Atomic decompositions for tensor products and polynomial spaces
A Schauder sequence space is a Banach space of scalar sequences \(Z\) such that the sequence of coordinate unit vectors \((e_i)_i\) is a basis for \(Z\). Given a Banach space \(X\), a Banach sequence space \(X_d\), a continuous linear operator \(S\colon X_d\to X\) and a sequence \((x_i')_i\) in \(X'\), the pair \(((x_i')_i, (Se_i)_i)\) is said to be an atomic decomposition for \(X\) with respect to \(X_d\) if for all \(x\) in \(X\), (a) \((x_i'(x))_i\in X_d\), (b) there are constants \(A\) and \(B\) such that \(A\| x\| \leq\| (x_i'(x))\| _{X_d}\leq B\| x\| \), (c) \(x=\sum_ix_i'(x) (S(e_i))\). The authors show that if \(((x_i')_i,(Se_i)_i)\) is an atomic decomposition for \(X\) with respect to \(X_d\) and \(\mu\) is a symmetric \(n\)-tensor norm, then \((\text{Perm}(\alpha)(x'_{\alpha_1}\otimes_s\dots\otimes_s x'_{\alpha_n})_{\alpha\in J},(Se_{\alpha_1}\otimes_s\dots\otimes_s Se_{\alpha_n}) _{\alpha \in J})\) is an atomic decomposition for the space of symmetric \(n\)-tensors \(\bigotimes_{\mu,s}^nX\) with respect to \(\bigotimes_{\mu,s}^nX_d\). In the above, the square decreasing ordering is taken on \(J\) and Perm\((\alpha)\) is the number of distinct multi-indices obtained from permuting \(\alpha\). They show that the atomic decomposition \(((x_i')_i,(Se_i)_i)\) for \(X\) with respect to \(X_d\) is shrinking if and only if \((Se_{\alpha_1}\otimes_s \cdots\otimes_s Se_{\alpha_n})_{\alpha\in J},\text{Perm}(\alpha)(x'_{\alpha_1}\otimes_s \cdots\otimes_s x'_{\alpha_n})_{\alpha\in J})\) is an atomic decomposition for \(\bigotimes_{\mu,s}^nX'\) with respect to \(\bigotimes_{\mu,s}^n(X_d)'\). The authors examine atomic decompositions for ideals of homogeneous polynomials showing, in particular, that the monomials with the square ordering will give an atomic decomposition for an ideal \(Q\) if and only if \(Q\) is a minimal ideal. They conclude the paper by showing that the reflexivity of the space of continuous \(n\)-homogeneous polynomials, \({\mathcal P}(^nX)\), can be characterised in terms of monomial atomic decompositions for \({\mathcal P}(^nX)\).
scientific article; zbMATH DE number 5312273
scientific article