In an Hibert space $H$ with orthonormal basis $\{e_k\}_{k\in\mathbb{Z}}$, let $T_n$ be the operator defined by the action $$ T_nx=(x, e_{n+1})e_n. $$ Compute its norm, eigenvalues and eigenvectors e show that for any $x\in H$, the sequence $\left\{\sum_{n=-N}^NT_nx\right\}_{N\in\mathbb{N}}$ converges.
My attempt. No problem about computing the norm, eigenvalues and eigenvectors. I find out that the norm is $1$. Now, to show that the sequence converges, I notice that $$ \sum_{n=-N}^N(x, e_{n+1})e_n=\sum_{n=-N}^Nx_{n+1}\leq M. $$ How can I conclude that the sequence converges?
Let $y_N:=\sum_{n=-N}^NT_nx$. Then for each $M\geqslant N$, $$y_M-y_N=\sum_{n=N+1}^M\left(x, e_{n+1}\right)e_n+\sum_{n=-M}^{-N-1}\left(x, e_{n+1}\right)e_n$$ and using orthogonality of the family $\{e_k\}_{k\in\mathbb{Z}}$, we derive that $$ \left\lVert y_M-y_N\right\rVert^2=\sum_{n=N+1}^M\left(x, e_{n+1}\right)^2+\sum_{n=-M}^{-N-1}\left(x, e_{n+1}\right)^2. $$ To conclude, it suffice to show that the series $\sum_{n\in\mathbb Z} \left(x, e_{n+1}\right)^2$ converges, which is usually done by using orthogonal projection on the vector space generated by $\{e_k\}_{k=-j}^j$.