Suppose \(\displaystyle{T}^{{2}}={T}_{{0}}\) :

Then for all \(\displaystyle{u}\in{R}{\left({T}\right)}\), there exist some \(\displaystyle{v}\in{V}\) such that,

\(\displaystyle{T}{v}={u}\)

\(\displaystyle{0}={T}^{{2}}{v}={T}{\left({T}{\left({v}\right)}\right)}={T}{\left({u}\right)}\), so \(\displaystyle{u}\in{N}{\left({T}\right)}\).

Thus,

\(\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}\)

Now suppose \(\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}:\)

Then for all \(\displaystyle{v}\in{V},{T}{\left({v}\right)}\in{R}{\left({T}\right)}.\)

So, \(\displaystyle{T}{\left({v}\right)}\in{N}{\left({T}\right)}.\)

Thus, \(\displaystyle{0}={T}{\left({T}{\left({v}\right)}\right)}={T}^{{2}}{\left({v}\right)}\).

Hence,

\(\displaystyle{T}^{{2}}={T}_{{0}}\)

Then for all \(\displaystyle{u}\in{R}{\left({T}\right)}\), there exist some \(\displaystyle{v}\in{V}\) such that,

\(\displaystyle{T}{v}={u}\)

\(\displaystyle{0}={T}^{{2}}{v}={T}{\left({T}{\left({v}\right)}\right)}={T}{\left({u}\right)}\), so \(\displaystyle{u}\in{N}{\left({T}\right)}\).

Thus,

\(\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}\)

Now suppose \(\displaystyle{R}{\left({T}\right)}\subseteq{N}{\left({T}\right)}:\)

Then for all \(\displaystyle{v}\in{V},{T}{\left({v}\right)}\in{R}{\left({T}\right)}.\)

So, \(\displaystyle{T}{\left({v}\right)}\in{N}{\left({T}\right)}.\)

Thus, \(\displaystyle{0}={T}{\left({T}{\left({v}\right)}\right)}={T}^{{2}}{\left({v}\right)}\).

Hence,

\(\displaystyle{T}^{{2}}={T}_{{0}}\)