How to delete the first column ( which is in fact row names) from a data file in linux?

idiomatic use of cut will be

cut -f2- input > output

if you delimiter is tab ("\t").

Or, simply with awk magic (will work for both space and tab delimiter)

 awk '{$1=""}1' input | awk '{$1=$1}1' > output

first awk will delete field 1, but leaves a delimiter, second awk removes the delimiter. Default output delimiter will be space, if you want to change to tab, add -vOFS="\t" to the second awk.

UPDATED

Based on your updated input the problem is the initial spaces that cut treats as multiple columns. One way to address is to remove them first before feeding to cut

sed 's/^ *//' input | cut -d" " -f2- > output

or use the awk alternative above which will work in this case as well.

You can use cut command with --complement option:

cut -f1 -d" " --complement input.file > output.file

This will output all columns except the first one.

@Karafka I had CSV files so I added the "," separator (you can replace with yours

cut -d"," -f2- input.csv  > output.csv

Then, I used a loop to go over all files inside the directory

# files are in the directory tmp/
for f in tmp/*
do
    name=`basename $f`
    echo "processing file : $name"
    #kepp all column excep the first one of each csv file 

    cut -d"," -f2- $f > new/$name
    #files using the same names are stored in directory new/  
done