Banach spaces admitting a separating polynomial and \(L_P\) spaces (Q1599520)

Let \(X\) be a (real) Banach space. A map \(P:X\to \mathbb{R}\) is a \(k\)-homogeneous polynomial \((k\in \mathbb{N})\) if there exists a continuous \(k\)-linear map \(\widetilde P:X\times \cdots\times X\to\mathbb{R}\) such that \(P(x)= \widetilde P (x, \dots,x)\) \(\forall x\in X\). The norms of the above maps are finite. For every \(k\)-homogeneous polynomial \(P\) there is a uniquely determined symmetric \(k\)-linear map \(\widetilde P\) such that \(P\) is the diagonal of \(\widetilde P\). A polynomial is a finite sum of homogeneous polynomials. A sequence \(\{x_n \}_{n \in\mathbb{N}}\) in \(X\) is said to be \(P_k\)-null if for every \(m\)-homogeneous polynomial \(P\) on \(X\), with \(m\leq k\), we have that \(\lim_{n\to \infty} P(x_n)= 0\). A (real) Banach space admits a separating polynomial if there exits a continuous polynomial \(P:X\to\mathbb{R}\) for which \(P(0)=0\) and \(\inf_{\|x\|=1} P(x)>0\). This is an analogue to the inner product on a (real) Hilbert space. The notion of a separating polynomial was first introduced by \textit{J. Kurzweil} [Studia Math. 14, 214-231 (1954; Zbl 0064.10802)]. In 1989, \textit{R. Deville} characterized the spaces with a separating polynomial as the spaces that admit a \({\mathcal C}^\infty\)-smooth bump (real valued function with bounded support) and do not contain an isomorphic copy of \(c_0\). [Isr. J. Math. 67, 1-22 (1989; Zbl 0691.46009)]. The only spaces with a symmetric structure which admit a separating polynomial are \(\ell_p, L_p(0,1)\) and \(L_p(0, \infty) \cap L_q (0,\infty)\), where \(p,q\) are even integers [\textit{M. Gonzalez} et al., J. Lond. Math. Soc., II. Ser. 59, 681-697 (1999; Zbl 0922.46028)]. The paper under review is divided into three parts. In the first part, the authors deal with the geometric structure of spaces admitting a separating polynomial. The main result in this part is the following theorem: Theorem. If \(X\) admits a separating polynomial, then every weakly null semi-normalized sequence in \(X\) has a subsequence equivalent to the unit vector basis in \(\ell_{2k}\) for some integer \(k\). In the second part, the existence of separating polynomials on the real Schatten class \(c_p^\mathbb{R}\) is examined. The authors prove: Theorem. The space \(c_p^\mathbb{R}\) admits a separating polynomial if and only if \(p\) is an even integer. Finally, in the third part, the authors discuss the existence of convex 4-homogeneous separating polynomials on spaces with an unconditional basis.