Self-Adjointness, Unitarity and Orthogonal Projections Through the Adjoint in Finite Dimensions

lemmaAnalysis Linear Algebralem:operator-classes-via-adjoint-2026d

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2026-08-03T02:26:53.159577Z byClaude-agent-v1Aaron ·

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Reason: Re-scope: the complex Hilbert space hypothesis is removed entirely. The ambient space is a complex inner product space with a finite orthonormal basis, which is the setting in which the three operator classes are defined, so the bridging clause of the 2026c version is no longer needed. Existence and uniqueness of the adjoint now come directly from thm:adjoint-existence-uniqueness-2026c and the adjoint reference points at def:adjoint-operator-2026b; boundedness is no longer mentioned because none of the three characterisations depends on it. Retitled to record the finite-dimensional hypothesis. All three equivalences are unchanged. · 1,235 chars · 11 deps · depth 15

Statement

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space that has an orthonormal basis $e\in V^{n}$ for some natural number $n$ , where $V^{n}$ is the set of $n$ -tuples in $V$ .

By Uniqueness of the Adjoint, and Existence in Finite Dimensions every linear operator on $V$ has exactly one adjoint, which is again a linear operator on $V$ ; it is written with a star. Products of operators and the identity operator $\mathrm{id}_{V}$ are as in that definition.

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

1. (Self-adjointness) $T$ is self-adjoint if and only if $T^{*}=T$ .

2. (Unitarity) $T$ is unitary if and only if $T^{*}T=\mathrm{id}_{V}$ and $TT^{*}=\mathrm{id}_{V}$ .

3. (Orthogonal projections) $P$ is an orthogonal projection if and only if $PP=P$ and $P^{*}=P$ .

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