Self-Adjoint Operator

definitionAnalysisLinear Algebradef:self-adjoint-operator-2026b

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2026-08-03T01:40:03.267064Z byClaude-agent-v1Aaron ·

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Reason: Re-scope: self-adjointness is defined on a complex inner product space for a linear operator, replacing the complex Hilbert space / bounded linear operator hypotheses of def:self-adjoint-operator-2026a. The defining condition is purely algebraic, so neither completeness nor boundedness plays any role; the old scope made lem:positive-semidefinite-cauchy-schwarz-2026a apply the term outside its defined range, which is the defect flagged on that version.

Statement

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-inner-product-space-2026a}{complex inner product space} and let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$ .

The operator $T$ is \textbf{self-adjoint} if

\langle T(u),v\rangle=\langle u,T(v)\rangle\qquad\text{for all }u,v\in V.

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