Strong time operators associated with generalized Hamiltonians

Let the pair of operators, $(H,T)$, satisfy the weak Weyl relation: $T{e}^{-itH}=e^{-itH}(T+t)$, where $H$ is self-adjoint and $T$ is closed symmetric. Suppose that g is a realvalued Lebesgue measurable function on $\text{RR}$ such that $g\in C^2(RK)$ for some closed subset $K\subset \text{RR}$ with Lebesgue measure zero. Then we can construct a closed symmetric operator $D$ such that $(g(H),D)$ also obeys the weak Weyl relation.