Help verifying the norm of the resolvent of a matrix

Question

I'm reading a document where it is said that if $$A=\begin{pmatrix}0 & 1\\0 & 0 \end{pmatrix}$$ then the norm of the resolvent for $z \neq 0$ is given by $$\|R(z,A)\|= \frac{\sqrt{2}}{\sqrt{1+2|z|^2-\sqrt{1+4|z|^2}}}.$$ I think that if $z\neq 0$ then $$ R(z,A)=(A-zI)^{-1}=\begin{pmatrix}-1/z & -1/z^2\\0 & -1/z \end{pmatrix}$$ and because of that $$\|R(z,A)\|=\frac{\sqrt{2|z|^2+1}}{|z|^2}.$$

Am I wrong?.

Answer 1

You have computed the Frobenius norm of the resolvent, whereas the first formula uses the spectral norm of the resolvent.