The Cayley transform of the generator of a polynomially stable $C_0$-semigroup

In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $C_0$-semigroup. To estimate the decay rate of the Cayley transform, for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations.

been well established whether and how polynomial decay of (T (t)) t≥0 in the continuous setting yields the decay of the corresponding Cayley transform in the discrete setting. The purpose of this paper is to show that polynomial decay of a C 0 -semigroup is preserved under the Cayley transformation in a certain sense.
Applications of the Cayley transform of a semigroup generator arise in numerical analysis [26] and system theory [32,Section 12.3]. In the finitedimensional case, a matrix and its Cayley transform share the same stability properties, but this does not hold in the infinite-dimensional case. In fact, in the Banach space setting, the Cayley transform of the generator of even an exponentially stable C 0 -semigroup may not be power bounded [15,Lemma 2.1]. For the case of Hilbert spaces, it is still unknown whether the corresponding Cayley transform is power bounded for every generator of a uniformly bounded C 0 -semigroup, as mentioned in Section 5.5 of [5]. However, some sufficient conditions for Cayley transforms to be power bounded have been obtained; see, e.g., [12,14,15,17]. In particular, it is well known that A generates a C 0semigroup of contractions on a Hilbert space if and only if the corresponding Cayley transform is a contraction [21,Theorem III.8.1]. We refer the reader to the survey [13] for more details.
In this paper, we prove that if (T (t)) t≥0 is a polynomially stable C 0semigroup with parameter α > 0 on a Hilbert space with generator A such that its Cayley transform A d is power bounded, then there exists M d > 0 such that for all n ∈ N, .
We also show that in some cases, such as when A is normal, the logarithmic correction can be omitted. Moreover, we give a simple example of a normal operator A for which the decay rate 1/n 1/(α+2) cannot be improved.
To obtain the decay estimate of Cayley transforms, we extend the Lyapunovbased approach developed by Guo and Zwart [17]. In [17], uniform boundedness and strong stability of a C 0 -semigroup have been characterized in terms of the solution of a certain Lyapunov equation. The integral conditions on resolvents obtained in [16,31] for uniform boundedness and in [33] for strong stability play an important role in this Lyapunov-based approach. Therefore, we first obtain a similar integral condition for polynomial stability. By means of this integral condition, we next relate polynomial stability to the Lyapunov equation used in [17]. Finally, we estimate the decay rate of the Cayley transform, by using the solution of the Lyapunov equation.
As another application of the Lyapunov-based approach, we consider the following robustness analysis of polynomial stability: If A generates a polynomially stable semigroup, then does A + rA −1 also generate a polynomially stable semigroup for every r > 0? Robustness of polynomial stability has been studied in [23][24][25]27]. In these previous studies, perturbations are not structured, but the norms of the perturbations are assumed to be bounded in a certain sense. In contrast, the class of perturbations we consider is limited to {rA −1 : r > 0}, but we do not place any norm conditions for perturbations. We show that if A generates a polynomially stable semigroup with parameter α > 0, then for every r > 0, A + rA −1 also generates a polynomially stable semigroup with the same parameter α in the case α > 2 and with parameter α + ε for arbitrary small ε > 0 in the case α < 2. If α = 2, then a logarithmic factor appears in the rate of decay.
This paper is organized as follows. In Section 2, we collect some preliminary results on polynomial stability. In Section 3, we present an integral condition on resolvents for polynomial stability and then connect this stability to Lyapunov equations. In Section 4, we study the decay rate of the Cayley transform of the generator of a polynomially stable C 0 -semigroup. Section 5 contains the robustness analysis of polynomial stability for the class of perturbations {rA −1 : r > 0}.
Notation Let C − := {λ ∈ C : Re λ < 0} and iR := {iη : η ∈ R}. The closure of a subset Ω of C is denoted by Ω. For real-valued functions f, g on for every t ≥ t 0 , and similarly, f (t) = o(g(t)) as t → ∞ if for every ε > 0, there exists t 0 ∈ J such that f (t) ≤ εg(t) for every t ≥ t 0 . Let X be a Banach space. For a linear operator A on X, we denote by D(A) and ran(A) the domain and the range of A, respectively. The space of bounded linear operators on X is denoted by L(X). For a closed operator A : D(A) ⊂ X → X, we denote by σ(A) and ̺(A) the spectrum and the resolvent set of A, respectively. Letσ(A) be the extended spectrum of A defined byσ For λ ∈ ̺(A), the resolvent operator is given by R(λ, A) := (λ − A) −1 . Let H be a Hilbert space. The inner product of H is denoted by ·, · . The Hilbert space adjoint for a linear operator A with dense domain in H is denoted by A * .

Background on polynomially stable semigroups
In this section, we review the definition and some important properties of polynomially stable C 0 -semigroups.
The spectrum of the generator of any uniformly bounded semigroup is contained in the closed left half-plane C − . Therefore, if A generates a polynomially stable semigroup, then σ(A) ⊂ C − .
Polynomial decay (2.1) of a C 0 -semigroup (T (t)) t≥0 on a Hilbert space can be characterized by orbits as well; see [6,Theorem 2.4].
Theorem 2.2 Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space H with generator A such that iR ⊂ ̺(A). For a fixed α > 0, (2.1) holds if and only if holds for every x ∈ H.
For the generator A of a polynomially stable C 0 -semigroup, −A is sectorial in the sense of [18,Chapter 2], and hence the fractional powers (−A) α are well defined for all α ∈ R. Using the moment inequality (see, e.g., Proposition 6.6.4 of [18]), we can normalize the decay rate in (2.1). See [2, Proposition 3.1] for the proof. Lemma 2.3 Let α > 0 and (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Banach space with generator A such that 0 ∈ ̺(A). Then for some/all γ > 0.
A similar normalization result holds also for the case of orbits (2.2). The proof is essentially same as that of Lemma 2.3, i.e., it is a consequence of the moment inequality as stated in the proof of Theorem 2.4 of [6]. However, to make our presentation self-contained, we give a short argument. Lemma 2.4 Let α > 0 and (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Banach space X with generator A such that 0 ∈ ̺(A). Then for all x ∈ X if and only if for all x ∈ X and some/all γ > 0.
Since sup for every x ∈ X, it follows from the uniform boundedness principle that there exists C > 0 such that Take x ∈ X, ε > 0, and k ∈ N. Let By (2.4), there exists t 0 > 0 such that For every t ≥ kt 0 , This implies that for every k ∈ N, By the moment inequality, for every k ∈ N and every ϑ ∈ (0, 1), there exists a constant L 1 > 0 such that for all t ≥ 0, where M := sup t≥0 T (t)x . This and (2.5) yield Suppose that (2.3) holds for all x ∈ X. Takeγ > 0 and x ∈ X. Substituting γ = 1 into (2.5), we have that for every k ∈ N, As in (2.6), we see that for every k ∈ N and every ϑ ∈ (0, 1), there exists L 2 > 0 such that In Lemma 2.4, we consider the global conditions on the decay of all orbits {(T (t)x) t≥0 : x ∈ X}. For individual orbits, a partial result holds. Since it can be obtained from the moment inequality as in (2.6), we omit the proof.

Polynomial stability and Lyapunov equation
In this section, we connect polynomial stability to a certain Lyapunov equation. To this end, we first develop an integral condition on resolvents for polynomial stability.
Proposition 3.1 Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space H with generator A such that iR ⊂ ̺(A). The following three assertions hold for a fixed α > 0: To prove Proposition 3.1, we study the decay rate of an individual orbit.
Lemma 3.2 Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space H with generator A such that 0 ∈ ̺(A). The following two assertions hold for fixed α > 0 and x ∈ H: for all γ ∈ (0, 1/2) and lim On the other hand, if (3.4) holds, then By the Plancherel theorem, First we consider the case γ ∈ (0, 1/2). We have that Since ε > 0 was arbitrary, it follows that (3.3) holds.
for every x ∈ H. By the uniform boundedness principle, there exists C 1 > 0 such that Hence Thus (3.2) holds.
Let ξ > 0 and A be the generator of a uniformly bounded C 0 -semigroup (T (t)) t≥0 on a Hilbert space H. Consider the Lyapunov equation It has a unique self-adjoint solution Q(ξ) ∈ L(H) given by (3.14) see, e.g., Theorem 4.1.23 of [8]. We restate Proposition 3.1 by using the self-adjoint solution of the Lyapunov equation (3.13). Lemma 3.3 Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space H with generator A such that iR ⊂ ̺(A), and let Q(ξ) ∈ L(H) be the self-adjoint solution of the Lyapunov equation (3.13) for ξ > 0. The following three assertions hold for a fixed α > 0: Proof Let ξ > 0 and x ∈ H. The Plancherel theorem shows that By the definition (3.14) of Q(ξ), we obtain Hence the assertions a)-c) immediately follow from Proposition 3.1.

Decay rate of Cayley transform
In this section, we study the decay rate of the Cayley transform of the generator of a polynomially stable C 0 -semigroup. We start by establishing the discrete version of the normalization results on polynomial decay rates developed in Lemmas 2.3-2.5.
Lemma 4.1 Let A be the generator of a uniformly bounded C 0 -semigroup on a Banach space X such that 0 ∈ ̺(A). Suppose that B ∈ L(X) is power bounded and commutes with A −1 . Then the following assertions hold for constants α, β > 0 and a nondecreasing function f : N → (0, ∞).
(i) ⇒ (ii): Takeγ > 0. Substituting γ = 1 and ϑ =γ/k with k >γ into (4.4), we obtain for some L 2 > 0. Hence we obtain (ii) with γ =γ by similar arguments as above. b) Suppose that (ii) holds, and set δ := αγ. Since for all x ∈ X, it follows from the uniform boundedness principle that there exists a constant C 1 > 0 such that This implies The rest of the proof is the same as the proof of a). We omit the details. c) This assertion directly follows from the application of the moment inequality as in (4.3).
Using the argument based on Lyapunov equations developed in Section 3, we estimate the decay rate of the Cayley transform. We use the following preliminary result obtained in the proof of [17,Theorem 4.3].

This implies that
for every x ∈ H. The first assertion (4.7) follows from Lemma 4.1. Applying the uniform boundedness principle to (4.7), we obtain (4.8).

Remark 4.4 In the proof of Theorem 4.3, we employ b) of Lemma 3.3. One can apply a) of Lemma 3.3 in a similar way and consequently obtain
for every 0 < γ < 1/2. However, this result is less sharp than (4.8).
We do not know whether the logarithmic factor in Theorem 4.3 may be dropped in general. In some cases, however, we can omit the logarithmic correction, as in Propositions 4.1 and 4.2 in [2]. It is worth mentioning that we do not need the assumption on the power boundedness of the Cayley transform.
Proposition 4.5 Consider the following two cases: a) Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space with generator A such that iR ⊂ ̺(A) and A is normal. b) Let Ω be a locally compact Hausdorff space and µ be a σ-finite regular Borel measure on Ω. Assume that either (i) X := L p (Ω, µ) for 1 ≤ p < ∞ and φ : Ω → C is measureable with essential range in the open left half-plane C − ; or that (ii) X := C 0 (Ω) and φ : Ω → C is continuous with φ(Ω) ⊂ C − . Let A be the multiplication operator induced by φ on X, i.e, Af = φf with domain D(A) := {f ∈ X : φf ∈ X} and (T (t)) t≥0 be the C 0 -semigroup on X generated by A.

In both cases a) and b), if T (t)(−A)
Proof The proof is divided up into two steps. In the first step, we characterize the norm of A n d (−A) −α for n ∈ N by the spectrum of A. In the second step, we obtain the decay estimate (4.10) from this characterization and the geometrical condition on σ(−A) for polynomial decay given in Propositions 4.1 and 4.2 of [2].
Step 1: First, we consider the case a). Define for λ ∈ C \ (−∞, 0] and n ∈ N. Then A n d (−A) −α−2 = f n (−A) by the product formula of functional calculi (see, e.g., Theorem 1.3.2.c) of [18]). The spectral mapping theorem (see, e.g., Theorem 2.7.8 of [18]) shows that where f n (∞) := lim λ→0 f n (1/λ) = 0. Since A is normal, we see that f n (−A) is also normal, by using a multiplication operator unitarily equivalent to A obtained from the spectral theorem; see, for example, Theorem 4.1 of [19] for the multiplicator version of the spectral theorem for unbounded normal operators. Moreover, f n (−A) is bounded. Hence the spectral radius of f n (−A) equals f n (−A) ; see, e.g., Theorem 5.44 of [34]. This yields A result similar to (4.12) is obtained in the case b). Let h ess (Ω) denote the essential range of a measurable function h : Ω → C. Define the function f n as in (4.11). In the L p -case (i), In the C 0 -case (ii), we also obtain (4.13), since by Proposition I.4.2 of [10].
where (e k ) k∈N is the standard basis of ℓ 2 (C). The semigroup (T (t)) t≥0 generated by A satisfies T (t)A −1 = O(1/t) as t → ∞ by Proposition 4.1 of [2]. We see from Proposition 4.5 that The Cayley transform A d is given by for all k ∈ N, it follows that A d is power bounded. By Lemma 4.1.a), the norm estimate (4.16) is equivalent to Define λ k := 1/k − ik ∈ σ(−A) for k ∈ N, and take m ∈ N. Then for every n ∈ N. We have that (4.18) By (4.18) with m = 1, the norm estimate (4.16) is optimal in the sense that lim sup n→∞ n A n d A −3 > 0. The norm estimate (4.17) and the substitution of m = 3 into (4.18) imply that the optimal decay rate of A n d A −1 is 1/n 1/3 .

Robustness analysis of polynomial stability
As another application of the argument based on Lyapunov equations established in Section 3, we here extend to the case of polynomial stability the following result (Lemma 2.6 of [17]) on the preservation of uniform boundedness.
Lemma 5.1 Let A be the generator of a uniformly bounded C 0 -semigroup on a Hilbert space H. If 0 ∈ ̺(A), then A + rA −1 also generates a uniformly bounded C 0 -semigroup semigroup on H for every r ≥ 0.
This robustness result can be extended to the case of strong stability and exponential stability; see p. 358 of [17].
It turns out that the class of perturbations {rA −1 : r > 0} results in at most only arbitrarily small loss of decay rates.
Proposition 5.2 Let A be the generator of a uniformly bounded C 0 -semigroup (T (t)) t≥0 on a Hilbert space H such that iR ⊂ ̺(A). If T (t)(−A) −α = O(1/t) as t → ∞ for some α > 0, then the following assertions hold for every r ≥ 0: a) A + rA −1 generates a uniformly bounded C 0 -semigroup (S r (t)) t≥0 on H; b) iR ⊂ ̺(A + rA −1 ); and c) for every ε ∈ (0, 1), We need three auxiliary results for the proof of Proposition 5.2. First, we present a simple result on the resolvent set of A + A −1 . Proof Let ω ∈ R. A routine calculation shows that where From iR ⊂ ̺(A), it follows that A − iω 1 I and A − iω 2 I are invertible in L(X). Hence Thus, iR ⊂ ̺(A + A −1 ).
Second, we present a result on the domain of fractional powers under bounded perturbations, which is used to prove (5.1) in the case 0 < α < 2.
Proof For α ∈ (0, 1), define the abstract Hölder spaces of order α by Since A and A + B generate C 0 -semigroups with negative growth bounds, it follows from Proposition II.5.33 of [10] that for 0 < β < α < 1. It suffices to show that X α ⊂ X B α for every α ∈ (0, 1). In fact, combining this inclusion with (5.2), we obtain To obtain X α ⊂ X B α , we use the identity We are now ready to prove Proposition 5.2.