# multiplication operator on $L^{2}$

Let $(X,\mathcal{A},\mu)$ be a measure space and $f\colon X\to\mathbb{K}$ a measurable function. Then $M_{f}\colon\phi\mapsto f\phi$ is the multiplication operator with $f$ defined on the subspace $Dom(M_{f})=\{\phi\in L^{2}_{\mathbb{K}}(X,\mathcal{A},\mu)\colon f\phi\in L^{2% }_{\mathbb{K}}(X,\mathcal{A},\mu)\}$. It plays an important role in quantum mechanics where the multiplication with the coordinates on $\mathbb{R}^{n}$ is the position operator.

Title multiplication operator on $L^{2}$ MultiplicationOperatorOnL2 2013-03-22 15:42:28 2013-03-22 15:42:28 scineram (4030) scineram (4030) 8 scineram (4030) Definition msc 47B38 operator multiplication operator