Stone's theorem on one-parameter unitary groups

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, Stone's theorem on <a href="/facts/One-parameter_group/Ep7zASA0">one-parameter</a> <a href="/facts/Unitary_group/lEvudIhU">unitary groups</a> is a basic theorem of <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a> that establishes a one-to-one correspondence between <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a> on a <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert space</a> 
 
 
 
 
 
 H
 
 
 
 
 {\displaystyle {\mathcal {H}}}
 
 and one-parameter families

(
        
          U
          
            t
          
        
        
          )
          
            t
            ∈
            
              R
            
          
        
      
    
    {\displaystyle (U_{t})_{t\in \mathbb {R} }}

of <a href="/facts/Unitary_operator/rkypfQQ9">unitary operators</a> that are <a href="/facts/Strong_operator_topology/awhSW5op">strongly continuous</a>, 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.
The theorem was proved by <a href="/facts/Marshall_Stone/TFzbE4GH">Marshall Stone</a> (1930, 1932), and <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a> (1932) showed that the requirement that 
 
 
 
 (
 
 U
 
 t
 
 
 
 )
 
 t
 ∈
 
 R
 
 
 
 
 
 {\displaystyle (U_{t})_{t\in \mathbb {R} }}
 
 be strongly continuous can be relaxed to say that it is merely <a href="/facts/Weakly_measurable_function/cAFxsUvL"> weakly measurable</a>, at least when the Hilbert space is <a href="/facts/Separable_space/2bmU73bk"> separable</a>.
This is an impressive result, as it allows one to define the <a href="/facts/Derivative/7Ito70Dh">derivative</a> of the <a href="/facts/Map_(mathematics)/FABF1l2f"> mapping</a> 
 
 
 
 t
 ↦
 
 U
 
 t
 
 
 ,
 
 
 {\displaystyle t\mapsto U_{t},}
 
 which is only supposed to be <a href="/facts/Continuous_function/sKbl02pB"> continuous</a>. It is also related to the theory of <a href="/facts/Lie_group/a9jCQyjF">Lie groups</a> and <a href="/facts/Lie_algebra/n2gAUkNq">Lie algebras</a>.

Stone's theorem on one-parameter unitary groups open-in-new

Stone's theorem on one-parameter unitary groups