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

Read More: https://epubs.siam.org/doi/abs/10.1137/110822840