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Kernel–range orthogonality for Hilbert adjoints

Statement

Assume the Axiom of Countable Choice. Let $T \in B (H, K)$ be a bounded linear operator between real or complex Hilbert spaces, with kernel $\ker T = {x : T x = 0}$ , range $ran T = T [H]$ and Hilbert adjoint $T^{*} \in B (K, H)$ . Then

$(ran T)^{⊥} = \ker T^{*}, (ran T^{*})^{⊥} = \ker T,$

and consequently

$\overline{ran T} = (\ker T^{*})^{⊥}, \overline{ran T^{*}} = (\ker T)^{⊥} .$

Facts & Assumptions

[A1]

$⟨ T x, y ⟩_{K} = ⟨ x, T^{*} y ⟩_{H}$ for all $x \in H$ , $y \in K$ , and the adjoint is a bounded linear operator (The Hilbert-space adjoint of a bounded operator, A bounded linear operator between normed spaces).

[A2]

$T^{* *} = T$ (Hilbert-adjoint identities).

[A3]

$S^{⊥} = {v : ⟨ v, s ⟩ = 0 \forall s \in S}$ ; if $⟨ v, w ⟩ = 0$ for every $w$ then $v = 0$ , and $S^{⊥ ⊥} = \overline{S}$ for every linear subspace $S$ (Orthogonality and the orthogonal complement, Real and complex inner-product spaces and their induced length, The double orthogonal complement of a subspace is its closure).

[A4]

The image of a linear map is a linear subspace, and the kernel of a linear map is a linear subspace (Linear subspace of a vector space).

[A5]

Countable Choice is the hypothesis under which adjoints and double complements are available (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Proof

technique · direct

Given: Countable Choice, Hilbert spaces $H, K$ and a bounded linear operator $T : H \to K$ .

1.1A1A3A5

For $y \in K$ , one has $y \in (ran T)^{⊥}$ exactly when $⟨ T x, y ⟩_{K} = 0$ for every $x$ , which by the adjoint identity is exactly when $⟨ x, T^{*} y ⟩_{H} = 0$ for every $x$ ; taking $x = T^{*} y$ shows this happens exactly when $T^{*} y = 0$ , that is $y \in \ker T^{*}$ .

2.1step 1.1A2

Replacing $T$ by $T^{*}$ in step 1.1 and using $T^{* *} = T$ gives $(ran T^{*})^{⊥} = \ker T$ .

3.1step 1.1step 2.1A3A4∎

The range of a linear map is a linear subspace, so the double-complement theorem applies to it: $\overline{ran T} = (ran T)^{⊥ ⊥} = ((ran T)^{⊥})^{⊥} = (\ker T^{*})^{⊥}$ , and likewise $\overline{ran T^{*}} = (\ker T)^{⊥}$ .

Kernel–range orthogonality for Hilbert adjoints

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources