The derivative of a $C_0$-semigroup with respect to a perturbation parameter

Question

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 $v \in H$.)

Question: Could you please help me with a bibliographic reference for the above?

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.

Furthermore, is it true that $\frac {\mathrm d} {\mathrm d \varepsilon} \ \mathrm e ^{-t (A + \varepsilon B)} = -t \ \mathrm e ^{-t (A + \varepsilon B)} B$ ?

Answer 1

I think Theorem 2.20 in Chapter IX of Kato,

Kato, Tosio, Perturbation theory for linear operators., Classics in Mathematics. Berlin: Springer-Verlag. xxi, 619 p. (1995). ZBL0836.47009.

has something to do with your question. A link to the section can be foud for example here.

Answer 2

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 $v \in H$.