Let $T:l^{2}\rightarrow l^{2}$ be a bounded operator defined as follows,
If $x=\left( x_{1},x_{2},...\right) $ and $Tx=\left( y_{1},y_{2},...\right) $ then
$y_{n}=0 \text{ if }n=1 $
$\text{ }=\frac{x_{2^{i}}}{i} \text{ if }3\leq n=2^{i}+1 $
$\text{ }=x_{n-1} \text{ else}$
So we have :
$Tx=\left( 0,x_{1},\frac{x_{2}}{1},x_{3},\frac{x_{4}}{2},x_{5},x_{6},x_{7},% \frac{x_{8}}{3},...\right) $
My questions are :
1) Is $\mathcal{R}\left( T\right) $ closed ?
2) Is $\mathcal{R}\left( T-\lambda I\right) $ closed for sufficiently small $\left\vert \lambda \right\vert >0$ ?
3) Do we have for some $x\in l^{2}\setminus \left\{ 0\right\},$ $\underset{n\rightarrow \infty }{% \lim }\left\Vert T^{n}x\right\Vert ^{\frac{1}{n}}=0$ ?
Thank you !