Algebraic Properties of the Adjoint

Let $H$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space} that has an \reftext{def:orthonormal-basis-2026a}{orthonormal basis} $e:[n]\to H$ for some \reftext{def:natural-numbers-2026a}{natural number} $n$, with $[n]$ the \reftext{def:initial-segment-natural-numbers-2026a}{initial segment} determined by $n$. By \ref{lem:finite-orthonormal-basis-operator-bounded-2026a} every \reftext{def:linear-operator-2026a}{linear operator} on $H$ is a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator}, and by \ref{thm:adjoint-existence-uniqueness-2026a} it has exactly one \reftext{def:adjoint-operator-2026a}{adjoint}, written with a star.

Let $S$ and $T$ be linear operators on $H$ and let $\lambda$ be a \reftext{def:complex-numbers-2026a}{complex number} with \reftext{def:complex-conjugate-2026a}{conjugate} $\overline{\lambda}$. The sum, scalar multiple, product and identity operator are as in \reftext{def:operator-operations-2026a}{that definition}. Then the following hold.

\textbf{1. (Sum)} $(S+T)^{*}=S^{*}+T^{*}$.

\textbf{2. (Scalar multiple)} $(\lambda S)^{*}=\overline{\lambda}\,S^{*}$.

\textbf{3. (Product)} $(ST)^{*}=T^{*}S^{*}$.

\textbf{4. (Involution)} $(S^{*})^{*}=S$.

\textbf{5. (Identity)} $(\mathrm{id}_{H})^{*}=\mathrm{id}_{H}$.