Generator of an analytic semigroup of operators

Question

I have an operator of the following form: $$ A = \begin{bmatrix} 0 & h_1 & h_2 \\ 0 & \Delta& h_3 \\ 0 & 0 & h_4 \end{bmatrix} $$ which results from a coupled PDE-ODE system and $\Delta$ is the Laplace operator. I would like to show that $A$ is a sectorial operator and hence generates an analytic semigroup.
Here $h_i$ for $i=1\ldots4$ are constants and $h_4 < 0$.

The definition of an analytic semigroup is here on Wikipedia.

I do not know how to show $$ \|R_{\lambda}(A)\|\le\frac{C}{|\lambda-\omega|} $$ for $\mathrm{Re}(\lambda)>\omega$ and $R_\lambda(A)$ is the resolvent of $A$, which is sufficient that A generates an analytic semigroup. But it is known that the Laplace operator is a sectorial operator and satisfy this property and the spectrum of A is $\sigma(A) = \{0\}\cup\sigma(\Delta) \cup \{h_4\} $.

Answer 1

As you write it, this is just a bounded perturbation of the sectorial operator

$$\begin{pmatrix} 0 & 0 & 0 \\ 0 & \Delta & 0 \\ 0 & 0 & 0 \end{pmatrix}.$$

It is quite standard that bounded perturbation does not destroy this property.