1

     \documentclass[xcolor=dvipsnames]{beamer}
    \usecolortheme[named=blue]{structure}

    %------------------------\AtBeginSubsection[]----------------------
    \AtBeginSubsection[]{
        \begin{frame}<beamer>[noframenumbering,plain]{Outline}
            \transsplitverticalout
            \tableofcontents[currentsection,currentsubsection]
        \end{frame}
    }

    \usetheme{Madrid}



    \setbeamercolor{author in head/foot}{bg=blue!30!black}
    \setbeamercolor{title in head/foot}{bg=yellow!30!white,fg=blue!30!black}
    \setbeamercolor{page number in head/foot}{bg=blue!30!black,fg=white}

    \title[The essential numerical range and the Olsen problem]{The
    essential numerical range and the Olsen problem}
    \author[{Student} (University of ....)]{Student}

       \useoutertheme{shadow}
          \usepackage{tikz}
         \usetikzlibrary{shadings}

        \colorlet{titleleft}{yellow!30!white}
         \colorlet{titleright}{blue!30!black}

         \setbeamercolor*{frametitle}{fg=blue!30!black}

       \makeatletter
       \pgfdeclarehorizontalshading[titleleft,titleright]
        {beamer@frametitleshade}
        {\paperheight}{%
         color(0pt)=(titleleft);
       color(\paperwidth)=(titleright)}
      \makeatother

    %-----------------------------footline-----------------------------
    \setbeamertemplate{footline}{
        \leavevmode%
        \hbox{%
            \begin{beamercolorbox}[wd=.3\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
                \usebeamerfont{author in head/foot}\insertshortauthor
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.6\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
                \usebeamerfont{title in head/foot}\insertshorttitle
    %            \hspace*{3em}
    %            \insertframenumber{} / \inserttotalframenumber\hspace*{1ex}
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,center]{page number in head/foot}%
    %            \usebeamerfont{title in head/foot}\insertshorttitle\hspace*{3em}
                \insertframenumber{} / \inserttotalframenumber
            \end{beamercolorbox}%
        }%
        \vskip0pt%
    }

    \begin{document}
\section{Introduction}
\section{Positive linear relations}
\section{Lattice linear relations}
\section{Domination by a positive compact relation.}
\subsection{Linear relation}
\subsection{Riesz spaces}
    \begin{frame}{Linear relations}

    \end{frame}

    \end{document} 

I want to change the frametitle as the style of {Student} (University of ....). Also I hope to change to colors of sections-subsections like the two colors of the foot.

enter image description here

1

With \colorlet{titleleft}{blue!30!black} \colorlet{titleright}{blue!30!black} \setbeamercolor*{frametitle}{fg=white} \setbeamercolor*{subsection in head/foot}{bg=yellow!30!white,fg=blue!30!black} you can achievce the following:

enter image description here

     \documentclass[xcolor=dvipsnames]{beamer}
    \usecolortheme[named=blue]{structure}

    %------------------------\AtBeginSubsection[]----------------------
    \AtBeginSubsection[]{
        \begin{frame}<beamer>[noframenumbering,plain]{Outline}
            \transsplitverticalout
            \tableofcontents[currentsection,currentsubsection]
        \end{frame}
    }

    \usetheme{Madrid}



    \setbeamercolor{author in head/foot}{bg=blue!30!black}
    \setbeamercolor{title in head/foot}{bg=yellow!30!white,fg=blue!30!black}
    \setbeamercolor{page number in head/foot}{bg=blue!30!black,fg=white}

    \title[The essential numerical range and the Olsen problem]{The
    essential numerical range and the Olsen problem}
    \author[{Student} (University of ....)]{Student}

       \useoutertheme{shadow}
          \usepackage{tikz}
         \usetikzlibrary{shadings}

        \colorlet{titleleft}{blue!30!black}
         \colorlet{titleright}{blue!30!black}

         \setbeamercolor*{frametitle}{fg=white}

         \setbeamercolor*{subsection in head/foot}{bg=yellow!30!white,fg=blue!30!black}


       \makeatletter
       \pgfdeclarehorizontalshading[titleleft,titleright]
        {beamer@frametitleshade}
        {\paperheight}{%
         color(0pt)=(titleleft);
       color(\paperwidth)=(titleright)}
      \makeatother

    %-----------------------------footline-----------------------------
    \setbeamertemplate{footline}{
        \leavevmode%
        \hbox{%
            \begin{beamercolorbox}[wd=.3\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
                \usebeamerfont{author in head/foot}\insertshortauthor
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.6\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
                \usebeamerfont{title in head/foot}\insertshorttitle
    %            \hspace*{3em}
    %            \insertframenumber{} / \inserttotalframenumber\hspace*{1ex}
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,center]{page number in head/foot}%
    %            \usebeamerfont{title in head/foot}\insertshorttitle\hspace*{3em}
                \insertframenumber{} / \inserttotalframenumber
            \end{beamercolorbox}%
        }%
        \vskip0pt%
    }

    \begin{document}
\section{Introduction}
\section{Positive linear relations}
\section{Lattice linear relations}
\section{Domination by a positive compact relation.}
\subsection{Linear relation}
\subsection{Riesz spaces}
    \begin{frame}{Linear relations}

    \end{frame}

    \end{document} 

For a more general approach, you might also be interested in defining your own colors. If you later want to change every blue element to have a different color, you only have to change one definition:

\colorlet{myblue}{blue!30!black}
\colorlet{myyellow}{yellow!30!white}
...
\setbeamercolor{author in head/foot}{bg=myblue}
\setbeamercolor{title in head/foot}{bg=myyellow,fg=myblue}
\setbeamercolor{page number in head/foot}{bg=myblue,fg=white}

\setbeamercolor*{subsection in head/foot}{bg=myyellow,fg=myblue}
...
\colorlet{titleleft}{myblue}  %<----- left half of frame title background color
\colorlet{titleright}{myblue} %<----- right half of frame title background color
\setbeamercolor*{frametitle}{fg=white} %<---- Color of the frame title text
  • Thank you very much for your great help. Please in general, If I want to change the color of the frametitle, what should I do? Thanks a lot. – Student Jul 20 at 16:45
  • @Student: Please see my edited answer for this. The background color of the frametitle is controlled by titleright and titleleft while {frametitle}{fg=... corresponds to the color of the frame title text. – leandriis Jul 20 at 16:57
  • Thank you for the details but perhaps you don't understand me very well. Please see my similar question:tex.stackexchange.com/questions/500753/… – Student Jul 20 at 17:38
0

Sorry,

your question is not very clear.

Maybe, do you want this results?

         \documentclass[xcolor=dvipsnames]{beamer}
    \usecolortheme[named=blue]{structure}

    %------------------------\AtBeginSubsection[]----------------------
    \AtBeginSubsection[]{
        \begin{frame}<beamer>[noframenumbering,plain]{Outline}
            \transsplitverticalout
            \tableofcontents[currentsection,currentsubsection]
        \end{frame}
    }

    \usetheme{Madrid}



    \setbeamercolor{author in head/foot}{bg=blue!30!black}
    \setbeamercolor{title in head/foot}{bg=yellow!30!white,fg=blue!30!black}
    \setbeamercolor{page number in head/foot}{bg=blue!30!black,fg=white}

    \title[The essential numerical range and the Olsen problem]{The
    essential numerical range and the Olsen problem}
    \author[{Student} (University of ....)]{Student}

       \useoutertheme{shadow}
          \usepackage{tikz}
         \usetikzlibrary{shadings}

        \colorlet{titleleft}{blue!30!black}
         \colorlet{titleright}{blue!30!black}

         \setbeamercolor*{frametitle}{fg=blue!30!white}

       \makeatletter
       \pgfdeclarehorizontalshading[titleleft,titleright]
        {beamer@frametitleshade}
        {\paperheight}{%
         color(0pt)=(titleleft);
       color(\paperwidth)=(titleright)}
      \makeatother

    %-----------------------------footline-----------------------------
    \setbeamertemplate{footline}{
        \leavevmode%
        \hbox{%
            \begin{beamercolorbox}[wd=.3\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
                \usebeamerfont{author in head/foot}\insertshortauthor
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.6\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
                \usebeamerfont{title in head/foot}\insertshorttitle
    %            \hspace*{3em}
    %            \insertframenumber{} / \inserttotalframenumber\hspace*{1ex}
            \end{beamercolorbox}%
            \begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,center]{page number in head/foot}%
    %            \usebeamerfont{title in head/foot}\insertshorttitle\hspace*{3em}
                \insertframenumber{} / \inserttotalframenumber
            \end{beamercolorbox}%
        }%
        \vskip0pt%
    }

    \begin{document}
\section{Introduction}
\section{Positive linear relations}
\section{Lattice linear relations}
\section{Domination by a positive compact relation.}
\subsection{Linear relation}
\subsection{Riesz spaces}
    \begin{frame}{Linear relations}

    \end{frame}

    \end{document} 

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