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I want to know how to centre-align a text or a number while using multicolumn in Table environment. I need to prepare a table as shown in the image below.

The MWE for the above table is as follows:

documentclass{article}



%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%

usepackage{multirow}

usepackage{calc}

usepackage{array}

newcolumntype{L}[1]{> 

{raggedrightletnewline\arraybackslashhspace{0pt}}m{#1}}

newcolumntype{C}[1]{> 

{centeringletnewline\arraybackslashhspace{0pt}}m{#1}}

    newcolumntype{R}[1]{> 

{raggedleftletnewline\arraybackslashhspace{0pt}}m{#1}}

newcommand{mc}[3]{multicolumn{#1}{#2}{#3}}

newcommand{mr}[3]{multirow{#1}{#2}{#3}}

usepackage{booktabs}

begin{document}



begin{table}[ht!]

 begin{center}

begin{tabular}{cccccccccccccc}hline 

 mc{2}{|C{0.75cm}|}{} & mc{3}{C{2cm}|}{} & mc{3}{C{3cm}|}{1} & mc{3}{C{3cm}|}{2} & mc{3}{C{3cm}|}{3}\ hline 

%-----------------------------------------------------------------------------------------------------------------------

 mc{2}{|C{0.75cm}|}{} & mc{3}{C{2cm}|}{} & mc{3}{p{3cm}|}{centering $T_{alpha}^{-1}$ exists and \ is bounded } & mc{3}{p{3cm}|}{centering $T_{alpha}^{-1}$ exists and \ is unbounded } & mc{3}{C{3cm}|}{$T_{alpha}^{-1}$ does not exists}\ hline

%-----------------------------------------------------------------------------------------------------------------------

mc{2}{|L{0.75cm}|}{$I$} & mc{3}{C{2.5cm}|}{$R(T-alpha I)=X$} & mc{3}{p{3cm}|}{$alpha in rho (T,X)$} & mc{3}{C{3cm}|}{$-$} & mc{3}{p{3cm}|}{centering$alpha in sigma_{p} (T,X)$ \ $alpha in sigma_{ap} (T,X)$}\ hline

end{tabular}

 end{center}

end{table}

%

 end{document}

The output of the MWE looks similar to the one in the image. The only issue is the alignment of the text or numbers.

Please help or provide suggestions?

I think you were almost there.

documentclass{article}

usepackage{array}

usepackage{amsmath}

usepackage{bbm}

begin{document}



newcolumntype{C}{>{centeringarraybackslash}p{3cm}}



begin{table}

renewcommand{arraystretch}{1.3}

begin{tabular}{|l|l|C|C|C|}

hline

             &                   & 1              &2                 & multicolumn{1}{c|}{3}                                                                                  \ hline

             &                   & $T_alpha^{-1}$ exists and is bounded & $T_alpha^{-1}$ exists and is unbounded & $T_alpha^{-1}$ does not exist                                                                         \ hline

$mathbbm{1}$ & $R(T-alpha mathbbm{1})=X$ & $alpha in rho (T,X)$               & -                                       & 

$begin{array}{r@{}l}

alpha& in sigma_{p} (T,X)\ alpha& in sigma_{ap} (T,X)

end{array}$ \ hline

end{tabular}

end{table}

end{document}

Using tabularx gives you more flexibility for line wrapping and total page width table. To center a cell use multicolumn{1}{c|}{..} and you can increase vertical space for rows by renewcommand{arraystretch}{1.3} for more readability. Cells of the last row can be vertically centered by using multirow but this will not make much difference.

documentclass{article}

usepackage{array,tabularx}

begin{document}



newcolumntype{L}{>{raggedrightarraybackslash}X}



begin{table}

renewcommand{arraystretch}{1.3}

begin{tabularx}{textwidth}{|l|l|L|L|L|}

hline

             &                   & multicolumn{1}{c|}{1}                & multicolumn{1}{c|}{2}                  & multicolumn{1}{c|}{3}                                                                                  \ hline

             &                   & $T_alpha^{-1}$ exists and is bounded & $T_alpha^{-1}$ exists and is unbounded & $T_alpha^{-1}$ does not exists                                                                         \ hline

        $I$  & $R(T-alpha I)=X$ & $alpha in rho (T,X)$               & -                                       & $alpha in sigma_{p} (T,X)$ newline $alpha in sigma_{ap} (T,X)$ \ hline

end{tabularx}

end{table}



end{document}

You may also want to improve the whole design using booktabs as follows:

documentclass{article}

usepackage{array,tabularx,booktabs}

begin{document}



newcolumntype{L}{>{raggedrightarraybackslash}X}



begin{table}

%renewcommand{arraystretch}{1.3}

begin{tabularx}{textwidth}{@{}llLLL@{}}

toprule

             &                   & multicolumn{1}{c}{1}                & multicolumn{1}{c}{2}                  & multicolumn{1}{c}{3}                                                                                  \ midrule

             &                   & $T_alpha^{-1}$ exists and is bounded & $T_alpha^{-1}$ exists and is unbounded & $T_alpha^{-1}$ does not exists                                                                         \ 

textit{$I$} & $R(T-alpha I)=X$ & $alpha in rho (T,X)$               & -                                       & $alpha in sigma_{p} (T,X)$ newline $alpha in sigma_{ap} (T,X)$ \ bottomrule

end{tabularx}

end{table}



end{document}

Avoid jailed tables:

documentclass{article}



%%%%%%%%%% PACKAGES %%%%%%%%%%%%%%%%%%%%%%

usepackage{array}

usepackage{booktabs}

usepackage{tabularx}



newcolumntype{C}{>{centering}X}



begin{document}



begin{table}[ht!]

centering

renewcommand{tabularxcolumn}{m}



begin{tabularx}{textwidth}{@{}ccCCC@{}}

toprule

&& 1 & 2 & 3 tabularnewline

cmidrule(lr){3-3} cmidrule(lr){4-4} cmidrule(l){5-5}

&& $T_{alpha}^{-1}$ exists and \ is bounded &

   $T_{alpha}^{-1}$ exists and \ is unbounded & 

   $T_{alpha}^{-1}$ does not exist tabularnewline

midrule

$I$ & $R(T-alpha I)=X$ & $alpha in rho (T,X)$ & --- & 

$alpha in sigma_{p} (T,X)$ \ $alpha in sigma_{ap} (T,X)$ tabularnewline

bottomrule

end{tabularx}



end{table}



end{document}

Define only the columns you need. Here two centered columns at their natural width, then three equalized columns. For all columns, both vertical and horizontal centering.

Since you want to specify the line breaks, I don't state arraybackslash and use tabularnewline for ending rows.

It's easy to make it jailed, though.

begin{tabularx}{textwidth}{|c|c|C|C|C|}

hline

&& 1 & 2 & 3 tabularnewline

hline

&& $T_{alpha}^{-1}$ exists and \ is bounded &

   $T_{alpha}^{-1}$ exists and \ is unbounded &

   $T_{alpha}^{-1}$ does not exist tabularnewline

hline

$I$ & $R(T-alpha I)=X$ & $alpha in rho (T,X)$ & --- &

$alpha in sigma_{p} (T,X)$ \ $alpha in sigma_{ap} (T,X)$ tabularnewline

hline

end{tabularx}