Stone's theorem on one-parameter unitary groups

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter families ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} of unitary operators that are strongly continuous, i.e., ∀ t 0 ∈ R , ψ ∈ H : lim t → t 0 U t ( ψ ) = U t 0 ( ψ ) , {\displaystyle \forall t_{0}\in \mathbb {R} ,\psi \in {\mathcal {H}}:\qquad \lim _{t\to t_{0}}U_{t}(\psi )=U_{t_{0}}(\psi ),} and are homomorphisms, i.e., ∀ s , t ∈ R : U t + s = U t U s . {\displaystyle \forall s,t\in \mathbb {R} :\qquad U_{t+s}=U_{t}U_{s}.} Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

Source: Wikipedia — Stone's theorem on one-parameter unitary groups (CC BY-SA 4.0)

Stone's theorem on one-parameter unitary groups

In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter families ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} of unitary operators that are strongly continuous, i.e., ∀ t 0 ∈ R , ψ ∈ H : lim t → t 0 U t ( ψ ) = U t 0 ( ψ ) , {\displaystyle \forall t_{0}\in \mathbb {R} ,\psi \in {\mathcal {H}}:\qquad \lim _{t\to t_{0}}U_{t}(\psi )=U_{t_{0}}(\psi ),} and are homomorphisms, i.e., ∀ s , t ∈ R : U t + s = U t U s . {\displaystyle \forall s,t\in \mathbb {R} :\qquad U_{t+s}=U_{t}U_{s}.} Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

This neuron ends here.

Source: Wikipedia "Stone's theorem on one-parameter unitary groups" · CC BY-SA 4.0

Share this article: X · Bluesky
Privacy Policy