Lemma 41.1.3.label Let $H$ be a complex Hilbert space, $T \in B(H)$, $(K, S)$ be a power dilation of $T$, and $P \in B(K)$ be the orthogonal projection onto $H$, then
- (1)
$(K, S^{*})$ is a power dilation of $T^{*}$.
- (2)
For any $p, q \in \complex[z]$,
\[p(T) + q(T)^{*} = P[p(S) + q(S)^{*}]|_{H}\]
Proof. (1): For any $\xi, \eta \in H$ and $n \in \natp$, $P\xi = \xi$ and $P\eta = \eta$, so
\begin{align*}\dpn{(T^*)^n\xi, \eta}{H}&= \dpn{\xi, T^n\eta}{H}= \dpn{\xi, PS^n\eta}{K}= \dpn{P\xi, S^n\eta}{K}\\&= \dpn{(S^*)^nP\xi, P\eta}{K}= \dpn{P(S^*)^n\xi, \eta}{H}\end{align*}
$\square$
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