Power-normed spaces

Abstract

To each power-norm $((E^n, \Vert \cdot \Vert _n):n\in {\mathbb N})$ based on a given Banach space E, we associate two maximal symmetric sequence spaces $L_\Phi ^E$ and $L_\Psi ^E$ whose norms $\Vert (z_k)\Vert _{L_\Phi ^E}$ and $\Vert (z_k)\Vert _{L_\Psi ^E}$ are defined by $\sup \{ \Vert (z_1x,\ldots ,z_nx)\Vert _n: \Vert x\Vert =1, n\in {\mathbb N}\}$ and $\sup \{ \Vert \sum _{k=1}^n z_kx_k\Vert : \Vert (x_1,\ldots ,x_n)\Vert _n=1, n\in {\mathbb N}\}$ respectively. For each $1\le p\le \infty$, we introduce and study the p-power-norms as those power-norms for which $L_\Phi ^E=\ell ^p$ and $L_\Psi ^E=\ell ^{p'}$, where $1/p+1/p'=1$. As a special cases of p-power-norms we introduce certain smaller class, to be called the class of $\ell ^p$-power-norms, which is shown to contain the p-multi-norms defined in (Dales et al., Multi-norms and Banach lattices, 2016), and to coincide with the multi-norms and dual-multi-norms defined in (Dales and Polyakov, Diss Math 488, 2012) in the cases $p=\infty$ and $p=1$ respectively. We give several procedures to construct examples of such p-power and $\ell ^p$-power-norms and show that the natural formulations of the (p, q)-summing, (p, q)-concave, Rademacher power norms, t-standard power norms among others are examples in these classes. In particular, for instance the Rademacher power norm is a 2-power norm and the (p, q)-summing power-norm is a $\ell ^r$-power-norm for $p>q$ with $\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$.