Operator ideal

<p>In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, an operator ideal is a special kind of <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of <a href="/facts/Continuous_linear_operator/KcqjDBrJ">continuous linear operators</a> between <a href="/facts/Banach_space/sDEIk7EC">Banach spaces</a>.  If an operator 
  
    
      
        T
      
    
    {\displaystyle T}
  
 belongs to an operator ideal 
  
    
      
        
          
            J
          
        
      
    
    {\displaystyle {\mathcal {J}}}
  
, then for any operators 
  
    
      
        A
      
    
    {\displaystyle A}
  
 and 
  
    
      
        B
      
    
    {\displaystyle B}
  
 which can be composed with 
  
    
      
        T
      
    
    {\displaystyle T}
  
 as 
  
    
      
        B
        T
        A
      
    
    {\displaystyle BTA}
  
, then 
  
    
      
        B
        T
        A
      
    
    {\displaystyle BTA}
  
 is class 
  
    
      
        
          
            J
          
        
      
    
    {\displaystyle {\mathcal {J}}}
  
 as well.  Additionally, in order for 
  
    
      
        
          
            J
          
        
      
    
    {\displaystyle {\mathcal {J}}}
  
 to be an operator ideal, it must contain the class of all finite-rank Banach space operators.
</p>

Operator ideal open-in-new

Operator ideal