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?
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$\begingroup$ 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? $\endgroup$– fedjaMay 20, 2019 at 12:22
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1$\begingroup$ 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. $\endgroup$– Christian RemlingMay 20, 2019 at 16:14
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$\begingroup$ @fedja: Yes, I mean what you say. Equivalently, $h^{-1}(A(t+h)-A(t))$ converges in the strong topology for $h\rightarrow 0$. $\endgroup$– LR235May 23, 2019 at 10:26
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$\begingroup$ @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. $\endgroup$– LR235May 26, 2019 at 8:32
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