Extremal homogeneous polynomials on real normed spaces
Let \(P\) be a continuous \(m\)-homogeneous polynomial on a normed space \(E\) over either the real or the complex scalars, and let \(\check P\) denote the unique associated symmetric \(m\)-linear form [see, e.g., \textit{S. Dineen}, ``Complex analysis on infinite-dimensional spaces'', Springer-Verlag (1999)] for background information on polynomials, multilinear forms, and their norms. It is well-known that \(1\leq{\|\check P\|\over\| P\|}\leq{m^m\over m!}\), with equality in the left-hand side for every \(P\) if \(E\) is Hilbert space. In [Proc. Am. Math. Soc. 99, 340-346 (1987; Zbl 0618.46022)], \textit{I. Sarantopoulos} showed that if the scalar field is \(\mathbb{C}\), then there are \(P\)'s for which we have equality in the right-hand side and for which \(\check P\) attains its norm if and only if \(E\) contains an isometric copy of \(\ell^m_1\). Here, the authors investigate the situation in which the space \(E= \ell^m_1\) over the scalar field \(\mathbb{R}\). They show that, as expected, the Nachbin polynomial \(P(x)= x_1\cdots x_m\) is the unique extremal polynomial in case \(m=2\) or 3, but that many small perturbations of this polynomial also yield extremal polynomials when \(m\geq 4\).
scientific article; zbMATH DE number 1278840
scientific article