I am looking for verification of my attempt in finding the adjoint operator of a linear operator.
Let $S(e_n)=e_{2n+1}$ be a linear operator in the Hilbert space $l^2(N)$, the space of all summable sequences, where $\{e_n\},n=0,1,2,...$, is the standard orthonormal basis.
Below is my work for finding the adjoint $S^*$ of $S$. Please tell me whether my solution is correct. I did this based on some other people's previous answers but they deleted them for some reason.
Continuing this preliminary work, we find that: $$ <S(x),y>=a_1y_3+a_2y_5+a_3y_7+...=\sum_{k=1}^\infty x_ky_{2k+1} $$ On the other hand, $$ <x,S^*(y)>=<x,z>=\sum_{k=1}^{\infty} x_kz_k $$ By equating $<Sx,y>$ and $<x,S^*y>$, we get: $$ z_1=y_3,z_2=y_5,z_3=y_7,... $$ So, $$ S^*(y)=(y_3,y_5,y_7,...) $$
$$ \langle Sx,y\rangle = \langle \sum_{n=0}^{\infty} \langle x,e_n\rangle e_{2n+1},y\rangle \\ = \sum_{n=0}^{\infty} \langle x,e_n\rangle\langle e_{2n+1},y\rangle \\ = \langle x,\sum_{n=0}^{\infty} \langle y,e_{2n+1}\rangle e_n\rangle \\ \implies S^*y=\sum_{n=0}^{\infty} \langle y,e_{2n+1}\rangle e_n. $$ So $S^*(y_0,y_1,y_2,\cdots)=(y_1,y_3,y_5,\cdots)$.