A separated block spectral projector is Frechet differentiable with a resolvent bound

Since $\mathrm{\parallel }A{\mathrm{\parallel }}_{op}\le \mathrm{\parallel }A{\mathrm{\parallel }}_F$ by [F6], every point of the operator-norm ball has a Frobenius neighborhood contained in it; hence the stated domain is open for the real Frechet definition. Diagonalize $H=U\mathrm{diag}({\mu }_1,\dots ,{\mu }_n)U^{\ast }$ with ${\mu }_1\le \cdots \le {\mu }_n$ by [F2]. Since [F1] places exactly ${\mu }_1,\dots ,{\mu }_r$ inside $\mathrm{\Gamma }$, the resolvent is $U\mathrm{diag}((z-{\mu }_j{)}^{-1})U^{\ast }$ on the contour.

For a sufficiently small Hermitian perturbation $E$ with $H+E$ still in the ball, the inverse identity gives $R_{H+E}=R_H+R_{H}E{R}_{H+E}$. Subtracting $R_H+R_{H}E{R}_H$ yields $R_{H+E}-R_H-R_{H}E{R}_H=R_{H}E{R}_{H}E{R}_{H+E}\mathrm{.}$ The identity follows by multiplying the two inverse equations for $zI-H$ and $zI-H-E$; all inverses exist by [F1].

Interpret a matrix-valued contour integral entrywise; each entry of the resolvent and derivative integrands is continuous on $\mathrm{\Gamma }$, so its scalar integral exists by [F4]. Finite matrix sums and constant left and right factors commute with this integral, and the Frobenius norm of an integral is at most the contour length times the supremum Frobenius norm of its integrand, by the triangle inequality for finite-dimensional Riemann sums. Integrating each diagonal entry and using [F3], linearity in [F4], and the finite spectral expansion gives $\frac{1}{2\pi i}{\int }_{\mathrm{\Gamma }}{R}_H(z)dz=U\mathrm{diag}(1,\dots ,1,0,\dots ,0)U^{\ast }={\mathrm{\Pi }}_r(H)\mathrm{.}$ Thus the contour expression represents the full occupied cluster projector, including repeated eigenvalues within that cluster.

For any matrix $A$ and matrix $X$ with columns $x_j$, the operator-norm definition gives $\mathrm{\parallel }AX{\mathrm{\parallel }}_F^2={\sum }_j\mathrm{\parallel }A{x}_j{\mathrm{\parallel }}_2^2\le \mathrm{\parallel }A{\mathrm{\parallel }}_{op}^2\mathrm{\parallel }X{\mathrm{\parallel }}_F^2$. Applying this also to adjoints gives the right-multiplication bound $\mathrm{\parallel }XB{\mathrm{\parallel }}_F\le \mathrm{\parallel }X{\mathrm{\parallel }}_F\mathrm{\parallel }B{\mathrm{\parallel }}_{op}$. Iterating these inequalities, then using [F1] and [F6], bounds the Frobenius norm of the right side by $(4\mathrm{/}g{)}^3\mathrm{\parallel }E{\mathrm{\parallel }}_F^2$, uniformly for $z\in \mathrm{\Gamma }$. The matrix-valued integral estimate in step 2.1 bounds the integrated remainder by $\mathrm{\ell }(\mathrm{\Gamma })(4\mathrm{/}g{)}^3\mathrm{\parallel }E{\mathrm{\parallel }}_F^2\mathrm{/}(2\pi )$. Dividing by $\mathrm{\parallel }E{\mathrm{\parallel }}_F$ makes this tend to zero, so the contour formula in step 2.1 is Frechet differentiable in the sense of [F5], with the stated derivative; the integral is real-linear in $E$ by entrywise linearity. Each directional derivative is a limit of Hermitian difference quotients, so it is Hermitian-valued because ${\mathbb{H}}_n$ is closed.

The same left/right multiplication inequalities give $\mathrm{\parallel }{R}_H(z)E{R}_H(z){\mathrm{\parallel }}_F\le (4\mathrm{/}g{)}^2\mathrm{\parallel }E{\mathrm{\parallel }}_F$ by [F1] and [F6]. Integrating over length $2\pi R$ yields $\mathrm{\parallel }D{\mathrm{\Pi }}_r(H)[E]{\mathrm{\parallel }}_F\le R(4\mathrm{/}g{)}^2\mathrm{\parallel }E{\mathrm{\parallel }}_F=16R\mathrm{\parallel }E{\mathrm{\parallel }}_F\mathrm{/}{g}^2$. For nearby $H^{\mathrm{\prime }}$ in the ball, the resolvent identity and [F1] give uniformly on $\mathrm{\Gamma }$ $\mathrm{\parallel }{R}_{H^{\mathrm{\prime }}}-R_H{\mathrm{\parallel }}_{op}\le (4\mathrm{/}g{)}^2\mathrm{\parallel }{H}^{\mathrm{\prime }}-H{\mathrm{\parallel }}_{op}$. Subtracting the two derivative integrands gives $R_{H^{\mathrm{\prime }}}E(R_{H^{\mathrm{\prime }}}-R_H)+(R_{H^{\mathrm{\prime }}}-R_H)E{R}_H$. Each summand has Frobenius norm at most $(4\mathrm{/}g{)}^3\mathrm{\parallel }{H}^{\mathrm{\prime }}-H{\mathrm{\parallel }}_F\mathrm{\parallel }E{\mathrm{\parallel }}_F$ by the left/right multiplication bounds and [F6], so their sum is at most twice this quantity. The matrix-valued integral estimate from step 2.1 then bounds the difference of derivative values by $2R(4\mathrm{/}g{)}^3\mathrm{\parallel }{H}^{\mathrm{\prime }}-H{\mathrm{\parallel }}_F\mathrm{\parallel }E{\mathrm{\parallel }}_F$. Taking the supremum over $\mathrm{\parallel }E{\mathrm{\parallel }}_F=1$ shows that the derivative varies continuously in operator norm, which by [F7] proves the claimed $C^1$ regularity; the bound was proved above.