We study super weakly compact operators through a quantitative method. We introduce a semi-norm $\sigma (T)$ of an operator $T:X\to Y$ , where X, Y are Banach spaces, the so-called measure of super weak noncompactness, which measures how far T is from the family of super weakly compact operators. We study the equivalence of the measure $\sigma (T)$ and the super weak essential norm of T. We prove that Y has the super weakly compact approximation property if and and only if these two semi-norms are equivalent. As an application, we construct an example to show that the measures of T and its dual $T^*$ are not always equivalent. In addition we give some sequence spaces as examples of Banach spaces having the super weakly compact approximation property.

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与超弱紧算子相关的半规范的等价性

我们通过定量的方法研究超弱紧算子。我们引入一个半范数$\sigma (T)$运营商的$T:X\to Y$， 在哪里X,是是 Banach 空间，即所谓的超弱非紧致性度量，它度量多远吨来自超弱紧致运算符家族。我们研究度量的等价性$\sigma (T)$和超弱本质范数吨. 我们证明是当且仅当这两个半范数等价时，具有超弱紧致逼近性质. 作为一个应用程序，我们构建了一个例子来表明吨及其对偶$T^*$并不总是等价的。此外，我们给出了一些序列空间作为具有超弱紧逼近似特性的 Banach 空间的例子。