PlanetMath: multiplication operator on $L^2$

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multiplication operator on $L^2$

(Definition)

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 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.

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See Also: operator

Also defines:

multiplication operator

Keywords:

multiplication operator

Attachments:: operator norm of multiplication operator on (Theorem) by rspuzio

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Cross-references: operator, coordinates, multiplication, subspace, measurable function, measure space
There are 5 references to this entry.

This is version 5 of multiplication operator on $L^2$

, born on 2006-02-25, modified 2006-04-01.
Object id is 7654, canonical name is MultiplicationOperatorOnMathbbL22.
Accessed 2017 times total.

Classification:

AMS MSC:

47B38 (Operator theory :: Special classes of linear operators :: Operators on function spaces )

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