Uniform Exponential Stability of Perturbed Semigroups: The Dyson–Phillips Formula Versus Gil’s Approach Via Commutators

We discuss the uniform exponential stability of strongly continuous semigroups generated by operators of the form A+B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ A+B $$\end{document}, where B is a bounded perturbation of a generator A. We compare two approaches to the problem: via the Dyson–Phillips formula and via the size of the norm of the commutator of A and B- the method recently developed by M. Gil’. We show that quite often the first approach is more powerful than the second one and, more importantly, easier to use.


Introduction
We say that a strongly continuous semigroup of operators e tA t≥0 , generated by an operator A in a Banach space X, is uniformly exponentially stable if the growth bound ω 0 := inf{ω ∈ R : e tA ≤ M ω e ωt for some M ω ≥ 1 and all t ≥ 0} of the semigroup is negative. There are several conditions equivalent to ω 0 < 0, see [5, V.1.7]. However, in the context of the paper, we note that, by the Datko and Pazy theorem [5, V.1.8], the semigroup generated by A is uniformly exponentially stable if and only if the map t → e tA is integrable over (0, +∞), e tA dt < +∞.
We concentrate on the uniform exponential stability of perturbed semigroups. Given a generator A and a bounded operator B in X, by the Bounded Perturbation Theorem [5,III.1.3] it follows that the operator A + B generates a strongly continuous semigroup, and we can ask under what conditions the semigroup is uniformly exponentially stable. The problem is well known and discussed by many authors-see for example Preda's survey [10] and references given there.
A direct way to approach the question of stability of perturbed semigroups is via the Dyson-Phillips formula [5, III.1.10], which we discuss in Sect. 3. On the other hand, the classical Liapunov Stability Theorem [5, I.2.10], which states that the semigroup generated by a bounded operator (matrix) A in C n is uniformly exponentially stable if and only if the real parts of all eigenvalues of A are negative, suggests that an analysis of the spectrum of a generator may be helpful. Indeed, see [5, V.1.10], if a semigroup is eventually norm-continuous, then it is uniformly exponentially stable if and only if the spectral bound s(A) := sup {Reλ : λ belongs to the spectrum σ(A)} of its generator A is negative. This idea can be carried over to perturbed semigroups. It turns out, see [4] and [10], that if at least one element of the Dyson-Phillips series, see (3.1), is norm-continuous, then the semigroup generated by a bounded perturbation A+B of A is uniformly exponentially stable if and only if the spectral bound s(A + B) is negative. In Hilbert spaces there is also a resolvent related condition that is equivalent to the uniform exponential stability, see the Gearhart-Prüss-Greiner theorem [5, V.1.11].
Recently, Gil' [6,7] found yet another approach to the uniform exponential stability of perturbed semigroups via the commutator of generators A and B. More precisely, he proved that for a generator A and a bounded operator B in a Banach space X, the semigroup generated by A + B is uniformly exponentially stable, provided that the following conditions hold: The result was slightly improved by A. Bobrowski, who showed in [3] that the conclusion of Gil's theorem still holds if we replace (c) with a weaker condition (d) K I < 1, where I := +∞ 0 e tB t 0 e (t−s)A e sA ds dt. Moreover, he provided an example of a uniformly exponentially stable perturbed semigroup such that conditions (a)-(b) hold, while the norm of K is arbitrarily large and condition (c) or (d) fails. This suggests that the size of the commutator of generators is not the best criterion for the stability of perturbed semigroups.
We try to understand how Gil's method relates to the approach via the Dyson-Phillips formula. First we show, see Sect. 2, that neither condition (a) nor (b) is necessary for the uniform exponential stability of perturbed semigroups, and, on the other hand, that there are (even bounded) generators A and B satisfying (a)-(b), such that the semigroup generated by A + B is not uniformly exponentially stable. This means that we cannot omit condition (c) or (d) in Gil's theorem, and that the size of the norm of K plays a role here. Next, in Sect. 3, we state a simple corollary, see Proposition 3.1, of the Dyson-Phillips theorem, that is useful for proving uniform exponential stability of perturbed semigroups. In particular, for one example considered by Gil' in [6], we give a simpler and simultaneously better estimate on when the discussed semigroup, depending on a parameter, is uniformly exponentially stable. In Sect. 4 we provide a class of examples of bounded perturbations of a semigroup resembling the translation semigroup in L 1 -type space, and compare in this case both approaches to the uniform exponential stability-via the Dyson-Phillips formula and via Gil's estimate of the norm of the commutator. Finally, in Sect. 5, we show that the examples discussed in Sect. 4 fall into a general scheme in an abstract Banach space. We prove that in this scheme the Dyson-Phillips series gives an explicit formula for the perturbed semigroup, which allows us to provide an efficient condition for its uniform exponential stability. As far as we understand, in many situations the approach via the Dyson-Phillips formula is at least as powerful as that of Gil'; see, however, Remark 4.3. What is even more important, the first one is easier to use.

Discussion of Gil's Conditions
First we notice that condition (b) is not necessary for the uniform exponential stability of a perturbed semigroup. Indeed, if A+B is a generator of a strongly continuous semigroup in a Banach space X, then e t(A+B) ≤ M e ωt for some M ≥ 1, ω ∈ R, and all t ≥ 0. Hence, where I is the identity operator in X, which implies that the semigroup generated by is uniformly exponentially stable, no matter what A and B are.
Next we show that condition (a) is not necessary either.
and clearly the semigroup generated by A+B is uniformly exponentially stable, since e t(A+B) 1 is of order e −t as t → +∞. As we indicated in Introduction there exist, see [3], uniformly exponentially stable semigroups generated by A + B with arbitrarily large norm of the commutator K of A and B. This naturally leads to the question if conditions (a)-(b), without (c) or (d), are sufficient for the uniform exponential stability of e t(A+B) t≥0 . In the following example we show that the answer is in the negative, and there are bounded operators A and B satisfying conditions (a)-(b), such that the semigroup generated by A + B is not uniformly exponentially stable. In other words, this means that the size of the norm of K plays a role in the discussed problem. Example 2.2. As in Example 2.1 we consider R 2 with the same norm, and define where A * is the transpose of A. Using [5, I.2.7] we calculate It follows that conditions (a)-(b) are satisfied, however, e t(A+B) = e −2t cosh 3t sinh 3t sinh 3t cosh 3t , t ≥ 0, which shows that the semigroup generated by A + B is not uniformly exponentially stable, since e t(A+B) 1 is of order e t . Note that in this case Consequently K 1 = 9, and K 1 J 2 > 1, which shows directly that condition (c) does not hold.

Dyson-Phillips Formula
Let A be a generator of a strongly continuous semigroup in a Banach space X. If B is a bounded operator in X, then the Dyson-Phillips theorem, see [5, III.1.10], says that the semigroup e t(A+B) t≥0 generated by A + B is given by the series which converges in the operator norm, where T 0 (t) := e tA and then as a simple consequence of the Dyson-Phillips formula we obtain the estimate This leads to the following result. We stress that Proposition 3.1 emphasizes the importance of the distance of B to the subspace spanned by the identity operator. What is interesting, in the context of Gil's commutator approach, in every Hilbert space for a given bounded operator B it is possible to find a bounded operator A with A = 1 such that the norm of their commutator is arbitrarily close to inf μ∈C 2 B + μI , see [11,Theorem 4]. This suggests that Gil's conditions and estimate (3.4) are somehow related, at least in Hilbert spaces. However, the nature of this relationship eludes the author.

Gil's Example
As an application of Proposition 3.1 we consider the example discussed by Gil', see Sect. 3, and in particular formula (3.7), in [6]. Let where (·, ·) 2 is the standard scalar product in C 2 . We introduce the generator A in X by By applying Gil's result we obtain that the semigroup generated by A + B is uniformly exponentially stable, provided that a ∈ (0, 1) and which holds for all a ∈ (0, α), where α is approximately 0.28.
On the other hand, for the identity operator I in X and the 2 × 2 identity matrix I 2 , we have here · is the operator norm in X, and · 2 is the operator norm in C 2 related to the standard scalar product (·, ·) 2 . The norm M (x) + I 2 2 is the spectral norm of M (x) + I 2 , see [8, p. 281], which is the square root of the that is for all a ∈ (−1, 1). In comparison to Gil's conclusion, which only gives a ∈ (0, α), where α ∼ 0.28, this is a stronger result. It is even possible to strengthened the result further, by choosing a different shift μ. We show that the semigroup generated by A + B is uniformly exponentially stable, provided that Indeed, let μ a := 1 + |a|. Then, as before, 1] M (x) + μ a I 2 2 .

More Examples
We provide a class of examples of bounded perturbations of a semigroup resembling the translation semigroup in L 1 -type space. In this setup we compare both approaches to the uniform exponential stability, via Gil's size of the commutator and via the Dyson-Phillips formula. We show, in Sect. 4.1, that even the simplest consequence of the latter, namely the estimate (3.3), is quite powerful and easy to apply. Let R + = [0, +∞) and let ψ be a real-valued locally integrable function on R + satisfying sup x∈R+ x+1 x |ψ(y)| dy < +∞.
This guarantees that (4.1) In the space L 1 (R + ) of Lebesgue integrable functions over R + equipped with the standard norm · 1 we introduce the operator A defined by Af We introduce the left translation operator τ t in L 1 (R + ) as and denote f ∞ := sup x∈R+ |f (x)| for every bounded function f on R + . Note that for φ defined above we have Moreover, We postpone the proof of Lemma 4.1 to "Appendix".

Bounded Perturbation of A
Here we consider a specific bounded perturbation of the generator A given by (4.2). To this end let φ, defined by (4.1), be such that 1/φ belongs to L 1 (R + ). Then, compare Sect. 3.2 in [3], where c is a continuous and compactly supported nonnegative function on R + satisfying c 1 = 1, defines a bounded operator in L 1 (R + ) with Note that B 2 = B, since . This implies that B 1 ≥ 1 and e tB = (e t − 1)B + I for all t ≥ 0, where I is the identity operator in L 1 (R + ). Consequently, since B is nonnegative, For each ω > 0 we denote and ask for which ω the semigroup generated by A + B ω is uniformly exponentially stable.
This implies that the semigroup generated by A+B ω is uniformly exponentially stable for every On the other hand, in order to apply Gil's theorem to A + B ω , we assume additionally (for simplicity) that c is continuously differentiable and c(0) = 0. By here we used the fact that c is differentiable, compactly supported and that c(0) = 0. This shows that AB − BA has a bounded extension K to L 1 (R + ) and By Gil's theorem, for every ω > 1 the semigroup generated by A + B ω is uniformly exponentially stable, provided that or, equivalently, that ω > ω κ : We showed in Example 4.2 that the semigroup generated by A + B ω is uniformly exponentially stable, provided that where the first estimate is obtained via the Dyson-Phillips formula, and the second one via Gil's theorem. To compare both approaches, or, in other words, to compare B 1 and ω K 1/2 1 we should somehow express K 1 in terms of B 1 , and this is not possible if we do not know c. However, we can make some estimates. Because cφ is continuous and compactly supported, we have, see (4.6), B 1 = cφ ∞ = c(x 0 )φ(x 0 ) for an x 0 ∈ [0, +∞). Assuming that c is differentiable, which we did when we used Gil's theorem, it follows that (cφ) (x 0 ) = 0, and consequently is chosen so that K 1 ≥ B 2 1 , then the Dyson-Phillips estimate gives better results no matter how big B 1 is. Finally, we also stress that similar analysis can be made for Gil's theorem with Bobrowski's improvement, however, the calculations are more involved here.

, then in particular
It is one of the advantages of the Dyson-Phillips approach that we do not need to calculate J or K explicitly, which is not easy in general, as we see in the following example.
and for A defined by (4.2) we have As before, by the Dyson-Phillips estimate, see (3.3), the semigroup generated by A + B ω = A + B − (1 + ω)I is uniformly exponentially stable, provided that ω > B 1 . On the other hand, in order to apply Gil's theorem, for ω > 0 we calculate where Γ is the Gamma function. Of course J ω converges for all ω > 0, since +∞ 0 e −ωt e t 1−γ dt does so, however, even if we know B 1 , it is really difficult to check, when condition (c) of Gil's theorem holds.

Abstract Setup
We consider the examples discussed in Sect. 4.1 as a special case of a general setup. We prove that in this setup it is possible to find an explicit formula for perturbed semigroups using the Dyson-Phillips theorem, which allows us to find a simple sufficient condition for the uniform exponential stability. Additionally, it turns out that in this setup the condition follows from condition (a) of Gil's theorem.
In the above setup we find an explicit formula for the perturbed semigroup generated by A + B (see [2, 8.2.23] and [1] for more examples in this fashion), and consequently we provide a sufficient condition for its uniform exponential stability.
Hence, it is easy to check by induction that Indeed, this holds for n = 0, and if we assume that T n+1 (t) is given by the above formula, then, by (5.5)  Since the right-hand side is of order e −ωt e tA as t → +∞, the result follows.
As a simple consequence of Theorem 5.1 we provide a much better estimate on when the semigroups generated by the operators A + B ω considered in Examples 4.2 and 4.4 are uniformly exponentially stable. converges for all ω > 0. Therefore, for such ω, the semigroup e t(A+Bω) t≥0 is uniformly exponentially stable by Theorem 5.1.
Finally, Theorem 5.1 applies also to the example considered by Bobrowski in [3]. We prove Proposition 1 from the cited paper.