On $l_{2}$ let $T$ be given by $Te_{n}=\frac{e_{n+1}}{n+1}$ where $(e_{n})_{n\ge1}$ is the canonical orthonormal basis. Find the spectral decomposition of $TT^*$.
I find that $T^*(e_{n})=\frac{e_{n-1}}{n}$ for $n$ greater than 1 and $T^*(e_{1})=0$ I find that the $e_{n}$ are the orthonormal eigenvectors for $TT^*$ with corresponding eigenvalues $\frac{1}{n^2}$ for $n\ge2$ and $0$ for $n=1$ so that the spectral decomposition is $TT^*x=\sum_{i=2}^{{\infty}}\frac{1}{n^2}\langle x,e_{n}\rangle e_{n}$.
Do you agree and if so, are there any which are a bit harder? Thanks