Dissipative operator

Question

Let $A$ a linear operator from his domaine $D(A)\subset H$ to $H$, with $H$ is a Hilbert space, such that $A$ is dissipative.

is it true that : if $\|y\|\leq \|z\|$ then $\|Ay\|\leq \|Az\|$?

Thank you

Answer 1

No.

Let $H = \mathbb{R}^2$ and set

$$ A = \begin{pmatrix} -1 & 0 \newline 0 & -100 \end{pmatrix} $$

which is clearly dissipative. Now take

$$ y = \begin{pmatrix} 0 \newline 1\end{pmatrix} \qquad z = \begin{pmatrix} 1 \newline 0 \end{pmatrix} $$

and you have a counter example.