Let $H$ be a Hilbert space, and $A : H \to H$ be the (semi-bounded) generator of the $1$-parameter $C\_0$-semigroup $[0, \infty) \ni t \mapsto \mathrm e ^{-t A}$. Let $B : H \to H$ be a bounded operator, and consider the "perturbation" $[0,1] \ni \varepsilon \mapsto A + \varepsilon B$. I would like to use the formula


$$\mathrm e ^{-t (A + B)} - \mathrm e ^{-t A} = \int \_0 ^1 \frac {\mathrm d} {\mathrm d \varepsilon} \ \mathrm e ^{-t (A + \varepsilon B)} \ \mathrm d \varepsilon$$


which I believe is true, but I do not know where to find. (In order to be true, it might be necessary to consider the formula in a strong sense, i.e. applied on some arbitrary $v \in H$.)



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> Question: Could you please help me with a bibliographic reference for the above?
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Davies' "One-parameter semigroups" doesn't have it. Please also notice that I am not interested in a proof, but only in a citable reference.



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> Furthermore, is it true that $\frac {\mathrm d} {\mathrm d \varepsilon} \ \mathrm e ^{-t (A + \varepsilon B)} = -t \ \mathrm e ^{-t (A + \varepsilon B)} B$ ?
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The derivative of a $C_0$-semigroup with respect to a perturbation parameter
The answer can be found in chap. IX of T. Kato's "Perturbation Theory for Linear Operators" and is, essentially, an elementary consequence of Duhamel's formula for inhomogeneous linear differential equations of order 1 in Hilbert spaces:


$$\mathrm e ^{-t(A+B)} v - \mathrm e ^{-tA} v = - \int \_0 ^t \mathrm e ^{-(t-s) (A+B)} B \ \mathrm e ^{-s A} v \ \mathrm d s$$


for all $v \in H$.


