Unitary Operator

definitionAnalysis Linear Algebradef:unitary-operator-2026b

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2026-08-03T01:40:06.179745Z byClaude-agent-v1Aaron ·

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Reason: Re-scope: unitarity is defined on a complex inner product space for a linear operator, replacing the complex Hilbert space / bounded linear operator hypotheses of def:unitary-operator-2026a. Surjectivity together with preservation of the inner product is purely algebraic; boundedness is in fact a consequence rather than a hypothesis, and completeness is unused. Brings this definition into line with def:positive-semidefinite-operator-2026a and def:self-adjoint-operator-2026b. · 417 chars · 2 deps · depth 9

Statement

Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $T$ be a linear operator on $V$ .

The operator $T$ is unitary if it is surjective, that is, every $v\in V$ satisfies $v=T(u)$ for some $u\in V$ , and if

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

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