For any complex square matrix $A$ and any $ε> 0$, the $ε$-pseudospectrum $Λ_ε(A)$ of $A$ is defined as the union of the spectra of $A+E$, where $E$ is any matrix with matrix 2-norm less than or equal to $ε$. The main result of this paper is a pseudospectral mapping theorem for complex analytic functions in the following form. Let $A$ be a matrix and $f$ be an analytic function on an open set containing the spectrum of $A$. For sufficiently small nonnegative numbers $ε$ and $s$, let $$ϕ(ε)= \sup_{ζ\inΛ_ε(A)}\text{inf}\{r\ge 0: f(ζ)\in Λ_r(f(A))\}$$ and $$ψ(s)= \sup_{z\inΛ_ε(f(A))}\text{ inf}\{r\ge 0: z\in f(Λ_r(A))\}.$$ Then $f(Λ_ε(A))\subseteq Λ_{ϕ(ε)}(f(A))\subseteq f(Λ_{ψ(ϕ(ε))}(A))$. When $ε= 0$, this reduces to the usual spectral mapping theorem. On the other hand, when $f$ is an affine function ($f(z)= α+βz$ for some complex $α$ and $β$), the set inclusions reduce to the equality $Λ_{ε|β|}(α+ βA)= α+ βΛ_ε(A)$, a case obtained before by {\it L. N. Trefethen} [Springer Ser. Comput. Math. 26, 217‒250 (1999; Zbl 0943.15003)]. Here the author also shows that this is the only class of functions $f$ for which the pseudospectral mapping theorem can be written as an equality for all matrices $A$: $f(Λ_ε(A))= Λ_{η(A,ε)}(f(A))$ for some nonnegative real-valued function $η= η(A,ε)$. For the case of computation, he also derives a weak form of the pseudospectral mapping theorem, which, unfortunately, cannot be further simplified even for a normal $A$. In the final section, some numerical computations with the Grcar and Kahan matrices are performed to illustrate the theory.
Reviewer: Pei Yuan Wu (Hsinchu)