Operators on Hilbert Space

• multiplication operators
• Hilbert adjoint

Theorem (Multiplication Operator)

Let $(X,\mathcal{A},\mu )$ be a $\sigma$-finite measure space. For any $\phi \in L^{\infty }(X,\mathcal{A},\mu )$, the multiplication operator $M_{\phi }$, defined on the Hilbert space $L^2(X,\mathcal{A},\mu )$ by $M_{\phi }f=\phi f$, is a bounded linear operator of norm $\Vert M_{\phi }\Vert =\Vert \phi {\Vert }_{\infty }$.