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Let $M_{\phi}$ be a multiplication operator $M_{\phi}:L^{2}\left(\mu\right)\rightarrow L^{2}\left(\mu\right)$ defined by $M_{\phi}f=\phi f$.

  1. Show that $\ker M_{\phi}=0$ if and only if $\mu\left(\left\{ x:\phi\left(x\right)=0\right\} \right)=0$.

  2. Give necessary and sufficient conditions on $\phi$ that $\mbox{ran}M_{\phi}$ be closed.

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    I'm too lazy to write the proof. Here are my results: $M_\phi$ have non trivial kernel iff $\mu(\phi^{-1}(\{0\}))>0$ $M_\phi$ have closed image iff $\mu(\phi^{-1}(\mathrm{Ball}_\mathbb{C}(0,r)\setminus\{0\}))=0$ for some $r>0$2012-11-22

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