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Ideals of homogeneous polynomials and weakly compact approximation property in Banach spaces. (English) Zbl 1174.46008
Summary: We show that a Banach space  $$E$$ has the weakly compact approximation property if and only if each continuous Banach-valued polynomial on  $$E$$ can be uniformly approximated on compact sets by homogeneous polynomials which belong to the ideal of homogeneous polynomials generated by weakly compact linear operators. An analogous result is established for the compact approximation property.
For an erratum to this paper see ibid. 60, No. 3, 887–887 (2010; Zbl 1224.46033).

##### MSC:
 46B28 Spaces of operators; tensor products; approximation properties 46G25 (Spaces of) multilinear mappings, polynomials 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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