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Duality of measures of non-𝒜-compactness

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Delgado J. , Piñeiro C.

Studia Mathematica

2015

tom 229

nr 2

95-112

Let 𝒜 be a Banach operator ideal. Based on the notion of 𝒜-compactness in a Banach space due to Carl and Stephani, we deal with the notion of measure of non-𝒜-compactness of an operator. We consider a map $χ_{𝒜}$ (respectively, $n_{𝒜}$) acting on the operators of the surjective (respectively, injective) hull of 𝒜 such that $χ_{𝒜}(T) = 0$ (respectively, $n_{𝒜}(T) = 0$) if and only if the operator T is 𝒜-compact (respectively, injectively 𝒜-compact). Under certain conditions on the ideal 𝒜, we prove an equivalence inequality involving $χ_{𝒜}(T*)$ and $n_{𝒜^{d}}(T)$. This inequality provides an extension of a previous result stating that an operator is quasi p-nuclear if and only if its adjoint is p-compact in the sense of Sinha and Karn.

An approximation property with respect to an operator ideal

100%

Delgado J. , Piñeiro C.

Studia Mathematica

2013

tom 214

nr 1

67-75

Given an operator ideal 𝒜, we say that a Banach space X has the approximation property with respect to 𝒜 if T belongs to $\overline{S∘T: S ∈ ℱ(X)}^{τ_{c}}$ for every Banach space Y and every T ∈ 𝒜(Y,X), $τ_{c}$ being the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting.