Perturbation of a maximal dissipative operator by a non-negative self-adjoint operator.

Question

Let $A$ be a maximal dissipative operator in a Hilbert space $\mathcal{H}$, and consider $B$ a self-adjoint operator such that $$ \langle B\xi,\xi \rangle \geq 0\ , \quad \xi\in \mathcal{H}\ .$$ Does $A+B$ remain a maximal dissipative operator if

• $B$ is a bounded operator?
• $B$ is an unbounded operator with $\operatorname{Dom}(B)\cap \operatorname{Dom}(A)=\operatorname{Dom}(A)$?

Answer 1

I find the answer in the book: "A short Course on operator semigroups" by Klaus-Jochen Engel and Rainer Nagel, in chapter 3.

• If $B$ is bounded.

• If $B$ is unbounded where

Note that there are more general theorems concerning the case where $B$ is an unbounded operator. We refer to sections 2 and 3 of chapter 3 of the same book.