Assumptions: Suppose $T$ is a self-adjoint operator (possibly unbounded) from the (dense) domain $D(T)$ on a Hilbert space $H$, hence $T:D(T)\rightarrow H$. Assume that $f$ is a continuous function on the spectrum of $T$, i.e. $f\in C(\sigma(T))$.
Question: We can define $f(T)$ (via spectral theorems), but what is the domain of $D(f(T))$, what is a good definition here. In particular, if $f$ is real valued, then $f(T)$ is self-adjoint (also by spectral theorems) but on which domain?
Thanks a lot.