The following is from Conway's Functional Analysis:
Why $[\operatorname{ker}(T - A)^*]^\perp = H$?
2026-03-31 14:24:59.1774967099
On 4.15. Corollary. Conway's Functional Analysis?
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Compactness of $T$ implies that of $T^{*}$. Since $\lambda \neq 0$ and $\overline {\lambda} \notin \sigma_p (T^{*})$ it follows from compacteness of $T^{*}$ that $T^{*}-\overline {\lambda}$ is invertible. Hence its kernel is $\{0\}$ and the orthogoanl complement of the kernel is $\mathcal H$.