# Strong Differentiability of Spectral Projections

Let $$H$$ be a Hilbert space and $$W$$ be a dense subspace, equipped with a different norm that turns it into a Hilbert space. Let $$(A(t))_{t\in[0,T]}$$ be a family of Operators in $$B(W,H)$$ (bounded operators from $$W$$ to $$H$$) that are self-adjoint with discrete spectrum as unbounded operators in $$H$$ with domain $$W$$. Assume that $$A(t)$$ is differentiable as a function of $$t$$ in the strong topology on $$B(W,H)$$ and that 0 is not in the spectrum of $$A(t)$$ for any $$t$$. Does this imply that the positive spectral projection of $$A(T)$$, i.e. $$\chi_{[0,\infty)}(A(t))$$, is differentiable in $$t$$ with respect to the strong topology on $$B(H,H)$$? Does someone know a reference where a statement like this might be found?

• When you say "$A(t)$ is differentiable in the strong topology of $B(W,H)$", do you mean that for every $x\in W$, $t\mapsto A(t)x$ is differentiable as a mapping from $[0,T]$ to $H$ or something else? May 20, 2019 at 12:22
• The standard reference for this kind of thing is Kato's book, though I have no idea if you'll find anything useful for this specific question there. May 20, 2019 at 16:14
• @fedja: Yes, I mean what you say. Equivalently, $h^{-1}(A(t+h)-A(t))$ converges in the strong topology for $h\rightarrow 0$. May 23, 2019 at 10:26
• @ChristianRemling: Thanks for the recommendation. I didn't find what I what looking for, but nevertheless I got some good inspirations for what I'm doing. May 26, 2019 at 8:32