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Non‐commutative f ‐divergence functional
Author(s) -
Moslehian Mohammad Sal,
Kian Mohsen
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201200194
Subject(s) - mathematics , divergence (linguistics) , commutative property , hilbert space , pure mathematics , invariant (physics) , convex function , operator (biology) , norm (philosophy) , regular polygon , philosophy , linguistics , biochemistry , geometry , chemistry , repressor , political science , transcription factor , law , mathematical physics , gene
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.