Definition 41.1.2 (Dilation).label Let $H$ be a complex Hilbert space and $T \in B(H)$, then a dilation of $T$ is a pair $(K, S)$ where:

1. (1)

   $K$ is a Hilbert space containing $H$.

2. (2)

   $S \in B(K)$.

3. (3)

   If $P \in B(K)$ is the orthogonal projection onto $H$, then $T = PS|_{H}$.

If $T^{n} = PS^{n}|_{H}$ for all $n \in \natz$, then $S$ is a power dilation of $T$.