Extendibility of polynomials and analytic functions on \(\ell_p\) (Q2717564)

A \(k\)-homogeneous polynomial \(P:E\to F\) is called extendible if for every Banach space \(E\subset G\) there is a polynomial \(Q: G\to F\) extending \(P\). Every extendible 2-homogeneous polynomial on a Banach space with cotype 2 is integral. As a consequence, for \(1< p\leq 2\), a 2-homogeneous polynomial on \(\ell_p\) is extendible if and only if it is nuclear. An idea of Defant and Floret is used to construct extendible non-nuclear 2-homogeneous polynomials on \(\ell_p\), \(p> 4\). All diagonal extendible homogeneous polynomials on \(\ell_p\), \(1< p<\infty\), are nuclear. These results permit the author to find non-extendable approximable homogeneous polynomials of any degree \(k>1\) on \(\ell_p\), \(1\leq p<\infty\). The author defines extendible holomorphic functions \(f: U\to F\) at a point \(a\in U\). A characterization in terms of the coefficients of the Taylor expansion of \(f\) at \(a\) is given. On each \(\ell_p\), \(1\leq p<\infty\), there is a holomorphic function of infinite radius of convergence with finite type Taylor coefficients which is not extendible.