Pseudospectral Gap Controls Operator Rank: A Phase Theory for Linear Dynamical Systems

We prove that the pseudospectral geometry of a linear dynamical system's state-transition matrix dictates the effective dimensionality of its Hankel operator. The central result is a pseudospectral rank bound: for any system whose pseudospectrum is confined strictly inside the unit disk with gap $\gamma >0$, the $\epsilon$-rank of the Hankel operator is $O(\frac{1}{\gamma }\log \frac{1}{\gamma })$, independent of the sequence length $N$. Consequently, the end-to-end mapping collapses onto a manifold of effective dimension $o(N)$. Conversely, for critical systems with pseudospectrum on the unit circle, the Hankel operator achieves full numerical rank $\Theta (N)$. This yields an intrinsic geometric separation: stability enforces not merely memory decay, but a collapse of the output manifold to low effective dimension. The bottleneck of long-range representation is therefore not the persistence of signal, but the dimensionality ceiling of the linear map itself. Our analysis rests on the Kreiss matrix theorem and the decay of Hankel singular values. We introduce a one-parameter phase diagram governed by the pseudospectral radius $r_{\mathrm{ps}}$, unifying dissipative, stable, and unitary limits as points on a continuous spectral axis.