Corollary 40.2.5.label Let $H$ be a complex Hilbert space and $T \in B(H)$, then $T$ is a partial isometry if and only if $T^{*}$ is a partial isometry.
Proof, [Corollary 12.7, Zhu93]. Suppose that $T$ is a partial isometry, then $P = T^{*}T$ is a projection onto $\ker(T)^{\perp}$ by Proposition 40.2.4. In which case, $T(T^{*}T) = T$ and $(TT^{*})^{2} = T(T^{*}T)T^{*} = TT^{*}$, so $TT^{*}$ is a projection, and $T^{*}$ is a partial isometry by Proposition 40.2.4.$\square$
Post a Comment