José G. Llavona

Homogeneous Orthogonally Additive Polynomials on Banach Lattices

The main result in this paper is a representation theorem for homogeneous orthogonally additive polynomials on Banach lattices. The representation theorem is used to study the linear span of the set of zeros of homogeneous real-valued orthogonally additive polynomials. It is shown that in certain lattices every element can be represented as the sum of two or three zeros or, at least, can be approximated by such sums. It is also indicated how these results can be used to study weak topologies induced by orthogonally additive polynomials on Banach lattices.

Composition operators between algebras of differentiable functions

Let E, F be real Banach spaces, U subset-or-equal-to E and V subset-equal-to F non-void open subsets and C(k)(U) the algebra of real-valued k-times continuously Frechet differentiable functions on U, endowed with the compact open topology of order k. It is proved that, for m greater-than-or-equal-to p, the nonzero continuous algebra homomorphisms A: C(m)(U) --> C(p)(V) are exactly those induced by the mappings g: V --> U satisfying phi . g is-an-element-of C(p)(V) for each phi is-an-element-of E*, in the sense that A(f) = fog for every f is-an-element-of C(m)(U). Other homomorphisms are described too. It is proved that a mapping g: V --> E** belongs to C(k)(V, (E**, w*)) if and only if phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*. It is also shown that if a mapping g: V --> E verifies phi . g is-an-element-of C(k)(V) for each phi is-an-element-of E*, then g is-an-element-of C(k-1)(V, E).

Compact and Weakly Compact Homomorphisms between Algebras of Continuous Functions

Abstract In this note we study the relationship between compactness and weak compactness of a continuous homomorphism A from C co ( S ) into C co ( T ) and the associated continuous function ϑ : T → S , where S and T are completely regular Hausdorff spaces.

Polynomial continuity on l(1)

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. A Banach space X has property(RP) if given two bounded sequences (u(j)), (v(j)) subset of X; we have that Q(u(j)) - Q(v(j)) --> 0 for every polynomial Q on X whenever P(u(j) - v(j)) --> 0 for every polynomial P on XI i.e., the restriction of every polynomial on X to each bounded set is uniformly sequentially continuous for the weak polynomial topology. We show that property (RP) does not imply that every scalar valued polynomial on X must be polynomially continuous.

Polynomially continuous operators

A mapping between Banach spaces is said to be polynomially continuous if its restriction to any bounded set is uniformly continuous for the weak polynomial topology. Every compact (linear) operator is polynomially continuous. We prove that every polynomially continuous operator is weakly compact.