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Let $A:D(A) \subseteq H \to H$ and $B:D(B) \subseteq H \to H$ be linear operators on a Hilbert space $H$ such that $A$ is a closed densely defined operator and $B$ is relatively bounded with respect to $A$ with relative bound $0$. We have to show that $D((A+B)^*)= D(A^*)$.

Since $B$ is $A$-bounded with $A$-bound $0$ and $A$ is closed, we have that $A+B$ is a densely defined closed operator with $D(A+B)=D(A)$. We know that $D(A^*) \subseteq D((A+B)^*)$ because $A^*+B^* \subseteq (A+B)^*$, but how can we show that $D((A+B)^*) \subseteq D(A^*)$.

Can you give me any hint or a reference for the adjoint of a relatively bounded perturbation, please?

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  • $\begingroup$ According to the definition I found in Kato, $B$ is $A$ bounded with relative bound $\xi$ if $D(B)\subset D(A)$ and there is a $\beta$ so that $\|Bu\|≤\beta \|u\| + \xi\|Au\|$. If the relative bound is $0$ then $B$ must be bounded by this definition. What definition are you working with? $\endgroup$ – s.harp Jul 30 at 20:53
  • $\begingroup$ @s.harp According to the definition of Kato, if $B$ is $A$- bounded with relative bound $\xi$, we can not guarantee that there exists $\beta>0$ such that $\|Bu\| \leq \beta \|u\| + \xi \| Au\|$ for $u \in D(A)$. $\endgroup$ – Mainkit Jul 30 at 21:44
  • $\begingroup$ Then what is the definition you are using? $\endgroup$ – s.harp Jul 30 at 21:47
  • $\begingroup$ I'm using definition from Kato: B is $A$-bounded if $D(A) \subseteq D(B)$ and there exists $a,b \geq 0$ such that $\|Bu \|\leq a \| u\|+ b\|Au \| ->(1)$ for all $u \in D(A)$. In that case we say that the relative bound is the infimum of $b\geq0$ such that (1) holds for some $a\geq 0$. $\endgroup$ – Mainkit Jul 30 at 21:52

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