The orthogonal complement of an invariant subspace under a unitary operator is invariant

Let $(𝑉, 𝕂, ⟨ \cdot | \cdot ⟩)$ be an Inner product space with invariant subspace $𝑊 \subseteq 𝑉$ under unitary endomorphism $𝑈 : 𝑉 \to 𝑉$ . Then the Orthogonal complement $𝑊 ⟂$ is also invariant under $𝑈$ . linalg

Proof

Let $𝑊 \subseteq 𝑉$ be an invariant subspace under $𝑈 : 𝑉 \to 𝑉$ . Then $𝑊$ is also invariant under $𝑈 - 1 = 𝑈 †$ , and thus for any $𝑣 \in 𝑊 ⟂$ and $𝑤 \in 𝑊$
$⟨ 𝑣 | 𝑈 𝑤 ⟩ = ⟨ 𝑈 - 1 𝑣 | 𝑤 ⟩ = 0$
as required.

This extends to a Unitary representation of a finite group easily. Since Every finite complex representation of a compact group is equivalent to a unitary representation, this doesn’t hold iff a representation is not unitary and non-finite.

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The orthogonal complement of an invariant subspace under a unitary operator is invariant

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