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Let  A  be a fixed normal operator and let      N ( Λ  ) \mathfrak {N}(\Lambda )       denote the normal operators with spectrum contained in      Λ  \Lambda      . Provided there is some  N  in      N ( Λ  ) \mathfrak {N}(\Lambda )       such that      A −  N A - N       belongs to the Schatten class      c p , p ⩾  2 {c_p},p \geqslant 2      , the main result of this paper obtains a best approximation for  A  from      N ( Λ  ) \mathfrak {N}(\Lambda )       with respect to the Schatten  p -norm. A necessary and sufficient condition is given for  A  to have a unique best approximation in that case.

Best approximation of a normal operator in the Schatten 𝑝-norm