If $A$ is a contraction, show that $$ex A^*(I-AA^*)^{1/2}=(I-A^*A)^{1/2}A^*. eex$$ Use this to show that if $A$ is a contraction on $scrH$, then the operators $$ex U=sex{a{cc} A&(I-AA^*)^{1/2}\ (I-A^*A)^{1/2}&-A^* ea}, eex$$ $$ex V=sex{a{cc} A&-(I-AA^*)^{1/2}\ (I-A^*A)^{1/2}&A^* ea} eex$$ are unitary operators on $scrHoplus scrH$.
Solution.
(1). By the singular value decomposition, there exist unitaries $W,Q$ such that $$ex A=WSQ^*,quad S=diag(s_1,cdots,s_n),quad s_igeq 0, eex$$ and hence $$ex A^*=QSW^*. eex$$ Consequently, $$eex a{rlrl} AA^*&=WS^2W^*,&A^*A&=QS^2Q^*,\ I-AA^*&=W(I-S^2)W^*,&I-A^*A&=Q(I-S^2)Q^*,\ (I-AA^*)^{1/2}&=WvLm W^*,& (I-A^*A^{1/2}&=QvLm Q^*, ea eeex$$ where $$ex vLm=diagsex{sqrt{1-s_1^2},cdots,sqrt{1-s_n^2}}. eex$$ Thus, $$eex ea A^*(I-AA^*)^{1/2}&=QSvLm W^*\ &=Qdiagsex{s_1sqrt{1-s_1^2},cdots, s_nsqrt{1-s_n^2}}W^*\ &=QvLm S W^*\ &=(I-A^*A)^{1/2} A^*. eea eeex$$
(2). As noticed in (1), $A$ is a contraction is equivalent to say that $A^*$ is a contraction. Direction computations with $$ex A^*(I-AA^*)^{1/2}=(I-A^*A)^{1/2}A^*,quad A(I-A^*A)^{1/2}=(I-AA^*)^{1/2}A eex$$ yields the fact that $U,V$ are unitary.