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Superposition operator on the space of sequences almost converging to zero

100%

Alekhno E.

Open Mathematics

2012

tom 10

nr 2

619-645

We study the superposition operator f on on the space ac 0 of sequences almost converging to zero. Conditions are derived for which f has a representation of the form f x = a+bx +g x, for all x ∈ ac 0 with a = f 0, b ∈ D(ac 0), g a superposition operator from ℓ∞ into I(ac 0), D(ac 0) = {z: zx ∈ ac 0 for all x ∈ ac 0}, and I(ac 0) the maximal ideal in ac 0. If f is generated by a function f of a real variable, then f is linear. We consider the conditions for which a bounded function f generates f acting on ac 0 and the conditions for which there exists a sequence y ∈ ac 0 such that y − f y ∈ ac 0. In terms of f, criteria for the boundedness, continuity, and sequential σ(ac 0ℓ1)-continuity of f on ac 0 are given. It is shown that the continuity of f is equivalent to the weak sequential continuity. Finally, properties of spaces D(ac 0) and I(ac 0) are studied, and in particular it is established that the inclusion I(ac 0) ⊕ {λe: λ ∈ ℝ} ⊂ D(ac 0) is proper, where e = (1, 1, …). By means of D(ac 0), a number of Banach-Mazur limit properties are derived.

A description of Banach space-valued Orlicz hearts

100%

Labuschagne C., Offwood T.

Open Mathematics

2010

tom 8

nr 6

1109-1119

Let Y be a Banach space, (Ω, Σ; μ) a probability space and φ a finite Young function. It is shown that the Y-valued Orlicz heart H φ(μ, Y) is isometrically isomorphic to the l-completed tensor product $$ H_\varphi \left( \mu \right)\tilde \otimes _l Y $$ of the scalar-valued Orlicz heart Hφ(μ) and Y, in the sense of Chaney and Schaefer. As an application, a characterization is given of the equality of $$ \left( {H_\varphi \left( \mu \right)\tilde \otimes _l Y} \right)* $$ and $$ H_\varphi \left( \mu \right)*\tilde \otimes _l Y* $$ in terms of the Radon-Nikodým property on Y. Convergence of norm-bounded martingales in H φ(μ, Y) is characterized in terms of the Radon-Nikodým property on Y. Using the associativity of the l-norm, an alternative proof is given of the known fact that for any separable Banach lattice E and any Banach space Y, E and Y have the Radon-Nikodým property if and only if $$ E\tilde \otimes _l Y $$ has the Radon-Nikodým property. As a corollary, the Radon-Nikodým property in H φ(μ, Y) is described in terms of the Radon-Nikodým property on H φ(μ) and Y.

Ograniczanie wyników

2 Open Mathematics

1 Alekhno E.

1 Labuschagne C.

1 Offwood T.

1 2012

1 2010

Wyniki wyszukiwania

Superposition operator on the space of sequences almost converging to zero

A description of Banach space-valued Orlicz hearts