- #1

mathzero

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I want to show $ T^{*}(Pr_{C}(y)) = Pr_{T^{*}(C)}(T^{*}y)$

where $T \in B(H)$ and $TT^{*}=I$ , $H$ is Hilbert space and $C$ is a closed convex non empty set.

but i don't know how to start, or what tricks needed to solve this type of problems.

also i want know how to construct $T$ to satisfying

$ Pr_{x + C}(y) = Pr_{C}(x + y)$ and $ Pr_{\lambda C}(\lambda y) = \lambda Pr_{C}(y)$

Thanks!