The Approximation Properties Determined by Operator Ideals

Dong Yang CHEN^{1}, Lei LI^{2}

1. School of Mathematical Sciences, Xiamen University, Xiamen 361005, P. R. China;
2. School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P. R. China

We introduce the notion of the right approximation property with respect to an operator ideal A and solve the duality problem for the approximation property with respect to an operator ideal A, that is, a Banach space X has the approximation property with respect to A^{d} whenever X^{*} has the right approximation property with respect to an operator ideal A. The notions of the left bounded approximation property and the left weak bounded approximation property for a Banach operator ideal are introduced and new symmetric results are obtained. Finally, the notions of the p-compact sets and the p-approximation property are extended to arbitrary Banach operator ideals. Known results of the approximation property with respect to an operator ideal and the p-approximation property are generalized.

The first author is supported by the Natural Science Foundation of Fujian Province of China (Grant No. 2015J01026); the second author is supported by the NSF of China (Grant No. 11301285)

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