Pseudospectrum (${\mathrm{Sp}}_{ϵ}(A)$)

For a bounded linear operator $A$ on a complex Banach space and $ϵ>0$, use the open conventionThus the pseudospectrum detects both noninvertibility and a large resolvent norm. A Neumann series shows that perturbations of operator norm less than $ϵ$ cannot create a new point of the spectrum where the displayed inverse norm is at most ${ϵ}^{-1}$.

For a square complex matrix, the same condition is $s_{\min }(A-zI)<ϵ$, where $s_{\min }$ is the least singular value. For general bounded linear operators, a small lower bound ${\inf }_{\Vert x\Vert =1}\Vert (A-zI)x\Vert$ 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.