Write the word 'M$\ddot{\rm u}$ller' in thebibliography

Your given code has a missing \end{frame} ...

With using \"u I can compile without errors:

\documentclass[xcolor=dvipsnames]{beamer}

\mode<presentation>
\usetheme{Madrid}

\begin{document}
\begin{frame}{}
     \begin{exampleblock}{\textcolor[rgb]{1.00,0.50,0.00}{Theorem} [M\"uller, Soltysiak (1992)]}
      For a $d$-tuple of ....
     \end{exampleblock}
\end{frame} % <=========================================================

 \begin{frame}
    \beamertemplatearticlebibitems
    \begin{thebibliography}{10}
     \bibitem{V. Muller, and A. Soltysiak}
    \alert{V. M\"uller, and A. Soltysiak} % <===========================
   \newblock  {Spectral radius formula for commuting Hilbert space operators}
   \newblock {\em Studia Math.103. (1992), 329-333}.
     \end{thebibliography}
 \end{frame}
\end{document}

with the result:

The accepted answer already solves your problem with the umlaut. To go one step further use latex's bibliography mechanism and beamer theorem environment and let them do the formatting for you:

\documentclass[xcolor=dvipsnames]{beamer}
\mode<presentation>
\usetheme{Madrid}


\makeatletter
\setbeamertemplate{theorem begin}
{%
    \setbeamercolor{block title}{bg=green!40!black}
  \begin{\inserttheoremblockenv}
  {%
    \textcolor{orange}{\inserttheoremname}
    \ifx\inserttheoremaddition\@empty\else\ \inserttheoremaddition\fi%
  }%
}
\makeatother

\begin{document}
\begin{frame}{}
\begin{theorem}[\cite{MS92}]
For a $d$-tuple of ....
\end{theorem}

\beamertemplatearticlebibitems
\begin{thebibliography}{9}
\bibitem[V. M\"{u}ller and A. Soltysiak (1992)]{MS92}
\alert{V. M\"{u}ller, and A. Soltysiak}
\newblock Spectral radius formula for commuting Hilbert space operators
\newblock \em Studia Math.103. (1992), 329-333.
\end{thebibliography}

\end{frame}
\end{document}