Generating classes of perfect Banach sequence spaces

Abstract:

A perfect sequence space $\lambda$ is said to be a step if ${l^1} \subset \lambda \subset {l^\infty }$ and $\lambda$ is a Banach space in its strong topology from ${\lambda ^{\text {x}}}$. In this paper a method is given to generate additional steps from a step $\lambda$. Precisely, ${\lambda ^p}$ is a step where ${\lambda ^p} \equiv \{ x = ({x_i})|{x_i} \in C$ and $|x{|^p} = (|{x_i}{|^p}) \in \lambda \}$, for $1 \leqq p < \infty$, with norm ${\left \| x \right \|_{{\lambda ^p}}} = {({\left \| {|x{|^p}} \right \|_\lambda })^{1/p}}$. It is shown that ${\lambda ^p}$, $1 < p < \infty$, is reflexive iff $\lambda$ has a Schauder basis. The space of diagonal maps of ${\lambda ^p}$ into $\lambda$ is characterized, as is the space of diagonal nuclear maps of $\lambda$ into ${\lambda ^p}$ when $\lambda$ has a Schauder basis.