On commuting operator exponentials, II

We prove that, under sufficient conditions on the spectra, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ e^Me^N\subseteq e^Ne^M\Rightarrow MN\subseteq NM , $$\end{document} where N is an unbounded normal operator and M is a bounded normal operator in the Hilbert space.