invariant
sublattices exist.
 ANTON SCHEP, University of South Carolina, Columbia, SC 29208, USA
Products of (weak) log convex operators are log convex
[PDF] 
A family {A(t);t Î I} of positive operators on a Banach lattice
E is called (weak) log convex if the function x® áA(t)x, x^{*} ñ is log convex for all 0 £ x Î E and all 0 £ x^{*} Î E^{*}. The main result is that the operator product of two such
families is again (weak)log convex. Applications to Kingman's Theorem
about log convexity of the spectral radius of such families follow.
 ADI TCACIUC, University of Alberta, Edmonton, Canada
On the existence of asymptoticl_{p} structures in Banach
spaces
[PDF] 
The asymptotic theory of infinite dimensional Banach spaces, developed
by Maurey, Milman and TomczakJaegermann, is concerned with the
structure of infinite dimensional Banach spaces manifested in the
finitedimensional subspaces that appear everywhere far away in the
space. The class of spaces that have a simple asymptotic structure,
in the sense that we can find a 1 £ p £ ¥ such that all
such finitedimensional subspaces as before are essentially l_{p}^{n}'s,
are of special interest and they are called asymptoticl_{p} spaces.
We prove that if a Banach space is saturated with infinite dimensional
subspaces in which all special ntuples of vectors are equivalent,
uniformly in n, then the space contains asymptoticl_{p} subspaces,
for some 1 £ p £ ¥. The proof reflects a technique used
by Maurey in the context of unconditional basic sequence problem and
extends a result by Figiel, Frankiewicz, Komorowski and
RyllNardzewski.
 VLADIMIR TROITSKY, University of Alberta, Edmonton, Alberta T6G 2G1
Norm closed algebraic ideals in L(l_{p} Ål_{q})
[PDF] 
It is well known that the only proper nontrivial normclosed
algebraic ideal in the algebra L(X) for X = l_{p}
(1 £ p < ¥) or X=c_{0} is the ideal of compact operators. The
next natural question is to describe all closed ideals of L(l_{p} Ål_{q}) for 1 £ p,q < ¥, p ¹ q, or,
equivalently, the closed ideals in L (l_{p},l_{q}) for
p < q. We show that for 1 < p < 2 < q < ¥ there are at least four
distinct proper closed ideals in L (l_{p},l_{q}),
including one that has not been studied before.
This is a joint work with B. Sari, Th. Schlumprecht and
N. TomczakJaegermann.
 MARTIN WEBER, Technische Universität Dresden
On Finite Elements in Vector Lattices of Operators
[PDF] 
In Archimedean vector lattices the notion of a finite element
simulates a continuous function with compact support in abstract
setting. Besides exhaustive studies of finite elements, in particular
vector lattices and Banach lattices, the investigation of finite
elements in vector lattices consisting of operators is also of
interest and yields some interesting results. The main results up to
now are obtained for the vector lattice of all regular operators
between two vector lattices. So the orthomorphisms in and the lattice
isomorphisms between Dedekind complete Banach lattices are finite
elements. If E,F are Banach lattices with F reflexive, then an
operator is a finite element if and only if its adjoint is finite.
Any regular operator defined on an ALspace which has its images in a
Dedekind complete AMspace with order unit is a finite element. Vice
versa, if for two Banach lattices E,F with F Dedekind complete
each regular operator mapping E into F is a finite element, then
E is lattice isomorphic to an ALspace and F lattice isomorphic to
an AMspace (not necessary with order unit). Not each finite rank
operator is a finite element. However, if the constitutes of such an
operator are finite elements in their corresponding vector lattices
then the operator is a finite element.

