Orbits of homogeneous polynomials on Banach spaces (original) (raw)

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.

Group invariant separating polynomials on a Banach space

2020

We study the group invariant continuous polynomials on a Banach space XXX that separate a given set KKK in XXX and a point zzz outside KKK. We show that if XXX is a real Banach space, GGG is a compact group of mathcalL(X)\mathcal{L} (X)mathcalL(X), KKK is a GGG-invariant set in XXX, and zzz is a point outside KKK that can be separated from KKK by a continuous polynomial QQQ, then zzz can also be separated from KKK by a GGG-invariant continuous polynomial PPP. It turns out that this result does not hold when XXX is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that XXX has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group invariant polynomial from the closed unit ball.

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.

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

Nonuniform polynomial behaviors in Banach spaces

INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2020

The aim of this paper is to present some characterization theorems for the nonuniform polynomial stability concept of evolution operators in Banach spaces. Also necessary and sufficient conditions for an evolution operator to be nonuniformly polynomially stable with respect to a family of norms are given.

A class of polynomials from Banach spaces into Banach algebras

Publications of the Research Institute for Mathematical Sciences, 2001

Let E be a complex Banach space and F be a complex Banach algebra. We will be interested in the subspace IP f (n E, F) of P (n E, F) generated by the collection of functions ϕ n (n ∈ AE, ϕ ∈ L(E, F)) where ϕ n (x) = (ϕ(x)) n for each x ∈ E.

Isometries between spaces of homogeneous polynomials

Journal of Functional Analysis, 2005

We derive Banach-Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of n-homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E onto F . With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of n-homogeneous integral polynomials and between the spaces of all n-homogeneous polynomials are also investigated.