Hilbert–Schmidt operator | Wikipedia audio article
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Date: 06/05/2019

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SUMMARY
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In mathematics, a Hilbert–Schmidt operator, named for David Hilbert and Erhard Schmidt, is a bounded operator A on a Hilbert space H with finite Hilbert–Schmidt norm

  
    
      
        ‖
        A
        
          ‖
          
            HS
          
          
            2
          
        
        =
        Tr
        ⁡
        
          (
          
            
              A
              
                ∗
              
            
            A
          
          )
        
        :=
        
          ∑
          
            i
            ∈
            I
          
        
        
          
            ‖
            
              A
              
                e
                
                  i
                
              
            
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            2
          
        
      
    
    {\displaystyle \|A\|_{\text{HS}}^{2}=\operatorname {Tr} \left(A^{*}A\right):=\sum _{i\in I}\left\|Ae_{i}\right\|^{2}}
  where  
  
    
      
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    {\displaystyle \|\ \|}
   is the norm  of H, 
  
    
      
        
          {
          
            
              e
              
                i
              
            
            :
            i
            ∈
            I
          
          }
        
      
    
    {\displaystyle \left\{e_{i}:i\in I\right\}}
   an orthonormal basis  of H, and Tr is the trace of a nonnegative self-adjoint operator. Note that the index set need not be countable. This definition is independent of the choice of the basis, and therefore

  
    
      
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    {\displaystyle \|A\|_{\text{HS}}^{2}=\sum _{i,j}\left|A_{i,j}\right|^{2}=\|A\|_{2}^{2}}
  for 
  
    
      
        
          A
          
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          ⟨
          
            
              e
              
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    {\displaystyle A_{i,j}=\left\langle e_{i},Ae_{j}\right\rangle }
   and 
  
    
      
        ‖
        A
        
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            2
          
        
      
    
    {\displaystyle \|A\|_{2}}
   the Schatten norm of 
  
    
      
        A
      
    
    {\displaystyle A}
   for p = 2. In Euclidean space 
  
    
      
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            HS
          
        
      
    
    {\displaystyle \|\ \|_{\text{HS}}}
   is also called Frobenius norm, named for Ferdinand Georg Frobenius.
The product of two Hilbert–Schmidt operators has finite trace class norm; therefore, if A and B are two Hilbert–Schmidt operators, the Hilbert–Schmidt inner product can be defined as

  
    
      
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