Consider $X=l_{2}.$ Let $T : l_{2}\longrightarrow l_{2}$ be defined by : $T(x_{1},x_{2},....)= (x_{1},\frac{x_{2}}{2},\frac{x_{3}}{3},...).$ And $S=I$ , the identity operator. Here $N(T)=N(S)=\lbrace{0\rbrace}.$ Suppose that $T$ majorizes $I.$ Then $T$ has a closed range, a contradiction. I have two questions :
- Why R(T) is closed ?
- what is the contradiction?
i need help please
It $T\geq I$, then $T$ is bounded below and this implies closed range.
In your setting that's a contradiction because $T$ is compact, and a compact operator only has closed range when its range is finite-dimensional. For your concrete case $T$ has dense range, so if its range would be close, it would be surjective (which it isn't).