DEPARTAMENTO DE MATEMÁTICA DOCUMENTO DE TRABAJO “K-Bounded Polynomials” (original) (raw)
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Mathematical Proceedings of the Royal Irish Academy, 1998
For a Banach space E we define the class PK ( N E) of K-bounded N -homogeneous polynomials, where K is a bounded subset of E . We investigate properties of K which relate the space PK ( N E) with usual subspaces of P( N E). We prove that K-bounded polynomials are approximable when K is a compact set where the identity can be uniformly approximated by finite rank operators. The same is true when K is contained in the absolutely convex hull of a weakly null basic sequence of E . Moreover, in this case we prove that every K-bounded polynomial is extendible to any larger space.
On the extensions of homogeneous polynomials
Proceedings of the American Mathematical Society, 2007
We investigate the problem of the uniqueness of the extension of n-homogeneous polynomials in Banach spaces. We show in particular that in a nonreflexive Banach space X that admits contractive projection of finite rank of at least dimension 2, for every n ≥ 3 there exists an n-homogeneous polynomial on X that has infinitely many extensions to X * * . We also prove that under some geometric conditions imposed on the norm of a complex Banach lattice E, for instance when E satisfies an upper p-estimate with constant one for some p > 2, any 2-homogeneous polynomial on E attaining its norm at x ∈ E with a finite rank band projection P x , has a unique extension to its bidual E * * . We apply these results in a class of Orlicz sequence spaces.
Bulletin of the Australian Mathematical Society, 1995
Consider the following possible properties which a Banach space X may have: (P): If (xi) and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj) − (yj) → 0, then Q(xj − yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynomial Q on X.(RP): If (xj)and (yj) are bounded sequences in X such that for all n ≥ 1 and for every continuous n-homogeneous polynomial P on X, P(xj − yj) → 0, then Q(xj) − Q(yj) → 0 for all m ≥ 1 and for every continuous m-homogeneous polynimial Q on X. We study properties (P) and (RP) and their relation with the Schur proqerty, Dunford-Pettis property, Λ, and others. Several applications of these properties are given.
Subspaces of polynomials on Banach spaces
2002
We study some subspaces of homogeneous polynomials which are defined as dual spaces to the symmetric tensor products of Banach spaces, endowed with special cross-norms.
A note on a class of Banach algebra-valued polynomials
International Journal of Mathematics and Mathematical Sciences, 2002
LetFbe a Banach algebra. We give a necessary and sufficient condition forFto be finite dimensional, in terms of finite typen-homogeneousF-valued polynomials.
Publications of the Research Institute for Mathematical Sciences, 2003
We show that for every Banach space X the set of 2-homogeneous continuous polynomials whose canonical extension to X * * attain their norm is a dense subset of the space of all 2-homogeneous continuous polynomials P( 2 X).
A Characterization of Polynomially Convex Sets in Banach Spaces
Results in Mathematics
Let E be a Banach space and △ * be the closed unit ball of the dual space E *. For a compact set K in E, we prove that K is polynomially convex in E if and only if there exist a unital commutative Banach algebra A and a continuous function f : △ * → A such that (i) A is generated by f (△ *), (ii) the character space of A is homeomorphic to K, and (iii) K = sp(f) the joint spectrum of f. In case E = C (X), where X is a compact Hausdorff space, we will see that △ * can be replaced by X. 1
Polynomially continuous operators
Israel Journal of Mathematics, 1997
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. Throughout, X and Y are Banach spaces, Sx the unit sphere of X, and N stands for the natural numbers. Given k E N, we denote by P(kX) the space of all khomogeneous (continuous) polynomials from X into the scalar field K (real or complex). We identify P(°X) = K, and denote P(X) := ~']~=o P(kX) • For the general theory of polynomials on Banach spaces, we refer to [11]. As usual, en
A Hahn-Banach theorem for integral polynomials
PROCEEDINGS-AMERICAN MATHEMATICAL …, 1999
Introduction A natural question concerning scalar-valued continuous homogeneous polynomi-als over a Banach space E is whether they can be extended to a larger space G, much as linear forms can be extended by using the Hahn-Banach theorem. It is well known that ...