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Let $H$ be a separable Hilbert space and $\Omega$ is a $\sigma$ finite measure space. Suppose that $U$ is a bounded linear operator on $L^2(\Omega , H)$ which commutes with the multiplication operator by a $g \in L^{\infty}(\Omega)$. I want to define a bounded linear operator $R(\omega)$ on $H$ such that \begin{equation*} R(\omega)(g(\omega)\varphi(\omega)) = g(\omega) U(\varphi)(\omega) \end{equation*} I define $R(\omega)$ as above exactly, linearity is clear , but i cant show that $R(\omega)$ is bounded unfortunately. Can you help me? Thanks.

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