Extreme polynomials and multilinear forms on \(\ell_1\) (Q1276360)

The authors' main interest is in the geometry of the Banach space \({\mathcal P}({^2\ell_1})\) of 2-homogeneous polynomials \(P\) on \(\ell_1\), endowed with the norm \(\| P\|= \sup\{| P(x)| | x\in \ell_1, \| x\|\leq 1\}\). They first examine the interesting special case of 2-homogeneous polynomials \(P(x,y)= ax^2+ by^2+ cxy\), where \(a,b,c\in\mathbb{R}\) and where \(x,y\in\mathbb{K}= \mathbb{R}\) or \(\mathbb{C}\). In this case, they show that \(\| P\|_{\mathbb{R}}=\| P\|_{\mathbb{C}}\) and they find necessary and sufficient conditions on the coefficients \(a\), \(b\), and \(c\) in order for \(P\) to have norm \(1\). Using this, they characterize the extreme points of the closed unit ball of \({\mathcal P}({^2\ell^2_1})\) in the real case, thereby complementing work of \textit{R. A. Ryan} and \textit{B. Turett} [J. Math. Anal. Appl. 221, No. 2, 698-711 (1998)]. The authors then use this to study extreme and strongly extreme points of the ball \({\mathcal P}({^2\ell_1})\). Among other things, they provide an example of a 2-homogeneous polynomial \(P\) which is not extreme, with the property that for the uniquely associated symmetric bilinear form, \(\check P\), \(\|\check P\|= 2\| P\|\). It is worth noting that this is an area of continued interesting open problems. For example, there does not appear to be a characterization of the extreme points of the ball of \({\mathcal P}({^2\ell^2_1})\) in the complex case. See also recent work by [\textit{C. Boyd} and \textit{R. A. Ryan}, ``Geometric theory of the space of integral polynomials and symmetric tensor products'' (to appear); \textit{Y. S. Choi} and \textit{S. G. Kim}, Arch. Math. 71, 472-480 (1998); and \textit{B. Grecu}, ``Extreme polynomials on Hilbert spaces'' (to appear)].