Non‐commutative f ‐divergence functional

We introduce the non‐commutative f ‐divergence functional Θ A ̃ , B ̃ : = ∫ T B t 1 2 f B t − 1 2A t B t − 1 2B t 1 2 d μ ( t )for an operator convex function f , whereA ̃ = ( A t ) t ∈ TandB ̃ = ( B t ) t ∈ Tare continuous fields of Hilbert space operators and study its properties. We establish some relations between the perspective of an operator convex function f and the non‐commutative f ‐divergence functional. In particular, an operator extension of Csiszár's result regarding f ‐divergence functional is presented. As some applications, we establish a refinement of the Choi–Davis–Jensen operator inequality, obtain some unitarily invariant norm inequalities and give some results related to the Kullback–Leibler distance.