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In this article, we first study, in the framework of operator theory, Pusz and Woronowiczs functional calculus for pairs of bounded positive operators on Hilbert spaces associated with a homogeneous two-variable function on $[0,infty)^2$. Our construction has special features that functions on $[0,infty)^2$ are assumed only locally bounded from below and that the functional calculus is allowed to take extended semibounded self-adjoint operators. To analyze convexity properties of the functional calculus, we extend the notion of operator convexity for real functions to that for functions with values in $(-infty,infty]$. Based on the first part, we generalize the concept of operator convex perspectives to pairs of (not necessarily invertible) bounded positive operators associated with any operator convex function on $(0,infty)$. We then develop theory of such operator convex perspectives, regarded as an operator convex counterpart of Kubo and Andos theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.
We improve the existing Ando-Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie-Trotter formula and consider the extension problem of operat
This paper concerns three classes of real-valued functions on intervals, operator monotone functions, operator convex functions, and strongly operator convex functions. Strongly operator convex functions were previously treated in [3] and [4], where
Recently the behavior of operator monotone functions on unbounded intervals with respect to the relation of strictly positivity has been investigated. In this paper we deeply study such behavior not only for operator monotone functions but also for o
In an earlier paper (A. N. Kochubei, {it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirovs fractional differentiation operator $D^alpha$, $alpha >0$, to radial functions on a non-Archimedean field. In particul
Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications for example in the formulation of quaterni