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 Orlicz sequence spaces.