Va\u{i}nberg--Br\`{e}gman relative entropy and quasinonexpansive operators

Abstract

We review the theory of Va\u{i}nberg--Br\`{e}gman relative entropies and quasinonexpansive operators on reflexive Banach spaces, and obtain several new results. We also develop an extension of this theory to nonreflexive Banach spaces, which is a joint generalisation of the reflexive Banach space approach and the finite-dimensional information geometric approach. In the reflexive case, we study generalised pythagorean inequality, as well as norm-to-norm, uniform, and Lipschitz--H\"{o}lder continuity, of (left and right) entropic projections, proximal maps, and resolvents. We also provide a detailed study of a special (`gauge') family of Va\u{i}nberg--Br\`{e}gman geometries and operators that is tightly related with the geometric properties of the underlying Banach space norm. The extended theory belongs to the intersection of convex theoretic and homeomorphic approaches to nonlinear analysis. Its models are constructed, using integration theory on order unit spaces, via nonlinear embeddings into reflexive rearrangement invariant spaces. E.g., we compute the exponent parameters of Lipschitz--H\"{o}lder continuity of the extended entropic projections and resolvents, and establish composability of a suitable class of nonlinear quasinonexpansive operators, over normal state spaces of JBW- and W*-algebras, determined by `gauge' Va\u{i}nberg--Br\`{e}gman geometries over, respectively, nonassociative and noncommutative L$_p$ spaces, and extended via Mazur embeddings. Other examples of extended Va\u{i}nberg--Br\`{e}gman geometries feature the (commutative and noncommutative) Lozanovski\u{i} factorisation map, generalised spin factors, finite dimensional base normed spaces, and convex spectral functions on unitarily invariant ideals of compact operators. We also discuss several categories of entropic projections and quasinonexpansive operators naturally appearing in this framework.