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

Abstract

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.