Symmetric finite representability of \(\ell^p\)-spaces in rearrangement invariant spaces on \((0,\infty)\) (Q2160215)

This paper is motivated by the classical result of Krivine on finite representability of \(\ell_p\) spaces, and more precisely on a version of this due to [\textit{H.~P. Rosenthal}, J. Funct. Anal. 28, 197--225 (1978; Zbl 0387.46016)] in the context of subsymmetric unconditional bases: Namely, if \((e_n)\) denotes such a basis and \(s_n\) are defined by \(\|\sum_{i=1}^{n}e_i\|=n^{1/s_n}\), then \(\ell_p\) is finitely representable in the span of \((e_i)\) provided \[ p\in[\liminf_{n\rightarrow \infty} s_n,\limsup_{n\rightarrow \infty} s_n]. \] In this paper the author provides a beautiful connection between finite representability of \(\ell_p\) on a separable rearrangement invariant space \(X\) on \((0,\infty)\) and the spectra of the doubling operator \(\sigma\) on \(X\) given by \(\sigma x(t)=x(t/2)\) for \(x\in X\), \(t\in(0,\infty)\). In particular, it is shown that \(\ell_p\) is \textit{symmetrically} finitely representable in \(X\) precisely when \(2^{1/p}\) is an approximate eigenvalue of the doubling operator \(\sigma\) on \(X\). Here, \(\ell_p\) is said to be symmetrically finitely representable in a space \(X\) if for every \(n\in\mathbb N\) and \(\varepsilon>0\), there exist equi-distributed and disjointly supported vectors \((x_k)_{k=1}^n\subset X\) such that for all scalars \((a_k)_{k=1}^n\) we have \[ (1+\varepsilon)^{-1}\Big(\sum_{k=1}^{n}|a_k|^p\Big)^{1/p}\leq\Big\|\sum_{k=1}^{n}a_kx_k\Big\|_X\leq (1+\varepsilon)\Big(\sum_{k=1}^{n}|a_k|^p\Big)^{1/p}. \] The author also shows that the set of approximate eigenvalues of the operator \(\sigma\) can only be a closed interval or the union of two disjoint closed intervals. We refer the reader to this very interesting paper for details.