  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.
