A Note on Integrability of the Norm of a Perturbed Semigroup

Recently, Michael Gil’ has provided a condition for integrability of norms of semigroups generated by operators of the form $$A+B$$

A
+
B

 where B is bounded, in terms of the norm of the commutator of A and B. We show that his argument, when slightly refined, leads to a bit more general result than the one presented in his paper. We also argue, by examining an example, that the size of the commutator may not be a good indicator of whether the perturbed semigroup has integrable norm or not.


Introduction
Recently, in [6], Michael Gil' has proved the following result. (See also his related article [7] devoted to the case where {e tA , t ≥ 0} is holomorphic.) Let B be a Banach space, and let L(B) be the algebra of bounded linear operators in B. Suppose that A is the generator of a strongly continuous semigroup {e tA , t ≥ 0} in B, and let B ∈ L(B) be such that (For the history of this kind of theorems, dating back to R.S. Phillips, see the survey article [10].) It is perhaps worth noting here that existence of the semigroup {e t(A+B) , t ≥ 0} is guaranteed by the Phillips Perturbation Theorem ( [1,2,4,8] or [9]); also, Lebesgue measurability of t → e tG for any strongly continuous semigroup {e tG , t ≥ 0} is not an issue (see [9] p. 306).
In this note we show that condition (c) may be relaxed: it suffices to assume, instead of (c), that (c') K I < 1, where This implication is established in Sect. 2. In Sect. 3, we discuss an illustrative example showing that in certain situations this criterion is significantly stronger than that of M. Gil'.
In Sect. 4, we argue that, unfortunately, 'the size of the commutator' seems to be a rather poor criterion for checking whether (1) holds. More specifically, we show that for any natural n there are semigroups {e tA , t ≥ 0} and {e tB , t ≥ 0} such that (i) conditions (a) and (b) of the theorem of Gil' are satisfied, (ii) the commutator of A and B, though bounded, has norm larger than n, and (iii) inequality (1) nevertheless holds.

Main Result
We start by noting that, by Tonnelli's Theorem and the fact that t → e tB is submultiplicative, with equality if t → e tB is multiplicative. This shows that assumption (c') is weaker than (c).
That (c') can be substantially weaker is shown by the following example: if A is chosen so that e tA = 1, t ≥ 0 and B is chosen so that, for a κ > 1 and ω > 0, e tB = κe −ωt , t ≥ 0, then I = κ ω 2 and J 2 = κ 2 ω 2 > I (see also our Sect. 3).
Our argument that (c') may successfully replace (c) is a straightforward refinement of that of M. Gil'. He proved (see equation (2.5) and the top of page 26 in [6]) that  By assumption (c'), w is integrable over R + with w L 1 := ∞ 0 w(t) dt = K I < 1. Introducing, for simplicity of notation, ψ(t) = e tA e tB and φ(t) = e t(A+B) we have thus Since, for any T > 0 (comp. p. 26 in [6]) We also note that M. Gil's estimates for the L 1 and supremum norms of e tA e tB − e t(A+B) (i.e., formulae (1.4) and (1.5) in [6]) may be similarly improved.

An Example
In this section we construct a specific example of semigroups for which I < J 2 , i.e. our criterion is significantly stronger than that of M. Gil'. Our starting point is a semigroup {e tA , t ≥ 0} of isometries; its generator A is then perturbed by a family of operators B ω , where ω > 0 is a parameter. We will show that (c') allows concluding (1) for a larger range of ω than (c).

Semigroup Generated by A
Let B be the space C[0, 1] of real-valued continuous functions f on the unit interval, and let for t ≥ 0, S(t) be the operator in B given by It can be checked that {S(t), t ≥ 0} is a strongly continuous semigroup of non-negative operators in B. As we shall see later (see part (c) of the proof of Proposition 1, later on), the domain D(A) of the generator of this semigroup is composed of f ∈ B such that the derivative of

Operators B ω , ω > 0 and the Related Semigroups
Next, given a continuously differentiable, non-negative function c ∈ B, such that c(1) = 0 and 1 0 c = 1, we consider C ∈ L(B) given by Then, for A straightforward argument (see e.g. [2] p. 6) based on the easy-to-establish formula C n = C, n ≥ 1 (use where I is the identity operator in B. Hence, arguing as above, we check that e tC = (e t − 1)κ + 1. Thus, introducing where ω > 0 is a parameter, we obtain

The Commutator
It is easy to check that C maps B into D(A) and that ACf = 0, f ∈ B. Also, assumption c(1) = 0 (used here for the first time) and integration by parts yield

Therefore the map D(A) f → CAf − ACf is a restriction of the bounded linear operator given by
Since B ω Af − AB ω f = CAf − ACf for all f ∈ D(A) and ω > 0 (i.e., a shift of C does not change the commutator) it follows that assumptions (a) and (b) of Sect. 2 are satisfied for our A and B = B ω for all ω > 0. Moreover, by (2) and (3), e tA e tBω dt = 1 1 + ω 1 + κ ω , e (t−s)A e sA ds dt = 1 (1 + ω) 2 and so J 2 ω and I ω differ by κ(κ−1) ω 2 (1+ω) 2 > 0. It is thus clear that (c') is satisfied for a larger range of ω than is (c).

Is 'The Size of the Commutator' a Good Criterion for (1)?
The criterion proved in Sect. 2 guarantees that for the operators A and B ω of our example, if i.e., for all sufficiently large ω. On the other hand, a look at the definitions of κ and K reveals that the function c may be modified, without affecting κ, so that K is made arbitrarily large (see the example at the end of this section for details). In such a case the criterion implies (4) only for really large ω. It is our goal in this section to establish the following statement.

Proposition 1. For the operators A and B ω of the previous section,
∞ 0 e t(A+Bω) dt < ∞ for all ω > 0 regardless of how large K is.
In view of our previous analysis, this proposition suggests that 'the size of the commutator' is probably not the best possible criterion for condition (1). We also note that this proposition holds even if we do not assume that c is differentiable and c(1) = 0; these two assumptions were used to check that the commutator is a bounded operator.
Proof (of Proposition 1). (a) We start by recalling (see e.g. [1,2,4]) that if {e tG , t ≥ 0} is a strongly continuous semigroup in a Banach space B and J ∈ L(B) is an isomorphism, then {J −1 e tG J, t ≥ 0} is also a strongly continuous semigroup, often termed similar or isomorphic to {e tG , t ≥ 0}. The generator, say G , of the latter semigroup is characterized as follows: f ∈ D(G ) iff Jf ∈ D(G), and then G f = J −1 GJf. Since e tG ≤ J −1 e tG J and the roles of the semigroups {e tG , t ≥ 0} and {e tG , t ≥ 0} are symmetric, it is clear that t≥ 0, for the J defined by or, which is the same, that (d) Next, since the operator C :=J −1 CJ is given by it is non-negative and has norm 1. Therefore, also the operators J −1 e t(C−I) J = e −t e tC , t ≥ 0 are non-negative and have norms equal to 1. This together with (8) implies that the so-called stability condition is satisfied and thus the Trotter Product Formula (see e.g. [1,2,4,5] or [8]; compare [3]) applies:  (7), completing the proof.
The operator A + C − I is in fact a Feller generator (see e.g. [1,5]). In the stochastic process described by this generator, a particle starting at an x ∈ [0, 1] travels along the trajectory t → xe −t for an exponential time with parameter 1 and then jumps to a randomly chosen y ∈ [0, 1], the probability that it jumps to a Borel set B ⊂ [0, 1] equaling B c, to continue his travel along t → ye −t , and so on.