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\(E^{\prime }\) and its relation with vector-valued functions on \(E\). (English) Zbl 1058.46024

This paper investigates the following question: If the duals of the Banach spaces \(E\) and \(F\) are isomorphic (isometric) and \(X\) is a Banach space, does this mean that the spaces of \(X\)-valued \(n\)-homogeneous polynomials on \(E\) and \(F\) are isomorphic (isometric)?
A Banach space \(E\) is said to be symmetrically regular if all symmetric linear operators from \(E\) into \(E'\) are weakly compact. When \(E\) and \(F\) are symmetrically regular, \(s: E'\to F'\) is an isomorphism (resp. isometry) and \(X\) is a Banach space, the authors observe that the mapping \({\bar s}(P)=\overline{P}\circ s'\circ J_F\) defines an isomorphism (resp. isometry) from the space of \(X\)-valued \(n\)-homogeneous polynomials on \(E\) onto the space of \(X''\)-valued \(n\)-homogeneous polynomials on \(F\). Moreover, when \(X=W'\) is a dual Banach space the range of \(\overline{s}\) is contained in the space of \(X\)-valued \(n\)-homogeneous polynomials on \(F\).
When \(s\) is an isomorphism (isometry) of \(E'\) onto \(F'\) and \(X\) is a Banach space, the authors show that \(\overline{s}\) is an isomorphism between the subspaces of \(X\)-valued finite-type (resp. nuclear, approximable, regular, integral, extendible, \(n\)-homogeneous polynomials which are weakly continuous on bounded sets) on \(E\) and spaces of \(X\)-valued finite-type (resp. nuclear, approximable, regular, integral, extendible, \(n\)-homogeneous polynomials which are weakly continuous on bounded sets) on \(F\).
When \(E\) does not contain a copy of \(\ell_1\), the same is true for the spaces of vector-valued weakly sequentially continuous polynomials. The authors show that when \(E\) and \(F\) are symmetrically Arens regular and \(W'\) is a dual Banach space, then \({\mathcal H}_b(E,W')\) and \({\mathcal H}_b(F,W')\) are isomorphic when \(E'\) and \(F'\) are isomorphic and that \({\mathcal H}_b(B_E,W')\) (resp. \({\mathcal H}^\infty(B_E,W')\)) and \({\mathcal H}_b(B_E,W')\) (resp. \({\mathcal H}^\infty (B_E,W')\)) are isomorphic when \(E\) and \(F\) are isometrically isomorphic. If \(W'\) is a Banach algebra, these isomorphisms are isomorphisms of Fréchet algebras. Results for spaces of weakly uniformly continuous holomorphc functions and boundedly integral functions are also presented.

MSC:

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
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References:

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