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  Volume 6, Issue 4, Article 110
 
Young's Inequality In Compact Operators - The Case Of Equality
    Authors: Renying Zeng,  
    Keywords: Young's Inequality, compact normal operator, Hilbert space.  
    Date Received: 24/08/05  
    Date Accepted: 22/09/05  
    Subject Codes:

47A63, 15A60.

  
    Editors: Fuzhen Zhang,  
 
    Abstract:

If $ a$ and $ b$ are compact operators acting on a complex separable Hilbert space, and if $ p,qin (1,infty )$ satisfy $ frac{1}{p}+frac{1}{q}=1$, then there exists a partial isometry $ u$ such that the initial space of $ u$ is $ (ker (vert ab^{ast} vert ))^bot $ and

$displaystyle uvert ab^{ast} vert u^{ast} leq frac{1}{p}vert avert ^p+frac{1}{q}vert bvert ^q.$    

Furthermore, if $ vert ab^{ast} vert $ is injective, then the operator $ u$ in the inequality above can be taken as a unitary. In this paper, we discuss the case of equality of this Young's inequality, and obtain a characterization for compact normal operators.

         
       
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