\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
In this paper we consider collections of compact (resp.
${\mathcal C}_p$ class) operators on arbitrary Banach (resp. Hilbert)
spaces. For a
subring $R$ of reals, it is proved that an $R$-algebra of compact
operators with spectra in $R$ on an arbitrary Banach space is
triangularizable if and only if every member of the algebra is
triangularizable. It is proved that every triangularizability result
on certain collections, e.g., semigroups, of compact operators on a
complex Banach (resp. Hilbert) space gives rise to its counterpart
on a real Banach (resp. Hilbert) space. We use our main results to
present new proofs as well as extensions of certain classical
theorems (e.g., those due to Kolchin, McCoy, and others) on
arbitrary Banach (resp. Hilbert) spaces.
\end{document}