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 $L^{2}$
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

multiplication operator on L2

multiplication operator on $L^{2}$