Spectrum of shift-operator

We have \begin{align} \lambda\in\sigma(T)&\iff \lambda I-T\ \text{not invertible }\\ \ \\ &\iff (\lambda I-T)^*=\bar\lambda I-T^*\text{not invertible }\\ \ \\ &\iff \bar\lambda\in\sigma(T^*). \end{align}

And $$ \bar\lambda\in\sigma_p(L)\iff \exists \text{ nonzero }v\in\ker(\bar\lambda I-L)=\text{ran}\, (\lambda I-R)^\perp\iff\lambda\in\sigma_r(R). $$ Note that the last "if and only if" requires the fact that $\sigma_p(R)=\emptyset$, since $$ \sigma_r(T)=\{\lambda:\ \text{ran}\,(\lambda I-T)^\perp\ne0\}\setminus\sigma_p(T). $$

See article of Helein https://www.imj-prg.fr/~frederic.helein/polyspec.pdf here the spectrum of the shift operator is worked out into detail.