Bounded linear operator Linear bounded operator T on $l^2$ is given by :
$T(x_1,x_2,x_3,..) := (x_1,x_1,x_1,x_2,x_2,x_2,x_3,x_3,x_3,..)$.
Prove that $\frac{T^*T}{3}$ and $\frac{TT^*}{3}$ are orthogonal projections.
Thanks in advance for the help

You have 
$$
\langle T^*Tx,y\rangle=\langle Tx,Ty\rangle= 3\sum_nx_n\overline{y_n}=\langle 3x,y\rangle.
$$
So $T^*T=3I$. Since $\ker TT^*=\ker T^*$, we have
$$
\overline{\text{ran}\,TT^*}=\ker(TT^*)^\perp=\ker (T^*)^\perp=\overline{\text{ran}\,T}.
$$
So the range of $TT^*/3$ equals the range of $T$. And it is a projection, since 
$$
\left(\frac{TT^*}3\right)^2=\frac{TT^*TT^*}9=\frac{3TT^*}9=\frac{TT^*}3. 
$$
So $TT^*/3$ is the orthogonal projection onto the range of $T$. 
