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arXiv · 1301.7349

Non-commutative f-divergence functional

Mohammad Sal Moslehian·Mohsen Kian
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Abstract

We introduce the non-commutative $f$-divergence functional $Θ(\widetilde{A},\widetilde{B}):=\int_TB_t^{\frac{1}{2}}f\left(B_t^{-\frac{1}{2}} A_tB_t^{-\frac{1}{2}}\right)B_t^{\frac{1}{2}}dμ(t)$ for an operator convex function $f$, where $\widetilde{A}=(A_t)_{t\in T}$ and $\widetilde{B}=(B_t)_{t\in T}$ are 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.

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Mohammad Sal Moslehian, Mohsen Kian. 2013-01-30. Non-commutative f-divergence functional. https://arxiv.org/abs/1301.7349

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