# multiplication operator on ${L}^{2}$

Let $(X,\mathcal{A},\mu )$ be a measure space^{} and $f:X\to \mathbb{K}$ a measurable function^{}. Then ${M}_{f}:\varphi \mapsto f\varphi $ is the multiplication operator with $f$ defined on the subspace^{} $Dom({M}_{f})=\{\varphi \in {L}_{\mathbb{K}}^{2}(X,\mathcal{A},\mu ):f\varphi \in {L}_{\mathbb{K}}^{2}(X,\mathcal{A},\mu )\}$. 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 ${L}^{2}$ |
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Canonical name | MultiplicationOperatorOnL2 |

Date of creation | 2013-03-22 15:42:28 |

Last modified on | 2013-03-22 15:42:28 |

Owner | scineram (4030) |

Last modified by | scineram (4030) |

Numerical id | 8 |

Author | scineram (4030) |

Entry type | Definition |

Classification | msc 47B38 |

Related topic | operator |

Defines | multiplication operator |