On Polynomial Properties in Banach Spaces

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In this paper some polynomial properties of Banach spaces are studied through the use of a general scheme referring to the relationship between different classes of subsets of a Banach space. More specifically, if G(E) is a class of subsets of E (bounded, weakly compact, limited …), a new class is defined on E by setting A∈GN(E) if θN(A)∈G(⨂E), where θN(x)=x⊗⋯⊗x and ⨂E is the N-fold symmetric projective tensor product of E. Thus, for example, just as E has the Dunford-Pettis property if W(E)⊂DP(E) (each relatively weakly compact set is a Dunford-Pettis set), it seems natural to define the N-Dunford-Pettis property (N-DPP) by WN(E)⊂DPN(E). Since every N-homogeneous polynomial on E can be written as T∘θN where T is a linear operator, it is possible in some instances to apply characterizations of the linear properties to the polynomial case. Thus, the following are proven to be equivalent: (1) E has the N-DPP. (2) For all F, each weakly compact N-homogeneous polynomial P:E→F sends sequences (xn) in E such that (θN(xn)) converges weakly, into norm convergent sequences. (3) Same as (2), but with F=c0. Other results are obtained for other polynomial properties.