Let $T:\ell^2\rightarrow \ell^2$ be defined by $T(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...)\rightarrow (0,0,\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...)$.Is $T$ Self Adjoint?Is $T$ Bounded?
$T$ is Self-Adjoint
$\langle T(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
=$\langle(0,0,\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
Now,
$\langle T^*(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
=$\langle (\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),T(\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
=$\langle(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(0,0,\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
Since,$\langle(0,0,\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle\neq \langle(\xi_1,\xi_2,\xi_3,\xi_4,\xi_5,...),(0,0,\eta_1,\eta_2,\eta_3,\eta_4,...)\rangle$
So,$T\neq T^* \implies T$ is not Self Adjoint.
Please Check the above proof whether it is correct or not?
I'm not getting how to show that $T$ is bounded...I've given a lots of time in proving the boundedness...If possible,please show how $T$ is bounded?