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Let $\Omega$ a bounded open set of $R^n$, $\omega$ a non-empty open subset of $\Omega$ and

$\chi_{\omega} : L^2(\Omega) \longrightarrow L^2(\omega)$ be the restriction operator to $\omega$, while $\chi_{\omega}^*$ denotes the adjoint operator of $\chi_{\omega}$ and given by \begin{equation}\label{v1} (\chi_{\omega}^*y)(x)=\left\{ \begin{array}{lll} y(x),\quad x\in \omega\\ 0, \quad otherwise \end{array} \right. \end{equation} and denote $i_{\omega} = \chi_{\omega}^*\chi_{\omega}$ ($i_{\omega}$ is a positive operator).

On the other hand, let $A :\mathcal{D}(A)\subset L^2(\Omega) \longrightarrow L^2(\Omega)$ be a linear unbounded operator such that $\langle Ay,y\rangle\leq 0, \forall y\in\mathcal{D}(A)$:

My question is : can we prove that $\langle i_{\omega}Ay,y\rangle\leq 0$ ?

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No, of course not. Even if $A$ is bounded, this would imply that $i_\omega A$ is self-adjoint. So you only have a chance if $A$ and $i_\omega$ commute.

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