On p.88, Definition 4.5 should read as follows: A point z in C is "degenerate" if every right singular vector u corresponding to the smallest singular value of A-zI satisfies u*(A-zI)u = 0.
On p.89, we observe: "Thus, the pseudospectrum is not pointed outward, except possibly at a degenerate point. In fact, a more detailed analysis due to Trefethen shows that the pseudospectrum is NEVER pointed outward. However, it can certainly be pointed inward, as the following example shows."
Although correct, it would be better to replace this by:
"Prop. 4.1 implies that at a nongenerate point, the pseudospectrum cannot be not pointed outward, that is, given any pointed closed convex cone whose vertex is that point, there are nearby points in the pseudospectrum lying outside the cone. In fact, a more detailed analysis due to Trefethen shows that the pseudospectrum is NEVER pointed outward [41]. However, it is certainly possible for the pseudospectrum to be pointed inwards, even at a nondegenerate point, as the following example shows."
On p.94, the first sentence after Corollary 7.2 should be deleted. In the next sentence, "At a nondegenerate smooth point z_0 with" should be replaced by, "With the above notation, if v*u is nonzero and".
On p.100, line 7, the word "choose" was omitted before "matrices".
Thanks to Juan-Miguel Gracia for drawing our attention to the flaw in our original definition of "degenerate".