The occupied-projector response map on its strict-gap domain

Let $n\ge 2$, $1\le r<n$, and use the real Hermitian matrix space ${\mathbb{H}}_n$ and projector set ${\mathcal{P}}_r$ of Rank-r orthogonal projector domain. Let $U\subseteq {\mathbb{H}}_n$ be open and let $F:U\to {\mathbb{H}}_n$ be a specified map. For a Hermitian matrix $H$, write its eigenvalues with multiplicity in ascending order as ${\lambda }_1(H)\le \cdots \le {\lambda }_n(H)\mathrm{.}$ Write $\sigma (H):=\{{\lambda }_j(H):1\le j\le n\}$ for its finite spectrum, with multiplicity omitted in this set notation. For any Hermitian $H$ with ${\lambda }_r(H)<{\lambda }_{r+1}(H)$, write ${\mathrm{\Pi }}_r(H)$ for the orthogonal projector onto the sum of eigenspaces belonging to its $r$ lowest eigenvalues. At a point $X\in U$ where ${\lambda }_r(F(X))<{\lambda }_{r+1}(F(X)),$ define $\mathrm{\Phi }(X):={\mathrm{\Pi }}_r(F(X))$. The strict gap separates the occupied cluster from the unoccupied cluster; eigenvalue multiplicities within either cluster are allowed. The construction's well-definedness as a unique member of ${\mathcal{P}}_r$ is proved in A strict occupied gap defines a unique rank-r spectral projector ↗.

A set $K\subseteq U$ has a uniform occupied/unoccupied gap $g>0$ when the displayed gap is at least $g$ for every $X\in K$. A self-consistent fixed point is a point $X_{\ast }$ with $\mathrm{\Phi }(X_{\ast })=X_{\ast }$. Its fixed-point residual is $\rho (X):=\mathrm{\parallel }X-\mathrm{\Phi }(X){\mathrm{\parallel }}_F$ whenever $\mathrm{\Phi }(X)$ is defined. The map is not asserted to be contractive merely because its gap is positive.