Let $\{e_{n}\}_{n \in \mathbb{N}}$ be an orthonormal sequence for a Hilbert space $H$, let $ T: H \rightarrow \ell^{2}$ be the analysis operator $ T x=\left\{\left\langle x, e_{n}\right\rangle\right\}_{n \in \mathbb{N}}$ and $T^{*}: \ell^{2} \rightarrow H $$ T^{*} c=\sum_{n=1}^{\infty} c_{n} e_{n}$ be the synthesis operator, I was wondering how to find explicit formula for $TT^*$. I have figured out that $T^{*} T=I$, however since $\left\{e_{n}\right\}_{n \in \mathbb{N}}$ is not complete so it's not clear to me how to find $TT^*$ from $T^*T$. Any help is appreciated.
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