Self-Adjointness, Unitarity and Orthogonal Projections Expressed Through 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 then a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator}, and by \ref{thm:adjoint-existence-uniqueness-2026a} it has an \reftext{def:adjoint-operator-2026a}{adjoint}. Products of operators and the identity operator $\mathrm{id}_{H}$ are as in \reftext{def:operator-operations-2026a}{that definition}.

Let $T$ and $P$ be linear operators on $H$. Then the following hold.

\textbf{1. (Self-adjointness)} $T$ is \reftext{def:self-adjoint-operator-2026a}{self-adjoint} if and only if $T^{*}=T$.

\textbf{2. (Unitarity)} $T$ is \reftext{def:unitary-operator-2026a}{unitary} if and only if $T^{*}T=\mathrm{id}_{H}$ and $TT^{*}=\mathrm{id}_{H}$.

\textbf{3. (Orthogonal projections)} $P$ is an \reftext{def:orthogonal-projection-2026a}{orthogonal projection} if and only if $PP=P$ and $P^{*}=P$.