multiplication operator on $L^2$

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