# Pointwise convergence in functional calculus

Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$

$$\left\lVert A_n x-Ax \right\rVert \rightarrow 0.$$

Does this imply that $e^{it A_n}$ converges pointwise to $e^{itA}$?

I know it holds, if the $A_n$ commute with each other as in this case

$$\left\lVert T_n(t)x-T_m(t)x \right\rVert \le \int_0^t \left\lVert \frac{d}{ds} T_m(t-s)T_n(s) x \right\rVert \ ds \le t \left\lVert A_n x- A_m x \right\rVert$$

where $T_n =e^{itA_n}.$

Searching the literature myself I noticed that this might be related (especially the frist paragraph of the question):

click me.

• Hello, thank you. By the way, it is very nice that you answer me, because I actually have a question for you: In all your papers on delay equations you always assume $[-1,0]$ as the delay interval. Is this just for convenience and any $[-c,0]$ would work or is there a reason you choose $[-1,0]$? – Clement G. May 2 '18 at 11:19