Let $A$ be the infinitesimal generator of an exponentially stable, strongly continuous semigroup on a Hilbert space. We show that the powers of the Cayley transform of $A$ are bounded by a constant times $\log (n+1)$. The proof is based on Lyapunov equations

Besseling, N.C.

Zwart, Heiko J.

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The growth of a semigroup and its Cayley transform

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The relation between the growth of a semigroupand its Cayley transformNiels Besseling, Hans ZwartDepartment of Applied MathematicsUniversity of TwenteP.O. Box 217, 7500 AE EnschedeThe NetherlandsEmail: {n.c.besseling, h.j.zwart}@math.utwente.nlApril 18, 2011AbstractLet A be the infinitesimal generator of an exponentially stable,strongly continuous semigroup on a Hilbert space. We show that thepowers of the Cayley transform of A are bounded by a constant timeslog(n + 1). The proof is based on Lyapunov equations.Mathematics Subject Classification (2000). 34G10, 39A11, 46N20,46N40, 47D06, 65J10.Keywords. C0-semigroups, Cayley transform, Crank-Nicolson scheme,Lyapunov equations, stability.1 Introduction to the main resultConsider the linear differential equation:ẋ(t) = Ax(t), x(0) = x0 (1)on the Hilbert space X. A standard numerical way of solving such a differen-tial equation is the Crank-Nicolson method. In this method, the differentialequation (1) is replaced by the difference equation:xd(n + 1) = (A + I) (A − I)−1 xd(n), xd(0) = x0. (2)1The operator (A + I)(A− I)−1 is known as the Cayley transform of A, andwe denote it by Ad.We are interested in understanding how the Cayley transform affectsthe solutions to the differential equation (1). For instance, what happens tostability and stable solutions. More specifically, if we know that the solutionsof the differential equation (1) are exponentially stable, so ‖ exp(At)‖ ≤Me−ωt, with ω > 0, what can be said about the solutions of the differenceequation (2) and ‖And‖?Van Dorsselaer, Kraaijevanger and Spijker [3] studied this question forX being a finite-dimensional Banach space. For these finite-dimensonalsystems, the Cayley transform does not change the stability properties of thesolutions. For example, a bounded semigroup corresponds with a boundedsolution of the difference equation. So, if for the semigroup holds:supt≥0‖eAt‖ =: M < ∞,then also the Cayley transform powers are bounded:‖And‖ ≤ min(s, n + 1) e M < ∞, for all n ∈ N,where s is the dimension of the space X.Since this bound depends on the dimension, it is not applicable forinfinite-dimensional systems. For infinite-dimensional Banach spaces, thereis the famous result of Brenner and Thomée [1], giving‖And‖ ≤ M · M1 ·√n.For Hilbert spaces, Gomilko [4] improved this result. He proved that thesolutions of equation (2) are bounded by a constant times log(n + 1), if thesolutions of equation (1) are bounded.In this paper, we obtain a similar result as Gomilko, but we present adifferent proof. The result is:Theorem 1.1. Let A generate an exponentially stable C0-semigroup on theHilbert space X, such that, ‖ exp(At)‖ ≤ Me−ωt with ω > 0, then for then-th power of its Cayley transform Ad, the following estimate holds:‖And‖ ≤ 1 + 2M +(M√2+ 1)M√ω+ (2log n − 1)(M2 +√2M). (3)For the proof, we use a technique which applies Lyapunov equations.This technique, used by Zwart in [5], is explained in section 2. Section 3is quite technical: we derive some lemma’s which we use in section 4 tocomplete the proof of the main result.22 Lyapunov equationsIn this section, we focus on Lyapunov equations. They will lead to a Lya-punov estimate: a first step to proof the main result.To every exponentially stable semigroup exp(At), one can find a uniquesolution Q to the corresponding Lyapunov equation. With this equation,we show that Q is also the solution to a discrete Lyapunov equation for Ad.This discrete Lyapunov equation leads to the following estimate:Lemma 2.1 (Lyapunov estimate). Let A generate an exponentially stableC0-semigroup, such that, ‖ exp(At)‖ ≤ Me−ωt with ω > 0, then for theCayley transform Ad of A and the operator C =√2(A− I)−1, the followingestimate holds:∞∑n=0‖CAndx‖2 ≤M22ω‖x‖2. (4)Proof. The semigroup exp(At) is exponentially stable, so there exists a Lya-punov function Q such that:A∗Q + QA = −I, (5)with ‖Q‖ ≤ M22ω . Equation (5) is called the continuous Lyapunov equation.We show that for the Cayley transform Ad a discrete Lyapunov equationexists with solution Q. For this we use equation (5):A∗dQAd − Q = (A − I)−∗[(A + I)∗Q(A + I)− (A − I)∗Q(A − I)](A − I)−1= C∗[A∗Q + QA]C= −C∗C.From this, the following estimate for Ad follows:∞∑n=0‖CAndx‖2 ≤ ‖Q‖‖x‖2 ≤M22ω‖x‖2.For further details on Lyapunov equations we refer to [2, Chapter 4].Remark 2.2. For A∗, a similar estimate holds:∞∑n=0‖C∗A∗nd x‖2 ≤M22ω‖x‖2. (6)These estimates are important for proving the main result. We build onthis in the next section.33 Estimates on operatorsIn this section we apply the Lyapunov estimates of the previous section, andwe make some technical steps towards the proof of the main result.Firstly, we define a sequence of operators which starts with And . Secondly,we estimate the last operator of the sequence. Finally, we estimate thedifference between two adjacent operators in the sequence. By repeatedlyapplying this last estimate we can prove the main result. This will be donein section 4.Definition 3.1. We define the operators Aj and Cj by:Aj := (γjA − εjI + I) (γjA − εjI − I)−1 , (7)Cj =√2 (γjA − εjI − I)−1 . (8)with:γj+1 =12γj, γ0 = 1 (9)εj+1 =12+12εj , ε0 = 0 (10)The operators Aj are the Cayley transform of a generator of an expo-nentially stable C0-semigroup. This means we can apply Lemma 2.1 withC = Cj.Remark 3.2. From Definition 3.1, it follows that A0 = Ad.Now, we consider the operator sequence:Anjj , with j ∈ {0, . . . , N}, N = ⌊2log n⌋, (11)nj+1 = ⌊nj2⌋, with n0 = n, nN = 1. (12)We will end the proof with an estimate on the norm of the first operator,‖An00 ‖ = ‖And‖. To arrive there, we start with an estimate on the norm ofthe last operator, ‖AnNN ‖ = ‖AN‖:Lemma 3.3. Let AN be the operator defined by Definition 3.1, then thefollowing estimate holds:‖AN‖ ≤ 1 + 2M. (13)4Proof. For the proof, we use the Hille-Yosida Theorem [2, Theorem 2.1.12]and that γN , ω and εN are positive:‖AN‖ = ‖ (γNA − εNI + I) (γNA − εN I − I)−1 ‖= ‖ (γNA − εNI − I + 2I) (γNA − εNI − I)−1 ‖= ‖I + 2 (γNA − (εN + 1)I)−1 ‖≤ 1 + 2 MγNω + εN + 1≤ 1 + 2M.Next, we want to estimate the difference between two successive opera-tors, Anjj and Anj+1j+1 . For this, we need the following lemma, which focusseson even nj.Lemma 3.4. Let Ak and Ak+1 be defined by Definition 3.1, then the fol-lowing estimate holds:‖A2mk − Amk+1‖ ≤M22√γkω + εk√γk+1ω + εk+1. (14)Proof. Firstly, we need the following result:A2k − Ak+1 = (γkA − εkI + I)2 (γkA − εkI − I)−2− (γk+1A − εk+1I + I) (γk+1A − εk+1I − I)−1= (γkA − εkI − I)−2[(γkA − εkI + I)2 (γk+1A − εk+1I − I)− (γkA − εkI − I)2 (γk+1A − εk+1I + I)]·(γk+1A − εk+1I − I)−1 . (15)We simplify the middle part:(γkA − εkI + I)2 (γk+1A − εk+1I − I)− (γkA − εkI − I)2 (γk+1A − εk+1I + I)= 4 (γkA − εkI) (γk+1A − εk+1I) − 2 (γkA − εkI)2 − 2I=(4γkγk+1 − 2γ2k)A2 +(4εkγk − 4εkγk+1 − 4εk+1γk)A+(4εkεk+1 − 2ε2k − 2)I=(2εk − 4εk+1)γkA +(4εkεk+1 − 2ε2k − 2)I= −2γkA + (2εk − 2) I= −2 (γkA − εkI + I) ,5where we used equation (9) and (10). Substituting this in equation (15),gives:A2k−Ak+1= −2 (γkA − εkI − I)−2 (γkA − εkI + I) (γk+1A − εk+1I − I)−1= −AkCkCk+1. (16)Secondly, we remark that we can write A2mk − Amk+1 as a finite sum in thefollowing way,A2mk − Amk+1 =m−1∑j=0A2(m−1−j)k[A2k − Ak+1]Ajk+1. (17)Now using equation (16) and equation (17), we get the following expression:‖A2mk −Amk+1‖= ‖m−1∑j=0A2(m−1−j)k[A2k − Ak+1]Ajk+1‖= ‖m−1∑j=0A2(m−j)−1k Ck Ck+1Ajk+1‖= sup‖x‖=1‖m−1∑j=0A2(m−j)−1k Ck Ck+1Ajk+1x‖= sup‖x‖,‖y‖=1|〈y,m−1∑j=0A2(m−j)−1k Ck Ck+1Ajk+1x〉|≤ sup‖x‖,‖y‖=1m−1∑j=0|〈y,A2(m−j)−1k Ck Ck+1Ajk+1x〉|= sup‖x‖,‖y‖=1m−1∑j=0|〈C∗kA∗2(m−j)−1k y,Ck+1Ajk+1x〉|6≤ sup‖x‖,‖y‖=1∞∑j=0|〈C∗kA∗2(m−j)−1k y,Ck+1Ajk+1x〉|≤ sup‖x‖,‖y‖=1∞∑j=0‖C∗kA∗2(m−j)−1k y‖2‖y‖212∞∑j=0‖Ck+1Ajk+1x‖2‖x‖212≤ M22√γkω + εk√γk+1ω + εk+1.In the penultimate step we used the Cauchy-Schwarz inequality on ℓ2(X).In the last step we used Lemma 2.1 and Remark 2.2.If nk is even, Lemma 3.4 gives an estimate for two successive operators.For an odd power nk, we need an extra step.Lemma 3.5. Let Ak be defined by Definition 3.1, then the following estimateholds:‖An+1k − Ank‖ ≤M√γkω + εk. (18)Proof. Using Lemma 2.1, gives:‖An+1k − Ank‖ = ‖ (Ak − I)Ank‖ = ‖(I +√2Ck − I)Ank‖=√2‖CkAnk‖ = supx 6=0√2‖CkAnkx‖‖x‖≤ supx 6=0(2∞∑n=0‖CkAnkx‖2‖x‖−2) 12≤ M√γkω + εk.The difference between to successive operators can be estimated as fol-lows:Lemma 3.6. Let Ak and Ak+1 be defined by Definition 3.1, then the fol-lowing estimate holds:‖Ankk − Ank+1k+1 ‖ ≤M22√γkω + εk√γk+1ω + εk+1+M√γkω + εk. (19)7Proof. In case nj is even, Lemma 3.4 implies equation (19). In case nj isodd, we combine Lemma 3.4 and Lemma 3.5:‖Ankk − Ank−12k+1 ‖ = ‖Ankk − Ank−1k + Ank−1k − Ank−12k+1 ‖≤ ‖Ankk − Ank−1k ‖ + ‖Ank−1k − Ank−12k+1 ‖≤ M22√γkω + εk√γk+1ω + εk+1+M√γkω + εk.Now we have the tools to prove the main result.4 Proof of the main resultIn this section we proof Theorem 1.1. For this we use Lemma 3.3 and Lemma3.6.Proof of the main result. We write And as a sum of operators of equation(11):‖And‖ = ‖An00 ‖ = ‖AN + (An00 − An11 ) +N−1∑j=1(Anjj − Anj+1j+1 )‖≤ ‖AN‖ + ‖An00 − An11 ‖ +N−1∑j=1‖Anjj − Anj+1j+1 ‖.By Lemma 3.3, we have an estimate for the norm of AN .‖AN‖ ≤ 1 + 2M.We also recall Lemma 3.6:‖Ankk − Ank+1k+1 ‖ ≤M22√γkω + εk√γk+1ω + εk+1+M√γkω + εk.From equation (9), (10) and (11), we know that N = ⌊2log n⌋,√γ0ω + ε0 =√ω and (γkω+εk)− 12 ≤√2 for k ≥ 1. Combining these, we find an estimatefor every operator in the sequence.‖And‖ ≤ ‖AN‖ + ‖An00 − An11 ‖ +N−1∑j=1‖Anjj − Anj+1j+1 ‖≤ 1 + 2M +(M√2+ 1)M√ω+ (2log n − 1)(M2 +√2M).8We remark that for this estimate ω has to be positive. This means thatthe semigroup has a negative growth bound, and the estimate does not holdfor bounded semigroups. Note that Gomilko proved ‖Anb ‖ ≤ M1 log(n + 1)for all bounded semigroups. The question whether with our techniques theestimate can be improved, is still open. Whether the logarithmic growth of‖And‖ is the worst growth, is another open problem.References[1] P. Brenner and V. Thomée, On rational approximations of groups ofoperators, SIAM Journal on Numerical Analysis, 17, no. 1 (1980), 119-125.[2] R.F. Curtain and H.J. Zwart, An Introduction to Infinite-DimensionalLinear Systems Theory, Springer-Verlag, 1995.[3] J.L.M. van Dorsselaer, J.F.B.M. Kraaijevanger, and M.N. Spijker, Lin-ear stability analysis in the numerical solution of initial value problems,Acta Numerica, (1993), 199-237.[4] A.M. Gomilko, The Cayley transform of the generator of a uniformlybounded C0-semigroup of operators, Ukrainian Mathematical Jour-nal, 56, no. 8 (2004), 1018-1029 (in Russian). English translation inUkrainian Math. J., 56, no. 8 (2004), 1212-1226.[5] H.J. Zwart, Growth estimates for exp(A−1t) on a Hilbert space, Semi-group Forum, 74, no. 3 (2007), 487-494.9

The growth of a semigroup and its Cayley transform

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