Variation of real powers of positive operators

Abstract

For each real number p consider the map f(A)=A^p on the set of positive operators in a Hilbert space. Let Df(A) denote the Fréchet derivative of this map at A. In an earlier paper it was shown that ||Df(A)||=||pA^p−1|| when 0≤p≤1. Here we show that this is true also when p &<0 and when p≥2 but not when 1&<p&<√2. Some other functions are also discussed.