A characterization of ${\ell }^p$-spaces symmetrically finitely represented in symmetric sequence spaces

Abstract

For a separable symmetric sequence space $X$ of fundamental type we identify the set $\mathcal{F}(X)$ of all $p\in [1,\infty ]$ such that ${\ell }^p$ is block finitely represented in the unit vector basis $\{e_k{\}}_{k=1}^{\infty }$ of $X$ in such a way that the unit basis vectors of ${\ell }^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint blocks of $\{e_k\}$ with the same ordered distribution. It turns out that $\mathcal{F}(X)$ coincides with the set of approximate eigenvalues of the operator $(x_k)\mapsto {\sum }_{k=2}^{\infty }{x}_{[k/2]}{e}_k$ in $X$. In turn, we establish that the latter set is the interval $[2^{{\alpha }_X},2^{{\beta }_X}]$, where ${\alpha }_X$ and ${\beta }_X$ are the Boyd indices of $X$. As an application, we find the set $\mathcal{F}(X)$ for arbitrary Lorentz and separable sequence Orlicz spaces.

submitted