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Duality, reflexivity and atomic decompositions in Banach spaces

100%

Carando D., Lassalle S.

Studia Mathematica

2009

tom 191

nr 1

67-80

We study atomic decompositions and their relationship with duality and reflexivity of Banach spaces. To this end, we extend the concepts of "shrinking" and "boundedly complete" Schauder basis to the atomic decomposition framework. This allows us to answer a basic duality question: when an atomic decomposition for a Banach space generates, by duality, an atomic decomposition for its dual space. We also characterize the reflexivity of a Banach space in terms of properties of its atomic decompositions.

The ideal of p-compact operators: a tensor product approach

81%

Galicer D., Lassalle S., Turco P.

Studia Mathematica

2012

tom 211

nr 3

269-286

We study the space of p-compact operators, $𝓚_{p}$, using the theory of tensor norms and operator ideals. We prove that $𝓚_{p}$ is associated to $/d_{p}$, the left injective associate of the Chevet-Saphar tensor norm $d_{p}$ (which is equal to $g_{p'}'$). This allows us to relate the theory of p-summing operators to that of p-compact operators. Using the results known for the former class and appropriate hypotheses on E and F we prove that $𝓚_{p}(E;F)$ is equal to $𝓚_{q}(E;F)$ for a wide range of values of p and q, and show that our results are sharp. We also exhibit several structural properties of $𝓚_{p}$. For instance, we show that $𝓚_{p}$ is regular, surjective, and totally accessible, and we characterize its maximal hull $𝓚_{p}^{max}$ as the dual ideal of p-summing operators, $Π_{p}^{dual}$. Furthermore, we prove that $𝓚_{p}$ coincides isometrically with $𝓠𝓝_{p}^{dual}$, the dual to the ideal of the quasi p-nuclear operators.

Ograniczanie wyników

2 Studia Mathematica

2 Lassalle S.

1 Carando D.

1 Galicer D.

1 Turco P.

1 2012

1 2009

Wyniki wyszukiwania

Duality, reflexivity and atomic decompositions in Banach spaces

The ideal of p-compact operators: a tensor product approach