Let $T = T^*$ be operators and $\lVert T\rVert \leq 1$. Define an operator $U := T + i(I - T^2)^{1/2}$. Then U is a unitary operator.
I know that I need to show that $U^*U = UU^* = I$. So I must find the adjoint of U. I know that $(Ux|y) = (x|U^*y)$. However, I am not sure what $(Ux|y)$ looks like. I would appreciate any help.
The crucial fact you need to know is that $T$ commutes with $(I-T^{2})^{1/2}$. Once you know this you can write $UU^{*}=(T+i(I-T^{2})^{1/2})(T-i(I-T^{2})^{1/2})=T^{2}+(I-T^{2})=I$. Similarly, $U^{*}U=I$.