A residual and a spectral separation bound an invariant-subspace gap

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a real or complex Hilbert space and let $B_h:H\to H$ be compact and self-adjoint. Put ${\mathrm{\Sigma }}_h:=\{0\}\cup \{\nu \mathrm{\ne }0:\nu \text{ is an eigenvalue of }{B}_h\}\mathrm{.}$ Let $\mathrm{\Lambda }\subseteq {\mathrm{\Sigma }}_h\setminus \{0\}$ be a finite cluster of nonzero eigenvalues, let $F$ be the sum of their eigenspaces, and assume $\mathrm{\Delta }:=\inf \{\mathrm{\mid }\mu -\nu \mathrm{\mid }:\nu \in {\mathrm{\Sigma }}_h\setminus \mathrm{\Lambda }\}>0$ for a real number $\mu$. The compact spectral theorem makes $F$ finite-dimensional. Define the directed Hilbert-space gap by $\mathrm{gap}(E,F):={\sup }_{\begin{array}{c}u\in E\\ \mathrm{\parallel }u\mathrm{\parallel }=1\end{array}}{\inf }_{v\in F}\mathrm{\parallel }u-v\mathrm{\parallel }\mathrm{.}$ If $E\subseteq H$ is a nonzero finite-dimensional subspace and $\mathrm{\parallel }(B_h-\mu I)u\mathrm{\parallel }\le \epsilon \mathrm{\parallel }u\mathrm{\parallel }(u\in E),$ then $\mathrm{gap}(E,F)\le \epsilon \mathrm{/}\mathrm{\Delta }\mathrm{.}$ The estimate compares subspaces and makes no choice of eigenvectors in a multiple eigenspace.