Trace inequalities of Lipschitz type for power series of operators on Hilbert spaces

Abstract

Let f (z) = ∑_(n=0)^∞▒∝_(n ) z^n be a function defined by power series with complex coefficient s and convergent n the open disk D(0;R) ⊂ ℂ, R > 0. We show, amongst other that, if T, V ∈ β_1(H), the Banach space of all trace operators on H, are such that ∥T∥₁ ,∥V∥₁ < R, then f(V ), f(T), f ′ ((1 − t)T + tV ) ∈ ℬ₁ (H) for any t ∈ [0; 1] and tr [f(V )] − tr [f(T)] = ∫_0^1▒tr [ (V − T)f ′( (1 − t)T + tV )] dt. Several trace inequalities are established. Applications for some elementary functions of interest are also given.