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 Incoming links: Double orthogonal complement

Use the convention that the inner product is linear in its first argument. By the defining identity for the adjoint operator,

x \in (Im T^{*})^{⊥} ⟺ ⟨ x, T^{*} y ⟩ = 0 for every y \in X ⟺ ⟨ T x, y ⟩ = 0 for every y \in X ⟺ T x = 0.

For the last implication choose

y = T x

Im T^{*}

gives

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

I = ker (U - I)

is closed. If

y \in I

and

w = Ux - x \in W

, then unitarity gives

⟨ y, w ⟩ = ⟨ y, Ux ⟩ - ⟨ y, x ⟩ = ⟨ U^{- 1} y, x ⟩ - ⟨ y, x ⟩ = 0,

I ⊥ W

Conversely,

z \in W^{⊥}

exactly when

0 = ⟨ z, Ux - x ⟩ = ⟨ U^{*} z - z, x ⟩

for every

x \in H

. Thus

U^{*} z = z

, equivalently

U z = z

, and

z \in I

. Hence

W^{⊥} = I

\overline{W} = (W^{⊥})^{⊥} = I^{⊥} .

H = I \oplus \overline{W} .

A_{n} = \frac{1}{n} \sum_{i = 0}^{n - 1} U^{i} .

For

y \in I

A_{n} y = y

. For a generator

w = U z - z

W

A_{n} w = \frac{U ^{n} z - z}{n}, ∥ A_{n} w ∥ \leq \frac{2∥ z ∥}{n} ⟶ 0.

Moreover

∥ A_{n} ∥ \leq 1

, so approximation extends this convergence from

W

\overline{W}

x = P x + (I - P) x

according to the orthogonal decomposition gives the Von Neumann mean ergodic theorem

n \to \infty lim \frac{1}{n} i = 0 \sum n - 1 U^{i} x = P x .