Lectures on Operator Semigroups

Yuri G. Kondratiev

2017 2 Contents

1 Linear Dynamical Systems5

2 Semigroups, Generators, and Resolvents7 2.1 Spectral Theory for Closed Operators...... 7 2.2 Generators of Semigroups and Their Resolvents...... 9 2.3 Hille-Yosida Generation Theorems...... 16

3 4 Contents Chapter 1 Linear Dynamical Systems

5 6 Chapter 1. Linear Dynamical Systems Chapter 2 Semigroups, Generators, and Resolvents

2.1 Spectral Theory for Closed Operators

Let pA, DpAqq be a closed operator in a Banach space X. Deﬁnition 2.1. We call

ρpAq :“ tλ P C | λ ´ A : DpAq Ñ X is bijectiveu the resolvent set and its complement σpAq :“ CzρpAq the spectrum of A. For λ P ρpAq, the inverse

Rpλ, Aq “ pλ ´ Aq´1 is, by the closed graph theorem, a bounded operator on X and will be called the resolvent. It follows from the deﬁnition

ARpλ, Aq “ λRpλ, Aq ´ I holds for every λ P ρpAq. Proposition 2.2 (Resolvent Equation=RE). For λ, µ P ρpAq, one has

Rpλ, Aq ´ Rpµ, Aq “ pµ ´ λqRpλ, AqRpµ, Aq. (2.1) Proof. The deﬁnition of the resolvent implies

rλRpλ, Aq ´ ARpλ, AqsRpµ, Aq “ Rpµ, Aq Rpλ, AqrµRpµ, Aq ´ ARpµ, Aqs “ Rpµ, Aq. If we subtract these equations and use the fact that Rpλ, Aq and Rpµ, Aq commute, we obtain (2.1).

7 8 Chapter 2. Semigroups, Generators, and Resolvents

The basic properties of the resolvent set and the resolvent map are now collected in the following proposition.

Proposition 2.3. For a closed operator A : DpAq Ñ X, the following properties hold. (i) The resolvent set ρpAq is open in C, and for µ P ρpAq one has

8 Rpλ, Aq “ pµ ´ λqnRn`1pµ, Aq n“0 ÿ for all λ P C satisfying |µ ´ λ| ă 1{}Rpµ, Aq}. The resolvent map is locally analytic with dn Rpλ, Aq “ p´1qnn!Rn`1pλ, Aq, n P . dλn N

(iii) Let λn P ρpAq with limnÑ8 λn “ λ0. Then λ0 P σpAq iﬀ

lim }Rpλ ,Aq} “ 8. nÑ8 n

Proof. (i) For λ P C write λ ´ A “ µ ´ A ` pλ ´ µq “ rpI ´ pµ ´ λqRpµ, Aqspµ ´ Aq.

This operator is bijective if rI ´ pµ ´ λqRpµ, Aqs is invertible, which is the case for |µ´λ|}Rpµ, Aq} ă 1. The inverse is then obtained as (von Neumann series!)

8 Rpλ, Aq “ Rpµ, AqrI ´ pµ ´ λqRpµ, Aqs´1 “ pµ ´ λqnRn`1pµ, Aq. n“0 ÿ Assertion piiq follows immediately from the series representation for the resolvent. To show piiiq we use piq, which implies 1 }Rpµ, Aq} ě ě 1 dist pµ, σpAq for all µ P ρpAq. This already proves one implication. For the converse, assume that λ0 P ρpAq. Then the continuous resolvent map remains bounded on the compact set tλn | n ě 0u. This contradicts the assumption; hence λ0 P σpAq. As an immediate consequence, we have that the spectrum σpAq is a closed subset of C. Nothing more can be said in general. However, if A is bounded, it follows that σpAq Ă tλ P C| | |λ| ď }A}u, 2.2. Generators of Semigroups and Their Resolvents 9 since 1 1 8 An Rpλ, Aq “ p1 ´ Aq´1 “ λ λ λn`1 n“0 ÿ exists for all |λ| ą }A}. In addition, an application of Liouville’s theorem to the resolvent map implies σpAq ‰ H.

Corollary 2.4. For a bounded operator A on a Banach space X, the spectrum σpAq is always compact and nonempty; hence its spectral radius

rpAq :“ supt|λ| | λ P σpAqu is ﬁnite and satisﬁes rpAq ď }A}.

2.2 Generators of Semigroups and Their Re- solvents

We recall that for a one-parameter semigroup a Banach space X uniform continuity implies diﬀerentiability of the map t Ñ T ptq P LpXq. The right derivative tA of T at t “ 0 then yields a bounded operator A with T ptq “ e , t P R`. We may hope that strong continuity of a semigroup might still imply some diﬀerentiability of the orbit maps

ξx : t Ñ T ptqx P X.

We first show that differentiability of ξx is already implied by right differentiability at t “ 0.

Lemma 2.5. Take a strongly continuous semigroup T ptq, t ě 0 and an element x P X. For the orbit map ξx : t Ñ T ptqx, the following properties are equivalent. paq ξxp¨q is diﬀerentiable on R`. pbq ξxp¨q is right diﬀerentiable at t “ 0.

Proof. We have only to show that pbq implies paq. For h ą 0, one has 1 1 lim pT pt ` hqxT ptqxq “ T ptq lim pT phqx ´ xq “ hÑ0`0 h hÑ0`0 h

1 T ptqξxp0q and hence ξx is right diﬀerentiable on R`. On the other hand, for ´t ď h ă 0, we write

1 1 1 1 pT pt ` hqx ´ T ptqxqT ptqξ p0q “ T pt ` hqp px ´ T p´hqxq ´ ξ p0qq h x h x 10 Chapter 2. Semigroups, Generators, and Resolvents

1 1 `T pt ` hqξxp0q ´ T ptqξxp0q. As h Ñ 0 ` 0, the ﬁrst term on the right-hand side converges to zero, since }T pt`hq} remains bounded. The remaining part converges to zero by the strong continuity of T ptq. Hence, ξx is also left diﬀerentiable, and its derivative is

1 1 ξxptq “ T ptqξxp0q, t ě 0. (2.2)

On the subspace of X consisting of all those x P X for which the orbit maps ξx are differentiable, the right derivative at t “ 0 then yields an operator A from which, in a sense to be specified later, we can hope to obtain the operators T ptq as the ”exponentials etA”. Definition 2.6. The generator A : DpAq Ă X Ñ X of a strongly continuous semigroup T ptq on a Banach space X is the operator

1 1 Ax :“ ξxp0q “ lim pT phqx ´ xq hÑ0`0 h deﬁned for every x in its domain

DpAq :“ tx P X | ξx is diﬀerentiableu. We observe that 1 DpAq “ tx P X | D lim pT phqx ´ xq. hÑ0`0 h Lemma 2.7. For the generator pA, DpAqq of a strongly continuous semigroup T ptq, t ě 0, the following properties hold. piq A : DpAq Ă X Ñ X is a linear operator. piiq If x P DpAq, then T ptqx P DpAq and d T ptqx “ T ptqAx “ AT ptqx dt for all t ě 0. piiiq For every t ě 0 and x P X, one has t T psqxds P DpAq. ż0 pivq For every t ě 0, one has t T ptqx ´ x “ A T psqxds, if x P X ż0 t “ T psqAxds, if x P DpAq. ż0 2.2. Generators of Semigroups and Their Resolvents 11

Proof. Assertion piq is trivial. To prove piiq take x P DpAq. Then it follows from (2.2) that 1 pT pt ` hqx ´ T ptqxq h converges to T ptqAx as h Ñ 0 ` 0. Therefore, 1 lim pT phqT ptqx ´ T ptqxq hÑ0`0 h exists, and hence T ptqx P DpAq with AT ptqx “ T ptqAx. The proof of assertion piiiq is included in the following proof of pivq. For x P X and t ě 0, one has

1 t t pT phq T psqxds ´ T psqxdsq “ h ż0 ż0 1 t 1 s “ T ps ` hqxds ´ T psqxds h h ż0 ż0 1 t`h 1 s “ T psqxds ´ T psqxds h h żh ż0 1 t`h 1 h “ T psqxds ´ T psqxds h h żt ż0 which converges to T ptqx ´ x as h Ñ 0 ` 0. Hence

t T ptqx ´ x “ A T psqxds, if x P X ż0 holds. T phqx´x If x P DpAq), then the functions s Ñ T psq h converge uniformly on h P r0.ts to the function s Ñ T psqAx as h Ñ 0 ` 0. Therefore,

1 t lim pT ptq ´ Iq T psqxds hÑ0`0 h ż0 t T phq ´ I “ lim T psq xds hÑ0`0 h ż0 t “ T psqAxds. ż0

Theorem 2.8. The generator of a strongly continuous semigroup is a closed and densely deﬁned linear operator that determines the semigroup uniquely. 12 Chapter 2. Semigroups, Generators, and Resolvents

Proof. Let T ptq, t ě 0 be a strongly continuous semigroup on a Banach space X. As already noted, its generator pA, DpAqq is a linear operator. To show that A is closed, consider a sequence pxnqnPN Ă DpAq for which limnÑinfty xn “ x and limnÑ8 Axn “ y exists. By the previous lemma, we have t T ptqxn ´ xn “ T psqAxnds ż0 for t ą 0.The uniform convergence of T p¨qAxn on r0, ts for n Ñ 8 implies that t T ptqx ´ x “ T psqyds. ż0 Multiplying both sides by 1{t and taking the limit as t Ñ 0 ` 0, we see that x P DpAq and Ax “ y, i.e., A is closed. By piiiq the elements 1 t T psqds t ż0 always belong to DpAq. Since the strong continuity of T ptq implies 1 t lim T psqxds “ x tÑ0`0 t ż0 for every x P X, we conclude that DpAq is dense in X. Finally, let Sptq be another strongly continuous semigroup having the same generator pA.DpAqq. For x P DpAq and t ą 0, we consider the map

s Ñ ηxpsq :“ T pt ´ sqSpsqx for s P r0, ts. Since for ﬁxed s the set Sps ` hqx ´ Spsqx t | h P p0, 1su Y tASpsqxu h is compact, the diﬀerence quotients 1 pη ps ` hq ´ η psqq h x x 1 1 “ T pt ´ s ´ hq pSps ` hqx ´ Spsqxq ` pT pt ´ s ´ hq ´ T pt ´ sqqSpsqx h h converge by piiq to d η psq “ T pt ´ sqASpsqx ´ AT pt ´ sqSpsqx “ 0. ds x

From ηxp0q “ T ptqx and ηxptq “ Sptqxwe obtain T ptqx “ Sptqx for all x in the dense domain DpAq. Hence, T ptq “ Sptq for each t ě 0. 2.2. Generators of Semigroups and Their Resolvents 13

Deﬁnition 2.9. A subspace D of the domain DpAq) of a linear operator A : DpAq Ñ X is called a core for A if D is dense in DpAq for the graph norm

}x}A :“}x} ` }Ax}.

We now state a useful criterion for subspaces to be a core for the generator.

Proposition 2.10. Let pA, DpAqq be the generator of a strongly continuous semigroup T ptq on a Banach space X. A subspace D Ă DpAq that is } ¨ }-dense in X and invariant under the semigroup pT ptqqtě0 is always a core for A.

Proof. For every x P DpAq we can ﬁnd a sequence pxnqnPN Ă D such that limnÑ8 “ x. Since for each n the map s Ñ T psqxn P D is continuous for the graph norm } ¨ }A (use T ptqAx “ AT ptqxq, it follows that t T psqxnds, ż0 being a Riemann integral, belongs to the } ¨ }A-closure of D. Similarly, the } ¨ }A-continuity of s Ñ T psqx for x P DpAq implies that

1 t } T psqxds ´ x} Ñ 0 t A ż0 as t Ñ 0 ` 0 and 1 t 1 t } T psqx ds ´ T psqxds} Ñ 0 t n t A ż0 ż0 as n Ñ 8 and for each t ą 0. This proves that for every ą 0 we can ﬁnd t ą 0 and n P N such that 1 t } T psqx ds ´ x} ă . t n A ż0 Hence, x P D¯ }¨}A .

Lemma 2.11. Let pA, DpAqq be the generator of a strongly continuous semigroup T ptq. Then, for every λ P C and t ą 0, the following identities hold:

t e´λtT ptqx ´ x “ pA ´ λq eλtT psqxds if x P X, ż0 t “ eλtT psqpA ´ λqxds if x P DpAq. ż0 14 Chapter 2. Semigroups, Generators, and Resolvents

Proof. It suﬃces to apply pivq to the rescaled semigroup Sptq :“ e´λtT ptq, t ě 0, whose generator is B “ A ´ λ with domain DpBq “ DpAq. Next, we give an important formula relating the semigroup to the resolvent of its generator. Theorem 2.12. Let T ptq be a strongly continuous semigroup on the Banach space X and take constants ω P R,M ě 1 such that }T ptq} ď Meωt, t ě 0.

For the generator pA, DpAq of T ptq the following properties hold. piq If λ P C 8 Rpλqx :“ e´sT psqxds ż0 exists for all x P X, then λ P ρpAq and Rpλq “ Rpλ, Aq. piiq If <λ ą ω, then λ P ρpAq, and the resolvent is given by the integral expression in piq. piiiq }Rpλ, Aq} ď M

8 Rpλ, Aq “ eλsT psqds. ż0 Proof. piq By a simple rescaling argument we may assume that λ “ 0. Then, for arbitrary x P X and h ą 0 we have T phq ´ I T phq ´ I 8 Rp0qx “ T psqxds h h ż0 1 8 1 8 “ T ps ` hqxds ´ T psqxds h h ż0 ż0 1 8 1 8 “ T psqxds ´ T psqxds h h żh ż0 1 h “ ´ T psqxds. h ż0 2.2. Generators of Semigroups and Their Resolvents 15

By taking the limit as h Ñ 0 ` 0, we conclude that RanpRp0qq Ă DpAq and ARp0q “ ´I. On the other hand, for x P DpAq we have

t lim T psqxds “ Rp0qx, tÑ8 ż0 t t lim A T psqxds “ lim T psqAxds “ Rp0qAx, tÑ8 tÑ8 ż0 ż0 where we have used lemma (iv) for the second equality. Since A, this implies Rp0qAx “ ARp0qx “ ´x and therefore Rp0q “ p´Aq´1 as claimed. Parts piiq and piiiq then follow easily from piq and the estimate

t t } e´λsT psqds} ď M epω´<λqsds, ż0 ż0 since for <λ ą ω the right-hand side converges to M <λ ´ ω as t Ñ 8.

Corollary 2.13. For the generator pA, DpAqq of a strongly continuous semigroup T ptq satisfying }T ptq} ď Meωt, t ě 0, one has, for <λ ą ω and n P N, that p´1qn´1 dn´1 Rnpλ, Aqx “ Rpλ, Aqx (2.3) pn ´ 1q! dλn´1

1 8 “ sn´1e´λsT psqxds (2.4) pn ´ 1q! ż0 for all x P X. In particular, M }Rnpλ, Aq} ď (2.5) p<λ ´ ωqn

for all n P N and <λ ą ω. Proof. Equation (2.3) is actually valid for every operator with nonempty resolvent set. On the other hand, by Theorem 2.12 (i), one has d d 8 Rpλ, Aqx “ e´λsT psqxds λ. dλ 0 ż 16 Chapter 2. Semigroups, Generators, and Resolvents

8 “ ´ se´λsT psqxds ż0 for <λ ą ω and x P X. Proceeding by induction, we deduce (2.4). Finally, the estimate (2.5) follows from

1 8 }Rnpλ, Aqx} ď } sn´1e´λsT psqxds} pn ´ 1q! ż0 1 8 ď }x}M sn´1epω´<λqsds pn ´ 1q! ż0 M “ }x}. p<λ ´ ωqn

Property piiq in Theorem 2.12 says that the spectrum of a semigroup generator is always contained in a left half-plane. The number determining the smallest such half-plane is an important characteristic of any linear operator and is deﬁned as follows.

Deﬁnition 2.14. To any linear operator A we associate its spectral bound deﬁned by spAq :“ supt<λ | λ P σpAqu. As an immediate consequence of Theorem 2.12 piiq the following relation holds between the growth bound of a strongly continuous semigroup and the spectral bound of its generator.

Corollary 2.15. For a strongly continuous semigroup T ptq with generator A, one has ´8 ď spAq ď ω0 ă `8.

2.3 Hille-Yosida Generation Theorems

We now turn to the fundamental problem of semigroup theory, which is to ﬁnd ways leading from the generator (or its resolvent) to the semigroup. More precisely, this means that we will discuss the following problem. Problem. Characterize those linear operators that are the generator of some strongly continuous semigroup. We already saw that generators

• are necessarily closed operators,

• have dense domain, and 2.3. Hille-Yosida Generation Theorems 17

• have their spectrum contained in some proper left half-plane. These conditions, however, are not sufficient. Lemma 2.16. Let pA, DpAqq be a closed, densely defined operator. Suppose there exist ω P R and M ą 0 such that rω, 8q Ă ρpAq and }λRpλ, Aq} ď M for all λ ě ω. Then the following convergence statements hold for λ Ñ 8: (i) λRpλ, Aqx Ñ x for all x P X. (ii) λARpλ, Aqx “ λRpλ, AqAx Ñ Ax for all x P DpAq. Proof. If y P DpAq, then λRpλ, Aqy “ Rpλ, AqAy`y. This expression converges to y as λ Ñ 8, since M }Rpλ, AqAy} ď }Ay}. λ Since }λRpλ, Aq} is uniformly bounded for all λ ě ω, statement (i) follows. The second statement is then an immediate consequence of the first one. Since for contraction semigroups the technical details of the subsequent proof become much easier (and since the general case can then be deduced from this one), we first give the characterization theorem for generators in this special case. Theorem 2.17 (Contracting Case, Hille, Yosida, 1948). For a linear operator pA, DpAqq on a Banach space X, the following properties are all equivalent. (a) pA, DpAqq generates a strongly continuous contraction semigroup. (b) pA, DpAqq is closed, densely defined and for every λ ą 0 one has λ P ρpAq and }λRpλ, Rq} ď 1. (2.6) (c) pA, DpAqq is closed, densely defined, and for every λ P C with <λ ą 0 one has λ P ρpAq and 1 }Rpλ, Aq} ď . (2.7) <λ Proof. In view of Theorem 2.8 and Theorem 2.12, it suffices to show pbq Ñ paq. To that purpose, we define the so-called Yosida approximants

2 An :“ nARpn, Aq “ n Rpn, Aq ´ nI which are bounded operators for each n P N and commute with one another. Consider then the uniformly continuous semigroups given by

tAn Tnptq :“ e , t ě 0.

Since An converges to A pointwisely on DpAq (by Lemma 2.16 (ii)), we antici- pate that the following properties hold. (i) T ptqx :“ limnÑ8 Tnptqx exists for each x P X. 18 Chapter 2. Semigroups, Generators, and Resolvents

(ii) T ptq is a strongly continuous semigroup on X. (iii) This semigroup has generator pA, DpAqq. By establishing these statements we will complete the proof. (i) Each Tnptq is a contraction semigroup, since

ńt n2}Rpn,Aq}t ńt nt }Tnptq} ď e e ď e e “ 1 for t ě 0. So, it suffices to prove convergence just on DpAq. By (the vector-valued version of) the fundamental theorem of calculus, applied to the functions s Ñ Tmpt ´ sqTnpsqx for 0 ď s ď t, x P DpAq and n, m P N, and using the mutual commutativity of the semigroups Tnptq for all n P N, one has t d T ptqx ´ T ptqx “ pT pt ´ sqT psqxqds n m ds m n ż0 t “ Tmpt ´ sqTnpsqpAnx ´ Amxqds. ż0 Accordingly, }Tnptqx ´ Tmptqx} ď t}Anx ´ Amx}. (2.8)

By Lemma 2.16 (ii), pAnxqnPN is a Cauchy sequence for each x P DpAq. Therefore, Tnptqx converges uniformly on each interval r0, t0s. (ii) The pointwise convergence of Tnptqx, n P N implies that the limit family T ptq satisﬁes the functional equation (FE), hence is a semigroup, and consists of contractions. Moreover, for each x P DpAq, the corresponding orbit map

ξ : t Ñ T ptqx, t P r0, t0s, is the uniform limit of continuous functions and so is continuous itself. This suﬃces to obtain strong continuity. (iii) Denote by pB,DpBqq the generator of T ptq and ﬁx x P DpAq. On each compact interval r0, t0s, the functions

ξn : tnptqx converge uniformly to ξp¨q by (2.8), while the diﬀerentiated functions

1 ξn : t Ñ TnptqAnx converge uniformly to η : t Ñ T ptqAx. This implies diﬀerentiability of ξ with ξ1p0q “ ηp0q, i.e., DpAq Ă DpBq and Ax “ Bx for x P DpAq. 2.3. Hille-Yosida Generation Theorems 19

Now choose λ ą 0. Then λ ´ A is a bijection from DpAq onto X, since λ P ρpAq by assumption. On the other hand, B generates a contraction semigroup, and so λ P ρpBq by Theorem 2.12. Hence, λ ´ B is also a bijection from DpBq onto X. But we have seen that λ ´ B coincides with λ ´ A on DpAq. This is possible only if DpAq “ DpBq and A “ B. If a strongly continuous semigroup T ptq with generator A satisfies, for some ω P R, an estimate }T pt} ď eωt, (2.9) then we can apply the above characterization to the rescaled contraction semigroup given by Sptq :“ e´ωtT ptq, t ě 0. Since the generator of Sptq is B “ A ´ ω Theorem 2.6 takes the following form. Let ω P R. For a linear operator pA, DpAq on a Banach space X the following conditions are equivalent. paq pA, DpAqq generates a strongly continuous semigroup T ptq pbq pA, DpAqq is closed, densely defined, and for each λ ą ω one has λ P ρpAq and }pλ ´ ωqRpλ, Aq} ď 1. (2.10) pcq pA, DpAqq is closed, densely defined, and for each λ P C with <λ ą ω one has λ P ρpAq and 1 }Rpλ, Aq} ď . (2.11) <λ ´ ω Semigroups satisfying ( 2.9 ) are called quasi-contractive. It is now a pleasant surprise that the characterization of generators of arbitrary strongly continuous semigroups can be deduced from the above result for contraction semigroups. However, norm estimates for all powers of the resolvent are needed. Theorem 2.18 (General Case, Feller, Miyadera, Phillips, 1952). . Let pA, DpAqq be a linear operator on a Banach space X and let ω P R,M ě 1 be constants. Then the following properties are equivalent. paq pA, DpAqq generates a strongly continuous semigroup T ptq satisfying }T ptq} ď Meωt. (2.12) pbq pA, DpAq is closed, densely defined, and for every λ ą ω one has λ P ρpAq and n }rpλ ´ ωqRpλ, Aqs } ď M, n P N. (2.13) pcq pA, DpAqq is closed, densely defined, and for every λ P C with <λ ą ω one has λ P ρpAq and

n M }R pλ, Aq} ď , n P N. (2.14) p<λ ´ ωqn 20 Chapter 2. Semigroups, Generators, and Resolvents

Proof. The implication paq Ñ pcq has been proved in the corollary, while pcq Ñ pbq is trivial. To prove pbq Ñ paq we use, as for Corollary 2.7, the rescaling technique. So, without loss of generality, we assume that ω “ 0, i.e.,

}λnRnpλ, Aq ď M for all ą 0, n P N. For every µ ą 0, deﬁne a new norm on X by

n n }x}µ “ sup }µ R pµ, Aqx}. ně0 These norms have the following properties. piq }x} ď }x}µ ď M}x}, i.e., they are all equivalent to } ¨ }. piiq }µRpµ, Aq}µ ď 1. piiiq }λRpλ, Aq}µ ď 1, @λ P p0, µs. pivq n n n n }λ R pλ, Aq} ď }λ R pλ, Aq}µ ď }x}µ for all λ P p0, µs, n P N. pvq

}x}λ ď }x}µ for λ P p0, µs. We give the proof only of piiiq. Due to the Resolvent Equation, we have that y :“ Rpλ, Aqx “ Rpµ, Aqx ´ pµ ´ λqRpλ, AqRpµ, Aqx “ Rpµ, Aqpx ´ pµ ´ λqyq.

This implies, by using piiq, that 1 λ ´ µ }y} ď }x} ` }y} , µ µ µ µ µ whence λ}y}µ ď }x}µ. On the basis of these properties one can deﬁne still another norm by

|||x||| :“ sup µ ą 0}x}µ, (2.15) which evidently satisﬁes

pviq }x} ď |||x||| ď M}x} and pviiq |||λRpλ, Aq||| ď 1 2.3. Hille-Yosida Generation Theorems 21 for all λ ą 0. Thus, the operator pA, DpAqq satisﬁes condition (2.6) for the equivalent norm |||¨||| and so, by the Theorem 2.6, generates a |||¨|||-contraction semigroup T ptq, t ě 0. Using pviq again, we obtain }T ptq} ď M.