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Explicit representation of compact linear operators in Banach spaces via polar sets

David E. Edmunds, Jan Lang (2013)

Studia Mathematica

We consider a compact linear map T acting between Banach spaces both of which are uniformly convex and uniformly smooth; it is supposed that T has trivial kernel and range dense in the target space. It is shown that if the Gelfand numbers of T decay sufficiently quickly, then the action of T is given by a series with calculable coefficients. This provides a Banach space version of the well-known Hilbert space result of E. Schmidt.

Extenders for vector-valued functions

Iryna Banakh, Taras Banakh, Kaori Yamazaki (2009)

Studia Mathematica

Given a subset A of a topological space X, a locally convex space Y, and a family ℂ of subsets of Y we study the problem of the existence of a linear ℂ-extender u : C ( A , Y ) C ( X , Y ) , which is a linear operator extending bounded continuous functions f: A → C ⊂ Y, C ∈ ℂ, to bounded continuous functions f̅ = u(f): X → C ⊂ Y. Two necessary conditions for the existence of such an extender are found in terms of a topological game, which is a modification of the classical strong Choquet game. The results obtained allow us...

Extension and lifting of weakly continuous polynomials

Raffaella Cilia, Joaquín M. Gutiérrez (2005)

Studia Mathematica

We show that a Banach space X is an ℒ₁-space (respectively, an -space) if and only if it has the lifting (respectively, the extension) property for polynomials which are weakly continuous on bounded sets. We also prove that X is an ℒ₁-space if and only if the space w b ( m X ) of m-homogeneous scalar-valued polynomials on X which are weakly continuous on bounded sets is an -space.

Extension of multilinear operators on Banach spaces.

Félix Cabello Sánchez, R. García, I. Villanueva (2000)

Extracta Mathematicae

These notes deal with the extension of multilinear operators on Banach spaces. The organization of the paper is as follows. In the first section we study the extension of the product on a Banach algebra to the bidual and some related structures including modules and derivations. Tha approach is elementary and uses the classical Arens' technique. Actually most of the results of section 1 can be easily derived from section 2. In section 2 we consider the problem of extending multilinear forms on a...

Extension of smooth subspaces in Lindenstrauss spaces

V. P. Fonf, P. Wojtaszczyk (2014)

Studia Mathematica

It follows from our earlier results [Israel J. Math., to appear] that in the Gurariy space G every finite-dimensional smooth subspace is contained in a bigger smooth subspace. We show that this property does not characterise the Gurariy space among Lindenstrauss spaces and we provide various examples to show that C(K) spaces do not have this property.

Extension operators on balls and on spaces of finite sets

Antonio Avilés, Witold Marciszewski (2015)

Studia Mathematica

We study extension operators between spaces of continuous functions on the spaces σ ( 2 X ) of subsets of X of cardinality at most n. As an application, we show that if B H is the unit ball of a nonseparable Hilbert space H equipped with the weak topology, then, for any 0 < λ < μ, there is no extension operator T : C ( λ B H ) C ( μ B H ) .

Extensions of the representation theorems of Riesz and Fréchet

João C. Prandini (1993)

Mathematica Bohemica

We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry...

Extensions uniformes des formes linéaires positives

Hicham Fakhoury (1973)

Annales de l'institut Fourier

Soit M un sous-espace fermé d’un espace de Banach ordonné V  ; ce travail propose des conditions nécessaires et suffisantes pour qu’il existe a 1 , tel que toute forme linéaire f positive et continue sur M admette une extension linéaire f ˜ positive et continue sur V , vérifiant f ˜ a f . On termine par l’exemple d’un couple ( M , V ) ne possédant pas la propriété précédente bien que toute forme linéaire positive continue sur M se prolonge en une forme linéaire du même type en V .

Currently displaying 101 – 120 of 140