= Pseudospectrum
{title2=$\operatorname{Sp}_\epsilon(A)$}
{wiki}
For a <bounded linear operator> $A$ on a complex <Banach space> and $\epsilon>0$, use the open convention
$$
\sigma_\epsilon(A)=\sigma(A)\cup\{z\notin\sigma(A):\|(A-zI)^{-1}\|>\epsilon^{-1}\}.
$$
Thus the <pseudospectrum> detects both noninvertibility and a large <resolvent norm>. A <Neumann series> shows that perturbations of <operator norm> less than $\epsilon$ cannot create a new point of the <spectrum> where the displayed inverse <norm> is at most $\epsilon^{-1}$.
For a square complex <matrix>, the same condition is $s_{\min}(A-zI)<\epsilon$, where $s_{\min}$ is the least <singular value>. For general <bounded linear operators>, a small lower bound $\inf_{\|x\|=1}\|(A-zI)x\|$ alone need not characterize the entire <spectrum of a bounded operator>: the <unilateral shift operator> is bounded below but is not surjective. The definition retains these spectral points explicitly.
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