Power series space representations of nuclear Fréchet spaces

Abstract:

Let $E$ be a nuclear graded Fréchet space such that the norms satisfy inequalities $||||_k^2 \leq {C_k}|||{|_{k - 1}}|||{|_{k - 1}}$ for all $k$, let $F$ be a graded Fréchet space such that the dual (extended real valued) norms satisfy inequalities $||||_k^{*2} \leq {D_k}||||_{k - 1}^*||||_{k + 1}^*$ for all $k$, and let $A$ be a tame (resp. linearly tame) linear map from $F$ to $E$. Then there exists a tame (resp. linearly tame) factorization of $A$ through a power series space $\Lambda _\infty ^2(\alpha )$. In the case of a tame quotient map, $E$ is tamely equivalent to a power series space of infinite type. This applies in particular to the range of a tame (resp. linearly tame) projection in a power series space $\Lambda _\infty ^2(\alpha )$. In this case one does not need nuclearity. It also applies to the tame spaces in the sense of the various implicit function theorems. If they are nuclear, they are tamely equivalent to power series spaces ${\Lambda _\infty }(\alpha )$.