\(E^{\prime }\) 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.


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


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