Distance to normal elements in $C^*$-algebras of real rank zero
  We obtain an order sharp estimate for the distance from a given bounded operator $A$ on a Hilbert space to the set of normal operators in terms of $\|[A,A^*]\|$ and the distance to the set of invertible operators. A slightly modified estimate holds in a general $C^*$-algebra of real rank zero. 
Ilya Kachkovskiy, Yuri Safarov
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