Residual distance and isolated-eigenspace certificate

Assume the Axiom of Choice and Countable Choice. Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space and let $T=T^{\ast }$ be self-adjoint on $H$, with spectral projection-valued measure $E_T$. Let $u\in D(T)$ satisfy $\mathrm{\parallel }u\mathrm{\parallel }=1$, let $\lambda \in \mathbb{R}$, and define the residual $r=(T-\lambda I)u$. For a Borel set $\mathrm{\Delta }\subseteq \mathbb{R}$, write $P_{\mathrm{\Delta }}=E_T(\mathrm{\Delta })$. If $g>0$ and $\mathrm{\mid }x-\lambda \mathrm{\mid }\ge g$ for every $x\in \sigma (T)\setminus \mathrm{\Delta }$, then

$\mathrm{dist}(\lambda ,\sigma (T))\le \mathrm{\parallel }r\mathrm{\parallel },\mathrm{\parallel }(I-P_{\mathrm{\Delta }})u\mathrm{\parallel }\le \frac{\mathrm{\parallel }r\mathrm{\parallel }}{g},$

where $\mathrm{dist}(\lambda ,S)={\inf }_{x\in S}\mathrm{\mid }x-\lambda \mathrm{\mid }$. If $\sigma (T)\setminus \mathrm{\Delta }=\mathrm{\varnothing }$, the second conclusion is $\mathrm{\parallel }(I-P_{\mathrm{\Delta }})u\mathrm{\parallel }=0$.

Spectral theorem for unbounded self-adjoint operators (PVM form)

Unbounded Borel functional calculus: domains, products, spectral mapping

The residual is the continuum quantity $(T-\lambda I)u$ and requires $u\in D(T)$. A raw discrete-matrix residual is not a continuum residual unless its vector and operator action are shown to satisfy these domain hypotheses. The distance estimate locates some spectrum; only the separate gap-and-cluster hypothesis bounds amplitude outside a specified spectral set.