M-ideals of homogeneous polynomials

• Autorzy: Verónica Dimant
• Języki publikacji: EN

Abstrakty

EN

We study the problem of whether $𝓟_{w}(ⁿE)$, the space of n-homogeneous polynomials which are weakly continuous on bounded sets, is an M-ideal in the space 𝓟(ⁿE) of continuous n-homogeneous polynomials. We obtain conditions that ensure this fact and present some examples. We prove that if $𝓟_{w}(ⁿE)$ is an M-ideal in 𝓟(ⁿE), then $𝓟_{w}(ⁿE)$ coincides with $𝓟_{w0}(ⁿE)$ (n-homogeneous polynomials that are weakly continuous on bounded sets at 0). We introduce a polynomial version of property (M) and derive that if $𝓟_{w}(ⁿE) = 𝓟_{w0}(ⁿE)$ and 𝒦(E) is an M-ideal in 𝓛(E), then $𝓟_{w}(ⁿE)$ is an M-ideal in 𝓟(ⁿE). We also show that if $𝓟_{w}(ⁿE)$ is an M-ideal in 𝓟(ⁿE), then the set of n-homogeneous polynomials whose Aron-Berner extension does not attain its norm is nowhere dense in 𝓟(ⁿE). Finally, we discuss an analogous M-ideal problem for block diagonal polynomials.

Kategorie tematyczne

• 47L22: Ideals of polynomials and of multilinear mappings
• 46G25: (Spaces of) multilinear mappings, polynomials
• 46B20: Geometry and structure of normed linear spaces
• 46B04: Isometric theory of Banach spaces

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 2011
• Tom: 202
• Numer: 1
• Strony: 81-104

Daty

wydano

2011

Twórcy

autor

Verónica Dimant

• Departamento de Matemática, Universidad de San Andrés, Vito Dumas 284, (B1644BID) Victoria, Buenos Aires, Argentina
• CONICET

Identyfikatory

DOI

10.4064/sm202-1-5