I am piggybacking off of How can I show that these two definitions of the pseudospectrum are equivalent?.
Given $A\in\mathbb{C}^{m\times m}$ with spectrum $\Lambda(A)\subseteq\mathbb{C}$ and $\epsilon>0$, define the 2-norm $\epsilon$-pseudospectrum of $A$, $A_\epsilon(A)$, to be the set of numbers $z\in\mathbb{C}$ satisfying any of the following conditions:
- (i) $z$ is an eigenvalue of $A+\delta A$ for some $\delta A$ with $||\delta A||_2\leq\epsilon$;
(ii) there exists a vector $u\in\mathbb{C}^m$ with $||(A-zI)u||_2\leq\epsilon$ and $||u||_2=1$;
(iii) $\sigma_m(zI-A)\leq\epsilon$; $\sigma_m$ is smallest singular value of $zI-A$.
(iv) $||(zI-A)^{-1}||_2\geq\epsilon^{-1}$.
Prove that conditions (i)–(iv) are equivalent.
I am trying to show how (ii) implies (iii).
I know $||(A-zI)u||_2\leq ||A-zI||_2 \cdot||u||_2 = ||A-zI||_2 = \sigma_n$, where $\sigma_n$ is the largest singular value, so that's of no use. Not to mention, the "direction" of the inequality is opposite to what I need it to be.
All other proofs (e.g. here) that I have seen have (i)->(ii)->(iv)->(i) and (i)<->(iii) specialized for the 2-norm. I'm curious about the specific implication of (ii)->(iii).
Any hints?