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

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  • $\begingroup$ A standard reference is Semigroups of Linear Operators and Applications to Partial Differential Equations by Pazy. $\endgroup$
    – user69642
    Jun 19, 2020 at 9:54
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    $\begingroup$ The last statement is not true, unless $A$ and $B$ commute. $\endgroup$ Jun 19, 2020 at 11:39
  • $\begingroup$ @MichaelRenardy: If it is not true, do you know the correct statement? Is it like the Duhamel formula: $\frac {\mathrm d} {\mathrm d \varepsilon} \ \mathrm e ^{-t (A + \varepsilon B)} = \int_0^1 \mathrm e ^{-s t (A + \varepsilon B)} (-t B) \mathrm e ^{-(1-s) t (A + \varepsilon B)} \ \mathrm d s$? $\endgroup$
    – Alex M.
    Jun 19, 2020 at 12:25
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    $\begingroup$ @AlexM. have a look at the Kato reference I gave. You can extract the answer from there. $\endgroup$ Jun 21, 2020 at 7:38

2 Answers 2

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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.

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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 $v \in H$.

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