Powers of Symmetric Sequence Ideals

Abstract

About 50 years ago it was observed that the relationship between $$L_1$$ and $$L_p$$ can be mimicked by assigning to certain function spaces A a “power” $$A^p$$ . We study this construction in the case of symmetric sequence ideals $$\mathfrak a$$ by letting $$\begin{aligned} \mathfrak a^p := \big \{ s \in \mathfrak l_\infty : |s|^{1/p} \in \mathfrak a\big \} \quad \text{ for } \text{ all }\quad p > 0. \end{aligned}$$ The purely algebraic level is simple. However, if $$\mathfrak a$$ is equipped with a norm $$\Vert \cdot |\mathfrak a\Vert $$ , then $$\begin{aligned} \Vert s|\mathfrak a^p\Vert := \big \Vert |s|^{1/p}\big |\mathfrak a\big \Vert ^p \quad \text{ for } \text{ all }\quad s \in \mathfrak a^p \end{aligned}$$ need not be a norm. This defect disappears in the setting of quasi-norms. Therefore quasi-Banach power scales $$\{ \mathfrak a^p \}_{p >0}$$ are well-behaved subjects, which deserve to be studied. In this paper, we are mainly dealing with $$\{ \mathfrak a^p \}_{p >0}$$ as a whole and not so much with the properties of its single members $$\mathfrak a^p$$ . Our main purpose is to improve the knowledge about the structure of the (very large) lattices of all symmetric sequence ideals. So specific one-parametric subsets, which look like intervals, are important. This paper should be a good illustration of the leitmotiv: Well-chosen concepts make mathematics easy.