I am interested in generalisations to Banach spaces of the following construction, which relates to the singular value decomposition of a finite-dimensional linear map. If $V$, $W$ are finite-dimensional real or complex vector spaces equipped with inner products, and $A \colon V \to W$ is a linear map, then the singular values $\sigma_1(A),\sigma_2(A),\ldots$ of $A$ can be defined to be the positive square roots of the (necessarily real and non-negative) eigenvalues of the positive semidefinite linear map $A^*A \colon V \to V$. The eigenspaces of $A^*A$ are pairwise orthogonal, and so are their images under $A$. Using these facts, if $\sigma_k(A)>\sigma_{k+1}(A)$ then we may choose orthogonal decompositions $V=E\oplus F$, $W=E'\oplus F'$ such that $AE=E'$, $AF=F'$, $\|Av\|\geq \sigma_k(A)\|v\|$ for all $v \in E$ and $\|Av\|\leq \sigma_{k+1}(A)\|v\|$ for all $v \in F$. Here, $E$ (resp. $F$) is the span of the eigenspaces of $A^*A$ corresponding to eigenvalues greater than or equal to $\sigma_k(A)^2$ (resp. less than or equal to $\sigma_{k+1}(A)^2$). A decomposition of this type continues to exist if $V$ and $W$ are Hilbert spaces, although in this case the definition of the quantities $\sigma_i(A)$ is more delicate.
My colleague and I are interested in the Banach space case, where we are motivated by infinite-dimensional generalisations of the Oseledets multiplicative ergodic theorem. We believe that we are able to prove the following result. Let $\mathfrak{X}_1$ and $\mathfrak{X}_2$ be Banach spaces, let $A \colon \mathfrak{X}_1 \to \mathfrak{X}_2$ be a bounded linear operator, and for each $n \geq 1$ define the s-number $c_n(A):=\inf\{\|A-F\|\colon \mathrm{rank } F<n\}$. Then for each $k \geq 1$ there exists a constant $K>1$, which does not depend on $A$, with the following property: if $c_k(A)>Kc_{k+1}(A)$, then there exist decompositions $\mathfrak{X}_1=E \oplus F$, $\mathfrak{X}_2=E'\oplus F'$ such that:
- $AE=E'$ and $AF=F'$
- $E$ and $E'$ have dimension $k$, and $F$ and $F'$ are closed with codimension $k$
- $\|Av\|\geq K^{-1}c_k(A)\|v\|$ for all $v \in E$ and $\|Av\|\leq Kc_{k+1}(A)\|v\|$ for all $v \in F$
- The projection operator with image $E$ and kernel $F$, and the projection operator with image $E'$ and kernel $F'$, are both bounded operators with norm at most $K$.
This result seemed to us to be both useful and relatively easy to prove, so we were wondering if it is already documented somewhere in the literature and perhaps even has a name. However, we have not been successful in finding a reference for this result. Is anyone aware of a resource in which the above result is proved?
Thanks in advance!
