I have tried to define Riemann zeta function for $ Re(s)=\frac12 $ in Hilbert space $ H=l^2(\mathbb{C, ds})$ using the well known formula of Riemann-Siegel function which it is defined as : $\zeta(0.5+it)=Z(t)\exp(-i \theta(t)) $ for every positive real $t$ with $ Z(t)$ is $Z$-function , I have tried to show that Riemann zeta function is unitary operator using the interesting property of Hilbert space which is inner product , I have accrossed the problem about rate growth of integral square of Z-function which it is $ t \log t $ is not bounded for to $t\to \infty $ the latter Botched me to juge wether $\zeta(0.5+it)$ is a unitary operator or not any idea to show that ?
Note:01 In Riemann Siegel function we have :$\exp(-i \theta(t)) $ present unitary operator
Edit:let us define my operator which i want it act on Riemann-siegel function . let $W(t)=\exp (itH)$, so $$ W(t) W(-t)=1\!\! 1, $$ We could define it as : $$ U(t)\equiv W(t) ~~e^{-\beta H/2}, $$
If $f$ is continuous $\Bbb{R} \to \Bbb{C}$ and $\forall t, |f(t)| =1$ and $$A(g)(t) = g(t)f(t)$$ then $A$ is an unitary linear operator $L^2(\Bbb{R}) \to L^2(\Bbb{R})$ because $$\|A(g)\|_{L^2} =\int_{-\infty}^\infty| A(g)(t)|^2dt=\int_{-\infty}^\infty| g(t)|^2dt= \|g\|_{L^2}$$ for all $g\in L^2(\Bbb{R}) $.