The $\epsilon$-pseudospectral abscissa $\alpha_\epsilon$ and radius $\rho_\epsilon$ of an n x n matrix are respectively the maximal real part and the maximal modulus of points in its $\epsilon$-pseudospectrum, defined using the spectral norm. It was proved in [A.S. Lewis and C.H.J. Pang. Variational analysis of pseudospectra. SIAM Journal on Optimization, 19:1048-1072, 2008] that for fixed $\epsilon>0$, $\alpha_\epsilon$ and $\rho_\epsilon$ are Lipschitz continuous at a matrix A except when $\alpha_\epsilon$ and $\rho_\epsilon$ are attained at a critical point of the norm of the resolvent (in the nonsmooth sense), and it was conjectured that the points where $\alpha_\epsilon$ and $\rho_\epsilon$ are attained are not resolvent-critical. We prove this conjecture, which leads to the new result that $\alpha_\epsilon$ and $\rho_\epsilon$ are Lipschitz continuous, and also establishes the Aubin property with respect to both $\epsilon$ and A of the $\epsilon$-pseudospectrum for the points z in the complex plane where $\alpha_\epsilon$ and $\rho_\epsilon$ are attained. Finally, we give a proof showing that the pseudospectrum can never be “pointed outwards”.
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Submitted. Jan 2011.