Let $H = l_2$ and consider the following operators: $T,S:H \to H$ $Tx = (0,x_1,x_2,\ldots)$ and $Sx = (x_2,x_3,x_4,\ldots)$ Show they are bounded, and find the adjoint of both:
For $T$, I have $\|Tx\|_2^2 = \|x\|_2^2$ so $\|T\| = 1$ hence it is bounded, as for finding the adjoint, I have $$\langle Tx,y\rangle = \sum_{k=1}^\infty x_k\overline{y_{k+1}} = \sum_{k=1}^\infty\overline{y_{k+1}\overline{x_k}}$$ but I am not sure how to conclude what the adjoint of $T$ is.
For $S$:
I have $\|Sx\|_2^2 \leq \|x\|_2^2 $ so $\|S\| \leq 1$, so it is bounded, as for the adjoint $$\langle Sx,y\rangle = \sum_{k=1}^\infty x_{k+1}\overline{y_k} = \sum_{k=1}^\infty \overline{y_k\overline{x_{k+1}}}$$ but I am with the same problem
any help please