projection-valued measure (Q339722)

mathematical operator-valued measure of interest in quantum mechanics and functional analysis

PVM
spectral measure

Language	Label	Description	Also known as
default for all languages	No label defined
English	projection-valued measure	mathematical operator-valued measure of interest in quantum mechanics and functional analysis	PVM spectral measure

Statements

subclass of

positive-operator-valued measure

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short name

PVM (English)

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defining formula

{\begin{aligned}&\pi \colon M\to B(H)\\&\forall E\in M\colon \pi (E)^{2}=\pi (E)=\pi (E)^{\dagger }\\&\pi (X)=\operatorname {id} _{H}\\&\pi (\varnothing )=0_{H}\\&\forall E\in M^{\mathbb {N} },x\in H\colon \left(\forall i\neq j\colon E_{i}\cap E_{j}=\varnothing \right)\implies \left\langle \pi \left(\bigcup _{i=0}^{\infty }E_{i}\right)x,x\right\rangle =\sum _{i=0}^{\infty }\langle \pi (E_{i})x,x\rangle \end{aligned}}

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in defining formula

(X,M)

symbol represents

measurable space

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\pi

symbol represents

projection-valued measure

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H

symbol represents

Hilbert space

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B(H)

symbol represents

bounded operator

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M

symbol represents

sigma-algebra

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X

symbol represents

set

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\operatorname {id}

symbol represents

identity function

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\langle ,\rangle

symbol represents

inner product

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^{\dagger }

symbol represents

Hermitian adjoint

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maintained by WikiProject

WikiProject Mathematics

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Identifiers

1 reference

28 October 2013

Microsoft Academic ID

28502450

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projection-valued measure (Q339722)

Statements

Identifiers

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