Berezin-Lieb Inequality: An Extension to Normal Operators

The Berezin-Lieb inequality, which is a Jensen’s type inequality for convex functions of self-adjoint operators, is considered. We find a similar type of inequality for convex functions of normal operators. The normal operator is assumed to be bounded and acting on an infinite dimensional separable Hilbert space H. Finally the minimal closed convex set upon which it is sufficient to define the convex function is determined.Keywords: Berezin-Lieb inequality, normal operator, convex combination, closed convex hul