Decay Rate of exp(A-1t)A-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\exp (A^{-1}t)A^{-1}}$$\end{document} on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data

This paper is concerned with the decay rate of eA-1tA-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{A^{-1}t}A^{-1}$$\end{document} for the generator A of an exponentially stable C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroup on a Hilbert space. To estimate the decay rate of eA-1tA-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$e^{A^{-1}t}A^{-1}$$\end{document}, we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_0$$\end{document}-semigroup whose generator is normal.


Introduction
Let A be the generator of a bounded C 0 -semigroup (e At ) t≥0 on a Hilbert space.Suppose that the inverse A −1 of A exists and also generates a C 0 -semigroup (e A −1 t ) t≥0 .The problem we study first is to estimate the decay rate of e A −1 t A −1 under additional assumptions on stability of (e At ) t≥0 .This is a variant of the so-called inverse generator problem raised by deLaubenfels [9]: "Let A be the generator of a bounded C 0semigroup on a Banach space.Assume that there exists an inverse A −1 as a closed, densely-defined operator.Does A −1 also generate a bounded C 0 -semigroup?"Positive answers to the inverse generator problem have been given for bounded holomorphic C 0 -semigroups on Banach spaces [9] and for contraction C 0 -semigroups on Hilbert spaces [29].Negative examples can be found in [20] with respect to the generation property of A −1 and in [29] with respect to the boundedness property of (e A −1 t ) t≥0 ; see also [11,16] for other counterexamples.For bounded C 0 -semigroups on Hilbert spaces, the answer to the inverse generator problem is still unknown, but it has been shown in [28] that if A generates an exponentially stable C 0 -semigroup on Hilbert spaces, then e A −1 t = O(log t) as t → ∞, that is, there exist M > 0 and t 0 > 1 such that e A −1 t ≤ M log t for all t ≥ t 0 .This result has been extended to bounded C 0 -semigroups with boundedly invertible generators in [5].We refer the reader to the survey article [14] for more detailed discussion of the inverse generator problem.
It is known that if A generates an exponentially stable C 0 -semigroup on a Banach space, then (1) for all k ∈ N.This norm-estimate has been obtained in [29] for k = 1 and in [10] for k ≥ 2. In this paper, we obtain an analogous estimate for the Hilbert case: if A generates an exponentially stable C 0 -semigroup on a Hilbert space, then (2) for all k ∈ N. To obtain the estimate (1) of the Banach case, an integral representation of (e A −1 t ) t≥0 (see [29,Lemma 3.2] and [16,Theorem 1]) has been applied.In contrast, to obtain the estimate (2) of the Hilbert case, we employ a functional-calculus approach based on the B-calculus introduced in [5].Another class of C 0 -semigroups we study are polynomially stable C 0 -semigroups with parameter β > 0 on Hilbert spaces, where (e At ) t≥0 is called a polynomially stable C 0 -semigroup with parameter β > 0 if (e At ) t≥0 is bounded and if e At (I − A) −1 = O(t −1/β ) as t → ∞.By a functional-calculus approach as in the case for exponentially stable C 0 -semigroups, we show that if, in addition, A is a normal operator, then (3) for all k ∈ N. We also present simple examples for which the norm-estimates (2) and (3) cannot be improved.Moreover, we prove that the (not necessarily normal) generator A of a polynomially stable C 0 -semigroup with any parameter on a Hilbert space satisfies sup t≥0 e A −1 t A −1 < ∞, inspired by the Lyapunov-equation technique developed in [28] .
It is our aim to estimate the decay rate of the solution (x n ) n∈N with smooth initial data x 0 ∈ D(A).
For a constant stepsize τ > 0, upper bounds on the growth rate of (A d (τ ) n ) n∈N0 have been obtained for bounded C 0 -semigroups and exponentially stable C 0 -semigroups on Banach spaces; see [6,15].The relation between (e A −1 t ) t≥0 and (A d (τ ) n ) n∈N0 in terms of growth rates has been investigated in [15].As in the inverse generator problem, (A d (τ ) n ) n∈N0 is bounded whenever C 0 -semigroups are bounded holomorphic on Banach spaces [7].Furthermore, it is well known that for contraction C 0 -semigroups on Hilbert spaces, A d (τ ) is a contraction; see [22].It is still open whether (A d (τ ) n ) n∈N0 is bounded for every bounded C 0 -semigroup on a Hilbert space.However, it has been shown independently in [1,13,17] that if A and A −1 both generate bounded C 0 -semigroups on a Hilbert space, then (A d (τ ) n ) n∈N0 is bounded.If in addition (e At ) t≥0 is strongly stable, then (A d (τ ) n ) n∈N0 is strongly stable [17].Without the assumption on A −1 , the norm-estimate A d (τ ) n = O(log n) as n → ∞ given in [13] remains the best so far in the Hilbert case.For polynomially stable C 0 -semigroups with parameter β > 0 on Hilbert spaces, the following estimates have been obtained in [25]: For more information on asymptotics of (A d (τ ) n ) n∈N0 , we refer to the survey [14].Some of the above results have been extended to the case of variable stepsizes.For bounded C 0 -semigroups on Banach spaces, the Crank-Nicolson scheme with variable stepsizes has the same growth bound as that with constant stepsizes; see [2].It has been proved that the solution (x n ) n∈N0 of the time-varying difference equation ( 4) is bounded if A generates a bounded holomorphic C 0 -semigroups on a Banach space [21,23,24] or if A and A −1 generate bounded C 0 -semigroups on a Hilbert space [23].The stability property, i.e., lim n→∞ x n = 0 for all initial data x 0 , has been also obtained under some additional assumptions in [23].
We show that if A and A −1 are the generators of an exponentially stable C 0 -semigroup and a bounded C 0semigroup on a Hilbert space, respectively, then the solution (x n ) n∈N0 of the time-varying difference equation (4) with x 0 ∈ D(A) satisfies (5) x Moreover, if A is a normal operator generating a polynomially stable C 0 -semigroup with parameter β > 0 on a Hilbert space, then (6) for all x 0 ∈ D(A).These estimates (5) and ( 6) follow from the combination of the norm-estimates (2) and ( 3) of e A −1 t A −1 and the Lyapunov-equation technique developed in [23].In the constant case τ n ≡ τ > 0, we give a simple example of an exponentially stable C 0 -semigroup showing that the decay rate A d (τ ) n A −1 = O(1/ √ n) cannot in general be improved.The estimate (5) includes the logarithmic factor √ log n, but its necessity remains open.On the other hand, the decay rate given in (6) is the same as that obtained for constant stepsizes in [25], and one cannot in general replace n −1/(2+β) in (6) by functions with better decay rates.This paper is organized as follows.In Section 2, we give long-time norm-estimates for e A −1 t A −1 .We consider first exponentially stable C 0 -semigroups and then polynomially stable C 0 -semigroups.In Section 3, the decay rate estimates of e A −1 t A −1 are utilized in order to examine the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data.Notation.Let C + := {λ ∈ C : Re λ > 0}, C + := {λ ∈ C : Re λ ≥ 0}, C − := {λ ∈ C : Re λ < 0}, and iR := {iη : η ∈ R}.Let N 0 and R + be the set of nonnegative integers and the set of nonnegative real numbers, respectively.For real-valued functions f, g on J ⊂ R, we write if there exist constants M > 0 and t 0 ∈ J such that f (t) ≤ M g(t) for all t ∈ J satisfying t ≥ t 0 .The gamma function is denoted by Γ, i.e., Let H be a Hilbert space.The inner product of H is denoted by •, • .The space of bounded linear operators on H is denoted by L(H).For a linear operator A on H, let D(A) and σ(A) denote the domain and the spectrum of A, respectively.For a densely-defined linear operator A on H, we denote by A * the Hilbert space adjoint of A. If A is the generator of a bounded C 0 -semigroup on a Hilbert space and if A is injective, the fractional power (−A) α of −A is defined by the sectorial functional calculus for α ∈ R as in [18,Chapter 3].Let ℓ 2 be the space of complex-valued square-summable sequences x = (x n ) n∈N endowed with the norm x :=

Norm-Estimates of Semigroups Generated by Inverse Generators
The aim of this section is to derive a norm-estimate for e A −1 t (−A) −α with α > 0. First we assume that A is the generator of an exponentially stable C 0 -semigroup on a Hilbert space.Next we focus on the generator A of a polynomially stable C 0 -semigroup on a Hilbert space.

Inverses of generators of exponentially stable semigroups.
Let A be the generator of an exponentially stable C 0 -semigroup on a Hilbert space.Then A is invertible, and A −1 generates a C 0 -semigroup (e A −1 t ) t≥0 .To obtain a norm-estimate for e A −1 t (−A) −α , we employ the B-calculus introduced in [5].
2.1.1.Basic facts on B-calculus.Let B be the space of all holomorphic functions f on C + such that We recall elementary properties of functions f in B. The proof can be found in [5, Proposition 2.2].Proposition 2.1.For f ∈ B, the following statements hold: A norm on B is defined by The space B equipped with the norm • B is a Banach algebra by [5,Proposition 2.3].Let M(R + ) be the Banach algebra of all bounded Borel measures on R + under convolution, endowed with the norm µ M(R+) := |µ|(R + ), where |µ| is the total variation of µ.For µ ∈ M(R + ), the Laplace transform of µ is the function We define Then the space LM endowed with the norm Lµ HP := µ M(R+) becomes a Banach algebra.As shown on [5, p. 42], LM is a subspace of B with continuous inclusion.
Let −B be the generator of a bounded C 0 -semigroup (e −Bt ) t≥0 on a Hilbert space H. Define K := sup t≥0 e −Bt .Using Plancherel's theorem and the Cauchy-Scwartz inequality, we obtain for all x, y ∈ H and ξ > 0; see [5,Example 4.1].From this observation, we define for f ∈ B and x, y ∈ H.This definition yields a bounded functional calculus; see [5,Theorem 4.4] for the proof.
Theorem 2.2.Let −B be the generator of a bounded C 0 -semigroup (e −Bt ) t≥0 on a Hilbert space H, and let f (B) be defined as in (7).Then the following statements hold: as defined in the Hille-Phillips calculus.
is called the B-calculus for B.

2.1.2.
Norm-estimate by B-calculus.By Theorem 2.2 (iii), the norm-estimate for a function f ∈ B implies the norm-estimate for the corresponding operator f (B).In [5,Lemma 3.4], the estimate where h t (z) := e −t/(z+1) , (8) has been derived for the norm-estimate of e A −1 t .To estimate e A −1 t (−A) −α for α > 0, the following result is used.
In [5,Corollary 5.7], the B-calculus and the norm-estimate (8) of h t (z) = e −t/(z+1) have been used to obtain the norm-estimate for e A −1 t .Analogously, the following theorem gives the norm-estimate for e A −1 t (−A) −α with α > 0.
Theorem 2.4.Let A be the generator of an exponentially stable C 0 -semigroup (e At ) t≥0 on a Hilbert space H. Then for all α > 0.
Proof.There exist constants K, ω > 0 such that e tA ≤ Ke −ωt for all t ≥ 0. We may assume that ω = 1 by replacing A by ω −1 A and t by ωt.Note that B := −A − I generates a bounded C 0 -semigroup (e −tB ) t≥0 with sup t≥0 e −tB ≤ K.
Take t, α > 0 arbitrarily, and consider the functions Then h t belongs to LM ⊂ B; see [5,Example 2.12].From the definition of the gamma function, we have that for all z > 0, Hence, the uniqueness theorem for holomorphic functions yields for all z ∈ C + .This means that r α is the Laplace transform of the function Hence r α also belongs to LM ⊂ B. Theorem 2.2 (i) shows that the function We obtain h t (B) = e A −1 t as shown in the proof of [5,Corollary 5.7].By Theorem 2.2 (ii), r α (B) coincides with the operator defined as the Hille-Phillips calculus.This and [18, Proposition 3.3.5]give Therefore, f t,α (B) = e A −1 t (−A) −α .Using Theorem 2.2 (iii), we have Thus, the desired estimate (12) holds by Lemma 2.3.
From the next example, we see that the norm-estimate (12) cannot be improved in general.
2.2.Inverses of generators of polynomially stable semigroups.We recall the definition of polynomially stable C 0 -semigroups.
Definition 2.2.A C 0 -semigroup (e At ) t≥0 on a Hilbert space is polynomially stable with parameter β > 0 if (e At ) t≥0 is bounded and if A C 0 -semigroup (e At ) t≥0 is simply called polynomially stable if it is polynomially stable with some parameter β > 0.
Let A be the generator of a polynomially stable semigroup on a Hilbert space H.By [4, Theorem 1.1], iR ∩ σ(A) = ∅.Then A is invertible, and therefore A −1 generates a C 0 -semigroup on H.When A is normal, we obtain the rate of decay of e A −1 t (−A) −α as in the case of exponentially stable C 0 -semigroups.We also show that sup t≥0 e A −1 t A −1 < ∞ without assuming that A is normal.
2.2.1.Case where generators are normal.When the generator A is a normal operator on a Hilbert space, a spectral condition equivalent to polynomial decay is known.The proof can be found in [3, Proposition 4.1].
Proposition 2.5.Let H be a Hilbert space and let A : D(A) ⊂ H → H be a normal operator.Assume that σ(A) ⊂ C − .For a fixed β > 0, the C 0 -semigroup (e At ) t≥0 satisfies if and only if there exist C, δ > 0 such that The next result gives an estimate for the rate of decay of e A −1 t (−A) −α .Proposition 2.6.Let H be a Hilbert space, and let A : D(A) ⊂ H → H be a normal operator generating a polynomially stable C 0 -semigroup (e At ) t≥0 with parameter β > 0 on H. Then for all α > 0.
As in the case of exponentially stable C 0 -semigroups, we present an example of an operator on ℓ 2 showing that the norm-estimate ( 14) cannot be in general improved.
By Proposition 2.5, A is the generator of a polynomially stable C 0 -semigroup with parameter β = 1.Let t, α > 0. We have Moreover, for all t ≥ α/3.This implies that for all k ∈ N and t = αk 3 /3, , and the estimate ( 14) cannot be improved for this operator A.

2.2.2.
Norm-estimate through Lyapunov equations.We show that if A is the generator of a polynomially stable C 0 -semigroup on a Hilbert space, then sup t≥0 e A −1 t A −1 < ∞.To this end, we use the following results on Lyapunov equations; see [8, Theorem 4.1.3,Corollary 6.5.1, and Theorem 6.5.2] for the proofs.Lemma 2.7.Let A be the generator of an exponentially stable semigroup (e At ) t≥0 on a Hilbert space H. Then there exists a unique self-adjoint, positive operator P ∈ L(H) such that P D(A) ⊂ D(A * ) and (15) A * P + P A = −I on D(A).
Moreover, the operator P is given by ( 16) for all x ∈ H.
To obtain sup t≥0 e A −1 t A −1 < ∞, we shall use the fact that P ∈ L(H) defined by ( 16) satisfies the Lyapunov equation ( 15).Lemma 2.8.Let A be the generator of a C 0 -semigroup on a Hilbert space H, and let C be a bounded linear operator from H to another Hilbert space Y .If there exists a self-adjoint, non-negative operator P ∈ L(H) such that P D(A) ⊂ D(A * ) and We shall need the estimate for adjoint operators in the following lemma; see [28,Lemma 2.1] for the proof.
Lemma 2.9.Let A be the generator of a bounded C 0 -semigroup (e At ) t≥0 on a Hilbert space H, and let K := sup t≥0 e At .Then for all x ∈ H and ξ, γ > 0, ( The same estimate holds for the adjoint. We are now in a position to prove that sup t≥0 e A −1 t A −1 < ∞ if A is the generator of a polynomially stable C 0 -semigroup.Actually, we will make the slightly weaker assumption that e At (I − A) −1 = O((log t) −β ) as t → ∞ for some β > 1.Notice that if the bounded C 0 -semigroup (e At ) t≥0 satisfies this estimate, then iR ∩ σ(A) = ∅ by [4,Theorem 1.1].The proof of the following theorem is inspired by the arguments in the proof of [28,Theorem 2.2].
Theorem 2.10.Let A be the generator of a bounded C 0 -semigroup on a Hilbert space H.If there exists β > 1 such that sup In particular, if A is the generator of a polynomially stable C 0 -semigroup on H, then the estimate (19) holds.
Proof.The proof consists of three steps.In the first step, we prove an estimate analogous to (17) for bounded C 0 -semigroups satisfying (18).In the next step, we obtain the norm-estimate of e (A−ξ1I) −1 t x − e (A−ξ2I) −1 t x for x ∈ D(A) and 0 < ξ 1 , ξ 2 < δ 0 , where δ 0 > 0 is some sufficiently small constant.This estimate is used to compare e (A−δI) −1 t x and e (A−δe −N I) −1 t x for x ∈ D(A), N ∈ N, and 0 < δ < δ 0 .Letting N → ∞, we show the desired estimate (19) in the last step.
Step 2: Take 0 < ξ 1 , ξ 2 < δ 0 .The variation of constants formula yields for all x ∈ H and t ≥ 0.Moreover, the resolvent equation gives Therefore, we obtain for all x ∈ H and t ≥ 0. Using the estimates ( 24) and ( 25), we have that for all x ∈ D(A), y ∈ H, and t ≥ 0, y, e (A−ξ1I) where for all x ∈ D(A) and t ≥ 0.

Crank-Nicolson Scheme with Smooth Initial Data
Let A be the generator of a bounded C 0 -semigroup on a Hilbert space H.For τ > 0, we define .
Let (τ n ) n∈N0 be a sequence of strictly positive real numbers.We consider the time-varying difference equation ( 28) In this section, we study the decay rate of the solution (x n ) n∈N0 of the difference equation (28) with smooth initial data.
The operators P (ξ) and Q(ξ) are given by for all x ∈ X.To estimate the decay rate of the solution (x n ) n∈N0 of the difference equation ( 28), we shall use that the operators P (ξ) and Q(ξ) defined as in (30) solve the Lypapunov equations (29).
Next we consider the case α = 1.The exponential integral satisfies Thus, (33) holds for all x ∈ D(A) and some suitable constant K > 0.
Combining Lemma 3.1 with the estimates (31)-(33), we obtain estimates for the decay rate of the solution (x n ) n∈N0 of the difference equation (28) with smooth initial data.Theorem 3.3.Let A be the generator of an exponentially stable C 0 -semigroup (e At ) t≥0 on a Hilbert space H. Suppose that A −1 generates a bounded C 0 -semigroup (e A −1 t ) t≥0 on H.If 0 < inf n∈N0 τ n ≤ sup n∈N0 τ n < ∞, then there exists K > 0 such that the solution (x n ) n∈N0 of the difference equation (28) satisfies the following statements: (i) For each 0 < α < 1, there exists K > 0 such that (34) x n ≤ K n α/2 (−A) α x 0 for all x 0 ∈ D((−A) α ) and n ≥ 1.
We compare the norm-estimates (34) and (35) with those in the time-invariant case τ n ≡ 2, using a simple example. .This implies that the norm-estimate (34) for the case 0 < α < 1 is optimal in the sense that one cannot obtain any better rates in general.
Next we show that (38) For all n ∈ N, 3.2.Normal generators of polynomially stable semigroups.Suppose that A is a normal operator on a Hilbert space H and generates a polynomially stable C 0 -semigroup.As in the case of exponentially stable C 0 -semigroups, one can obtain an estimate for the decay rate of the solution (x n ) n∈N0 of the difference equation (28) with smooth initial data.Note that A −1 generates a contraction C 0 -semigroup on H; see [29,Lemma 4.1].
The following lemma gives estimates analogous to those in Lemma 3.2.