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Relation Between Best Approximant and Orthogonality in $C_1$-classes  
 
  Authors: Mecheri Salah,  
  Keywords: Best approximant, Schatten $p$-classes, Orthogonality, $varphi$-Gateaux derivative.  
  Date Received: 21/05/05  
  Date Accepted: 13/03/06  
  Subject Codes:

Primary 41A52, 41A35, 47B47; Secondary 4

  
  Editors: Gradimir V. Milovanovic,  
 
  Abstract:

Let $E$ be a complex Banach space and let $M$ be subspace of $E$. In this paper we characterize the best approximant to $Ain E$ from $M$ and we prove the uniqueness, in terms of a new concept of derivative. Using this result we establish a new characterization of the best-$mathcal{C}_1$ approximation to $Ain mathcal{C}_1$ (trace class) from $M$. Then, we apply these results to characterize the operators which are orthogonal in the sense of Birkhoff.;


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