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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 $\operatorname{Dom}(B)\cap \operatorname{Dom}(A)=\operatorname{Dom}(A)$?
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1 Answer 1

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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.enter image description here

  • If $B$ is unbounded enter image description here whereenter image description here

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.

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