Square of V Operator on Hilbert Space

Theorem

Let $\struct {\HH, \innerprod \cdot \cdot}$ be a Hilbert space over $\C$.

Let $\struct {\HH \times \HH, \innerprod \cdot \cdot_{\HH \times \HH} }$ be the Hilbert space direct sum of $\HH$ with itself.

Define $V : \HH \times \HH \to \HH \times \HH$ by:

$\map V {x, y} = \tuple {-y, x}$

for each $\tuple {x, y} \in \HH \times \HH$.

Then $V^2 = -I$.

For each $x, y \in \HH$ we have:

$\ds \map {V^2} {x, y}$

$=$

$\ds \map V {-y, x}$

$\ds $

$=$

$\ds \tuple {-x, -y}$

$\ds $

$=$

$\ds -\tuple {x, y}$

$\blacksquare$