Assume we have a family of self-adjoint (unbounded) operators $(T_s)$, $s \in \mathbb R$, defined on a common domain. Assume that they are bounded from below and let denote $E(s)$ the infimum of the spectrum of $T_s$. I'm looking for sufficient conditions for the function $s \mapsto E(s)$ to be differentiable.
Is it sufficient that the resolvent is differentiable in the norm topology? That is, assume there exists some $\lambda$ in the resolvent set of all $T_s$, $s$ in a neighborhood of some $s_0$, and then assume that the function $s \mapsto (T_s - \lambda)^{-1}$ is differentiable in the space of bounded operators.
I've read something like this in a paper but without further explanation, so I wonder if this is true more or less in general or does depend on their specific situation.