Invariant subspaces and rational approximation

Let $T$ be a linear operator in a Banach space $X$ with the complete set of eigen- and root-vectors. Each of formulas (1)–(3) defines a “capacity” $\operatorname{cap}k$ of the integer valued function (the divisor) $k$, a capacity $\operatorname{cap}E\overset{\text{def}}=\operatorname{cap}k$ of the subspace $E\overset{\text{def}}=E^k$, generated by the root subspaces $\operatorname{Ker}(T-\lambda I)^s$, $0\le s<k(\lambda)$, $\lambda\in\mathbb C$, or a capacity $\operatorname{cap}x\overset{\text{def}}=\operatorname{cap}k$ of the vector $x$ $\operatorname{span}(T^nx:n\ge0)=E^k$. It is proved that
$$ \varliminf E^{k_n}\overset{\text{def}}= \{x:\lim\operatorname{dist}(x,E_{k_n})=0\}\neq X\Longleftrightarrow \varliminf\operatorname{cap}E^{k_n}<\infty $$
and that $x$ is not cyclic $(V(T^nx:n\ge0)\ne x)$, if $x=\lim_nx_n$, $\sup_n\operatorname{cap}x_n<\infty$. The principal special case $T=Z^*$, $Z^*f\overset{\text{def}}=\frac{f-f(0)}z$ is considered in detail. In this case root-vectors are rational functions. Bilateral estimates of capacities are given for the Hardy spaces $H^p$, $1\le p\le\infty$, and the spaces $C_A^{(n)}\overset{\text{def}}=\{f:f^{(n)}\in C_A\}$ ($C_A$ being the disc-algebra). These results imply known theorems of G. Tumarkin, H. Douglas–H. Shapiro–A. Shields and of H. Hilden–L. Wallen.