Pusz--Woronowicz functional calculus and extended operator convex perspectives

TL;DR

This paper extends the Pusz--Woronowicz functional calculus for pairs of positive operators, introduces a generalized operator convexity concept, and develops a theory of operator convex perspectives as an operator mean framework.

Contribution

It introduces a novel functional calculus for operator pairs with minimal assumptions and generalizes operator convexity to extended real-valued functions, expanding the operator mean theory.

Findings

Extended functional calculus for bounded positive operators.

Generalized operator convexity for functions with extended real values.

Development of operator convex perspectives as operator means.

Abstract

In this article, we first study, in the framework of operator theory, Pusz and Woronowicz's 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 Ando's theory of operator means. Among other results, integral expressions and axiomatization are discussed for our operator perspectives.