Tame Factorization Property I

Abstract

We introduce the \emph{tame factorization property} for triples of Fr\'echet spaces: a triple $(E,G,F)$ has the tame factorization property, denoted by $(E,G,F) \in \mathfrak{TF}$, if there exists a nondecreasing function $S:\mathbb{N}\to\mathbb{N}$ such that every operator from $E$ to $F$ factoring through $G$ is $S$-tame. We give a complete characterization of $\mathfrak{TF}$ in terms of operator seminorm estimates. We specialize this characterization to triples in which one or more of the spaces are K\"othe spaces, putting explicit conditions on the K\"othe matrices, and we describe how $\mathfrak{TF}$ behaves under projective tensor products of K\"othe spaces. Finally, we show that a triple of K\"othe spaces has the tame factorization property if and only if the family of operators that factor as products of two quasi-diagonal operators is tame.