A refinement of Baillon's theorem on maximal regularity
aa r X i v : . [ m a t h . F A ] A ug A REFINEMENT OF BAILLON’S THEOREM ONMAXIMAL REGULARITY
BIRGIT JACOB, FELIX L. SCHWENNINGER, AND JENS WINTERMAYR
Abstract.
By Baillon’s result, it is known that maximal regularitywith respect to the space of continuous functions is rare; it implies thateither the involved semigroup generator is a bounded operator or theconsidered space contains c . We show that the latter alternative can beexcluded under a refined condition resembling maximal regularity withrespect to L ∞ . Introduction
The question whether the solutions to an abstract Cauchy problem dd t x ( t ) = Ax ( t ) + f ( t ) , t ∈ [0 , τ ] ,x (0) = 0 , (1.1)where A generates a strongly continuous semigroup S = ( S ( t )) t ≥ on aBanach space X , preserve the regularity of the inhomogeneity f : [0 , τ ] → X is omnipresent in the study of parabolic equations. They turn out tobe particularly useful for investigating nonlinear equations, see e.g. [1, 16,28, 32] and the references therein. More precisely, maximal regularity ofthe semigroup (or, equivalently, the generator) requires that ddt x and Ax have the same regularity as f , e.g. that ddt x and Ax are well-defined in L p ((0 , τ ) , X ) for any f ∈ L p ((0 , τ ) , X ) , τ > , where x : [0 , τ ] → X refersto the mild solution to (1.1). This property is equivalent to an inequality ofthe form (cid:13)(cid:13)(cid:13)(cid:13) A Z · S ( · − s ) f ( s )d s (cid:13)(cid:13)(cid:13)(cid:13) L p ((0 ,τ ) ,X ) κ τ k f k L p ((0 ,τ ) ,X ) , (1.2)for some constant κ τ > and all f ∈ L p ((0 , τ ) , X ) . The theory on maximalregularity has started with the works by Simon and Sobolevskii [15, 35], (BJ, JW) University of Wuppertal, School of Mathematics and NaturalSciences, IMACM, Gaußstraße 20, D-42119 Wuppertal, Germany. (FLS)
Department of Applied Mathematics, University of Twente,P.O. Box 217, 7500 AE Enschede, The Netherlands andDepartment of Mathematics, Center for Optimization and Approximation,University of Hamburg, Bundesstr. 55, 20146 Hamburg, Germany
E-mail addresses : [email protected],[email protected],[email protected] .2020 Mathematics Subject Classification.
Primary 47D06, 35K90; Secondary 47B37.
Key words and phrases.
Maximal regularity, Baillon’s theorem, strongly continuoussemigroup, admissible operator, uniformly continuous semigroup. who showed that analyticity of the semigroup is a sufficient condition onHilbert spaces when p ∈ (1 , ∞ ) . In fact, also on general Banach spaces, ana-lyticity is necessary for maximal regularity and the property is independentof the particular choice p ∈ (1 , ∞ ) , [10, 12, 17], see also [29] for the caseof continuous functions. However, this characterization fails to be true forgeneral non-Hilbert spaces, as was shown by Kalton–Lancien [26], see also[20, 21]. The appropriate replacement for UMD spaces was found by Weisto be the property that A is R -sectorial , [40].On the other hand, the cases p = 1 and p = ∞ are exotic in a way. In1980, Baillon [4] proved that maximal regularity with respect to the spaceof the continuous functions—i.e. replacing “ L p ((0 , τ ) , X ) ” by “ C([0 , τ ] , X ) ”in (1.2)—implies that A must be bounded when X does not contain anisomorphic copy of the sequence space c . A rather simple example, dueto T. Kato, of an unbounded operator A on c —which had been knownprior to Baillon’s work—shows that the latter assumption on X cannot bedropped in general. Note that a simplified proof of Baillon’s result can befound in [18] and that the case X = L had even been treated earlier in[13]. Moreover, it is not hard to see that in Baillon’s result “ C -maximalregularity” may be replaced by “ L ∞ -maximal regularity”, see also [22]. Thedual situation of L -maximal regularity was covered by Guerre-Delabrière[22]. In [37], see also the comments in [11], Travis (implicitly) showed that C -maximal regularity is equivalent to the property that S is of boundedsemivariation on some interval [0 , τ ] , i.e., var τ ( S ) := sup k x i k , t
In the following S = ( S ( t )) t ≥ al-ways refers to a strongly continuous semigroup on a Banach space X withgenerator A . In this paper all Banach spaces are assumed to be complex.For normed spaces X and Y , L ( X, Y ) denotes the space of bounded lin-ear operators from X to Y , with the convention L ( X ) = L ( X, X ) . Thedomain and range of a linear, possibly unbounded, operator B will be de-noted by D ( B ) and ran B respectively. Furthermore, let ρ ( B ) refer to theresolvent set of B and for λ ∈ ρ ( B ) we write R ( λ, B ) = ( λ − B ) − for theresolvent operator. We associate the following abstract Sobolev spaces withthe semigroup S ; the space X = ( D ( A ) , k · k D ( A ) ) , where k · k D ( A ) refersto the graph norm of A , and X − , which is the completion of X with re-spect to the norm k R ( λ, A ) · k for some fixed λ ∈ ρ ( A ) . It is well-knownthat S can be uniquely extended to a strongly continuous semigroup S − on X − whose generator A − extends A and has domain D ( A − ) = X . Foran interval I ⊆ R + := [0 , ∞ ) , p ∈ [1 , ∞ ] and some Banach space U , let L p ( I, U ) , Reg(
I, U ) and C( I, U ) refer to the spaces of Lebesgue-Bochner B. JACOB, F.L. SCHWENNINGER, AND J. WINTERMAYR p -integrable (equivalence classes of) functions, the regulated functions andthe continuous functions, f : I → U , respectively. We equip both C( I, U ) and Reg(
I, U ) with the supremum norm. Sometimes we will use the placeholder F in F( I, U ) to formulate a statement for either of the above spaces. Definition 1.1.
Let F be either L p , p ∈ [1 , ∞ ] or C or Reg . A stronglycontinuous semigroup S := ( S ( t )) t > (or its generator A ) is said to satisfy maximal regularity property with respect to F or F -maximal regularity , if forsome τ > and all f ∈ F([0 , τ ] , X ) , it holds that ( S ∗ f )( t ) ∈ D ( A ) foralmost every t ∈ (0 , τ ) and A ( S ∗ f ) ∈ F([0 , τ ] , X ) , where ( S ∗ f )( t ) := Z t S ( t − s ) f ( s )d s. (1.4)It is easy to see that S has the C -maximal-regularity property if andonly if S ∗ f ∈ C([0 , τ ] , X ) , see also [18]. Furthermore, whenever S has the C -maximal-regularity for some τ > , then this holds for every τ > .The following notions are central for this work. Definition 1.2.
Let
U, Y be Banach spaces and F be either L p , Reg or C .(1) An operator B ∈ L ( U, X − ) is called an F -admissible control opera-tor or F -admissible for S , if for some (hence all) τ > the mapping Φ τ : F([0 , τ ] , U ) → X − , u Z τ S − ( τ − t ) Bu ( t )d t has range in X , i.e. ran(Φ τ ) ⊂ X .(2) We call C ∈ L ( X , Y ) an F -admissible observation operator or F -admissible for S , if for some (hence all) τ > the operator Ψ τ : X → F([0 , τ ] , Y ) , x CS ( · ) x has a bounded extension to X , again denoted by Ψ τ .If lim sup τ → + k Φ τ k L (F([0 ,τ ] ,U ) ,X ) = 0 or lim sup τ → + k Ψ τ k L ( X, F([0 ,τ ] ,Y )) = 0 ,we say that B or C are zero-class F -admissible , respectively.The above notion of admissible operators, first coined by G. Weiss [41,42], plays an important role in the context of infinite-dimensional linearsystems theory, and particularly, in the context of boundary control andobservation, see also [36]. Note that by the Closed-Graph Theorem, B ∈L ( U, X − ) is F -admissible if and only if Φ τ is bounded from F([0 , τ ] , U ) to X , i.e. there exists K τ > such that (cid:13)(cid:13)(cid:13)(cid:13)Z τ S − ( τ − s ) Bu ( s )d s (cid:13)(cid:13)(cid:13)(cid:13) K τ k u k F([0 ,τ ] ,U ) , u ∈ F([0 , τ ] , U ) . This also shows that in the definition the norm k Φ τ k L (F([0 ,τ ] ,U ) ,X ) in the abovedefinition of zero-class admissibility is well-defined. Since for p ∈ (1 , ∞ ) , C( I, U ) ⊂ Reg(
I, U ) ⊂ L ∞ ( I, U ) ⊂ L p ( I, U ) ⊂ L ( I, U ) REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 5 for bounded intervals I ⊂ R + , with continuous embeddings, there is a natu-ral chain of implications for the property of admissible operators. In partic-ular, B being a C -admissible control operator is the weakest property in thethe scale of F -admissibility. Dually, any F -admissible observation operatoris L -admissible. In the following we will only be interested in the caseswhere B = A − is an admissible control operator or C = A is an admissibleobservation operator.The following result, which is a slight extension of [25, Proposition 16],marks the point of departure for Section 2. The proof follows the same linesas in the cited reference and is therefore omitted. Proposition 1.3.
Let A be the generator of a strongly continuous semigroup S . If either • A is a zero-class L -admissible observation operator, or • A − is a zero-class C -admissible control operator,then A is bounded. In Example 2.3 below we will show that the assumption of zero-class ad-missibility cannot be dropped in the above proposition (in either case). Weconclude this preparatory section with a result that gives an indication howthe seemingly strong condition of A − being an admissible control operatorrelates to admissibility of general control operators. Proposition 1.4.
Let S be a strongly continuous semigroup on a Banachspace X with generator A . Let F be a placeholder for either C , Reg or L ∞ .The following assertions are equivalent. (1) For every Banach space U , every operator B ∈ L ( U, X − ) is F -admissible. (2) A − is F -admissible.Proof: Without loss of generality assume that ∈ ρ ( A ) and fix t > .Note that the mapping f A − f ( · ) is an isomorphism from L ∞ ((0 , t ) , X ) to L ∞ ((0 , t ) , X − ) . Assume that A − is L ∞ -admissible and consider B ∈L ( U, X − ) for some Banach space U . Since for any u ∈ L ∞ ((0 , t ) , U ) it holdsthat Bu ( s ) = A − ˜ u ( s ) where g = A − − Bu ∈ L ∞ ((0 , t ) , X ) , we conclude that B is L ∞ -admissible. The converse is clear since A − ∈ L ( X, X − ) .Note that the variant of Proposition 1.4 for F = L p , p < ∞ , is triv-ial, since, by Hölder’s inequality, Proposition 1.3 implies that A − is L p -admissible if and only if A is bounded.2. Maximal regularity and admissible generators
The following two propositions show that maximal regularity and admis-sibility with respect to continuous, regulated functions and L -functions isclosely related. Proposition 2.1.
Let S be a strongly continuous semigroup on a Banachspace X with generator A . Then the following assertions hold: (1) S satisfies L -maximal-regularity if and only if A is L -admissible. B. JACOB, F.L. SCHWENNINGER, AND J. WINTERMAYR (2) A − is L p -admissible for some p ∈ (1 , ∞ ) if and only if A ∈ L ( X ) .Proof: The first assertion was proved in [27, Theorem 3.6] (without ex-plicitly using the notion of admissible operators) by rescaling the semigroup.Assertion (2) is immediate from Proposition 1.3 and Hölder’s inequality.Note that by Guerre-Delabrière’s result it holds that if S satisfies L -maximal-regularity and A is unbounded, then X must contain a comple-mented copy of ℓ , see [22] and [27]. Proposition 2.2.
Let S be a strongly continuous semigroup on a Banachspace X with generator A . Then the following assertions are equivalent: (1) S satisfies C -maximal-regularity, (2) A − is C -admissible, (3) S is of bounded semivariation, i.e. (1.3) holds for all τ > , (4) S satisfies Reg -maximal-regularity, (5) A − is Reg -admissible.Proof:
The equivalences (1) ⇔ (2) ⇔ (3) are shown in [37, Lemma 3.1 andProposition 3.1]. The implication (4) ⇒ (5) is easy to see from the definitionsand (5) ⇒ (4) is a consequence of [36, Theorem 4.3.1], which even impliesthat Ax is continuous for any f ∈ Reg([0 , τ ] , X ) , where x is the solution toequation (1.1). Since (5) trivially implies (2), it follows that (5) ⇒ (3).It thus remains to show that (3) ⇒ (5). Let f : [0 , τ ] → X , τ > , be anarbitrary regulated function and suppose that S is of bounded semivariationon the interval [0 , τ ] . For f there exists a sequence of step functions ( f n ) n ∈ N represented by f n ( s ) := n X i =1 f ( d ni ) χ ( d ni − ,d ni ) ( s ) , s ∈ [0 , τ ] , which converges uniformly to f and where d n < d n < · · · < d nn = τ .Define g n ( s ) = S ( τ − s ) f ( d ni ) for d ni − < s d ni , i = 1 , . . . , n , and g n (0) = S ( τ ) f (0) . Because f is bounded and S is strongly continuous, we have that ( g n ) n ∈ N is uniformly bounded and converges uniformly to s S ( τ − s ) f ( s ) for n → ∞ . Therefore, for lim n →∞ Z τ g n ( s )d s = Z τ S ( τ − s ) f ( s )d s. Because R τ g n ( s )d s ∈ D ( A ) we can calculate, A Z τ g n ( s )d s = A n X i =1 Z d ni d ni − g n ( s )d s = n X i =1 (cid:2) S ( τ − d ni ) − S ( τ − d ni − ) (cid:3) f n ( d ni )=: h n . Since S is of bounded semivariation, we have for n, m ∈ N that k h n − h m k X var τ ( S ) k f n − f m k ∞ , REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 7 where var τ ( S ) is defined as in (1.3). Since ( f n ) n ∈ N is a Cauchy sequence,it follows that ( h n ) n ∈ N is a Cauchy sequence of X and thus converges to alimit in X . Hence, as A is closed, Z τ S ( τ − s ) f ( s )d s ∈ D ( A ) , and therefore, Z τ S ( τ − s ) A − f ( s )d s = A Z τ S ( τ − s ) f ( s )d s ∈ X. The following example has been used in the context of C -maximal regu-larity several times, [4, 18], and seems—in the context of maximal regularity—to go back T. Kato . We use it to show that an analogous statement asProposition 2.2 for L ∞ does not hold. Example 2.3.
Let X = c ( N ) and Ax = P ∞ n =1 − nx n e n with D ( A ) = { x ∈ X : P ∞ n =1 − nx n e n ∈ c ( N ) } and where ( e n ) n ∈ N refers to the canonical basis.It is easy to see that ( A, D ( A )) generates an exponentially stable stronglycontinuous semigroup S = ( S ( t )) t > given by S ( t ) x = ∞ X n =1 e − nt x n e n (see e.g. [19, Example 4.7 iii), Chapter I]). Let now B = − A − ∈ L ( X, X − ) .Define u ∈ L ∞ ([0 , τ ] , c ( N )) by u ( s ) = P ∞ n =1 ( u ( s )) n e n , where ( u ( s )) n := ( if s ∈ [ τ − n , τ − n ] and n < τ, otherwise.The element f = R τ S − ( τ − s ) Bu ( s )d s = R τ S − ( s ) Bu ( τ − s )d s in X − canbe represented by a sequence ( f n ) n and we can calculate for all n ∈ N with n < τ , f n = Z τ e − ns n ( u ( τ − s )) n d s = Z n n ne − ns d s = − ( e − − e − ) > . This shows f / ∈ c ( N ) and therefore B is not L ∞ -admissible. On the otherhand, it is easy to see that S satisfies F -maximal-regularity for F = C , Reg and L ∞ , see e.g. [18], and thus B = − A − is Reg -admissible and hence C -admissible by Proposition 2.2. Since A is obviously unbounded, Proposition1.3 shows that B is not zero-class F -admissible with F equal to Reg or C . Remark 2.4.
It seems to have been unnoticed in the literature that the sim-ple Example 2.3 answers an old question posed by G. Weiss in [42, Remark6.10] about whether L -admissibility of B ∗ with respect to S ∗ implies that B is L ∞ -admissible for S for general B ∈ L ( U, X − ) and where B ∗ is to beunderstood as an admissible observation operator. In the setting of the ex-ample, the dual semigroup on X ∗ = ℓ ( N ) is given by T ∗ ( t ) x = P n e − nt x n e n see A. Pazy’s review of [4], MR0577152, on MathSciNet B. JACOB, F.L. SCHWENNINGER, AND J. WINTERMAYR for any x = P n x n e n and where ( e n ) n ∈ N (again) refers to the canonical ba-sis. It is easy to check that A ∗ = ( A − ) ∗ is an L -admissible observationoperator for S ∗ . On the other hand, B = A − is not L ∞ -admissible for S asshown above. Furthermore, the example shows that Reg -admissibility doesnot imply zero-class
Reg admissibility. Therefore, the assumption of “zeroclass admissibility” in Proposition 1.3 cannot be relaxed to plain “admis-sibility”. On the other hand, the existence of a
Reg -admissible operator B which is not zero-class Reg -admissible also establishes a counterexample inthe context of input-to-state stability for infinite-dimensional linear systems,see e.g. [33, 24]: By [24, Proposition 2.10], it shows that there exists a system ( A, B ) which is input-to-state stable, but not integral input-to-state stableboth with respect to respect to Reg -input functions.In order to proceed to our main result, we need to discuss Baillon’s resulton maximal regularity with respect to C in more detail. The following resultwas derived within the proof of Baillon’s theorem [4], see also [18]. Sincewe need it formulated explicitly, we sketch a short argument based on aclassical characterization of spaces containing c due to Bessaga–Pełcyński[6], see also [18]. Proposition 2.5 (Baillon’s theorem and Baillon spaces) . Let A generatea strongly continuous semigroup S on a Banach space X . If S satisfies C -maximal-regularity and A is unbounded, then X contains and isomorphiccopy of c . More precisely, there exists a sequence ( z n ) n ∈ N in X with thefollowing properties: (1) Z := span( z n ) n ∈ N is isomorphic to c , (2) ( z n ) n ∈ N is a Schauder basis of Z , (3) < inf n k z n k sup n k z n k < ∞ , (4) lim n →∞ R ( λ, A ) z n = 0 for any λ ∈ ρ ( A ) .We call such a space Z a Baillon space .Proof:
To show the existence of a sequence ( z n ) n ∈ N satisfying (2)–(3),it suffices to find a sequence ( x n ) n ∈ N and a constant M > such that inf n k x n k > and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X j =0 x n j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M, m ∈ N , for any increasing sequence ( n j ) j of positive integers, see [18, Theorem 0.1].Following [4] or [18]—using that A is unbounded and that S satisfies C -maximal-regularity—a possible choice is given by x n = t n AS ( t n ) y n withsuitably chosen sequences ( t n ) n ∈ N and ( y n ) n ∈ N with lim n →∞ t n = 0 and k y n k = 1 , n ∈ N . By [6, Cor. 1 and Lemma 3], see also [2, Lemma D.2 andTheorem D.3], the sequence ( z n ) n ∈ N can then be derived as a block basisfrom ( x n ) n ∈ N , i.e. there exists an increasing sequence of positive integers ( p k ) k ∈ N and a sequence of positive numbers ( s k ) k ∈ N such that z k = p k +1 X p k +1 s k x k , k ∈ N , REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 9 satisfies (2)–(3). By the proof of Lemma D.2 in [2], the sequence ( s k ) k ∈ N can be chosen to be bounded. To see (4), assume without loss of generalitythat λ = 0 ∈ ρ ( A ) . Then A − x n = t n S ( t n ) y n converges to as n → ∞ , andthus lim n →∞ A − z n = 0 .In particular, Baillon’s result shows that if A − is L ∞ -admissible and A isunbounded, then X must contain c and therefore, e.g., fails to be reflexive.This, however, does not exploit the difference between L ∞ -admissibility and C -admissibility. A step towards this is achieved in the following result. Theorem 2.6.
Let S be a strongly continuous semigroup on a Banach space X . If A − is L ∞ -admissible then F av ( S ) = D ( A ) , where F av ( S ) := { x ∈ X : lim sup t → + t k S ( t ) x − x k < ∞} refers to the Favard space of S .Proof: Without loss of generality we may assume that ∈ ρ ( A ) . Let x ∈ F av ( S ) . It is well-known, e.g. [38, Theorem 3.2.8], that there existsa bounded sequence ( y n ) n ∈ N in X such that lim n →∞ A − y n = x for some x ∈ X . Let t n be a strictly decreasing sequence of positive numbers with t = 1 and lim n →∞ t n = 0 . Define u : [0 , → X by u ( s ) = y n for s ∈ [1 − t n , − t n +1 ) . Clearly, u ∈ L ∞ ((0 , , X ) since ( y n ) is bounded. We have by assumptionthat Z S − (1 − s ) A − u ( s )d s = Z A − S − ( s ) u (1 − s )d s = ∞ X n =0 Z t n +1 t n A − S − ( s ) u (1 − s )d s = ∞ X n =0 Z t n t n +1 dd s S − ( s ) u (1 − s )d s = ∞ X n =0 ( S ( t n ) y n − S ( t n +1 ) y n ) , where the involved integrals and sums converge in X − with limit in X .Upon considering a subsequence, assume that P ∞ n =0 t − n k y n +1 − y n k − < ∞ .Without loss of generality we can set y = 0 . By using that the semi-group is in fact analytic, which follows from the assumption that A − is L ∞ -admissible, e.g. by [25, Proposition 9] and Proposition 1.4, k S ( t n +1 )( y n − y n +1 ) k k AS ( t n +1 ) kk y n − y n +1 k − Ct − n +1 k y n − y n +1 k − , and thus P ∞ n =0 S ( t n +1 )( y n − y n +1 ) converges in X absolutely. Combiningthis with the above shows that S ( t N ) y N = N − X n =0 S ( t n +1 ) y n +1 − S ( t n ) y n = N X n =0 ( S ( t n ) y n − S ( t n +1 ) y n ) + N X n =0 S ( t n +1 )( y n − y n +1 ) converges for N → ∞ in the X − -norm with a limit in X . Since k S ( t N ) y N − A − x k − k S ( t N ) y N − S ( t N ) A − x k − + k ( S ( t N ) − I ) A − x k − M k y N − A − x k − + k S ( t N ) A − x − A − x k − , we have that the X − -limit of S ( t N ) y N equals A − x which is the X − -limitof the sequence ( y n ) . Thus, A − x ∈ X , and therefore x = A − A − x ∈ D ( A ) .Hence, F av ( S ) = D ( A ) since the other inclusion holds trivially.Note that the assumption of L ∞ -admissibility in Theorem 2.6 cannot berelaxed to C -admissibility; see Example 2.3 where F av ( S ) = D ( A ) , whichcan be checked directly, or by Theorem 2.9 below. For what follows it willbe crucial to introduce the following norm on X , ||| x ||| = sup k x ⊙ k X ∗ |h x, x ⊙ i| , (2.1)which is known to be equivalent to the norm on X , see e.g. [38, p. 7].Furthermore, it is clear that the mapping j : X → X ⊙∗ , x ( x ⊙
7→ h x ⊙ , x i ) (2.2)is an isometry when X is equipped with k| · k| . Note that j is in general notisometric when the norm k · k is considered on X . However, if X ⊙ = X ∗ ,then j equals the canonical isometry from X in its bidual.By a result due to van Neerven, [39, Theorem 3.2], see also [38, Theorem3.2.9], the property that F av ( S ) = D ( A ) is equivalent to the condition thatthe set R ( λ, A ) K ( X, |||·||| ) is closed in X for some (hence all) λ ∈ ρ ( A ) , where K ( X, |||·||| ) refers to { x ∈ X : ||| x ||| } . We will employ this fact in the fol-lowing. We emphasize that the use of the ||| · ||| -norm is crucial here as thecorresponding statement involving the k · k -norm does not hold in general,see [38, Example 3.2.11].It is not hard to see that F av ( S ) = D ( A ) is satisfied for all semigroupswhenever X is reflexive, see e.g. [38] and [19, Corollary II.5.21]. This alsoshows that the converse of Theorem 2.6 is not true. Furthermore, the case X = c is special as F av ( S ) = D ( A ) implies that S is uniformly continuousthen, [38, Theorem 3.2.10]. The latter result rests on non-trivial fact of thegeometry on c , which, loosely speaking, guarantees that a given operator R : c → Y , with some arbitrary Banach space Y , is an isomorphism un-der comparably little information on R . To make this more explicit, let usintroduce the following notions, which will be used subsequently. Definition 2.7 (Semi-embeddings and G δ -embeddings) . Let X and Y beBanach spaces. An injective bounded linear operator R : X → Y is called REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 11 • a semi-embedding if R ( { x ∈ X : k x k } ) is closed in Y ; or • a G δ -embedding if R ( M ) is a G δ -set for any closed bounded set M in X .Semi-embeddings were first studied by Lotz, Peck and Porta in [31] andfurther investigated by Bourgain and Rosenthal in [9], who introduced thenotion of a G δ -embedding. The latter was partially motivated by the factthat the property of R being a semi-embedding is neither inherited by re-strictions to closed subspaces nor invariant under isomorphisms. We collectthe following facts for later reference. Lemma 2.8 (Bourgain–Rosenthal [9]) . Let X , Y be Banach spaces and R ∈ L ( X, Y ) . Then the following assertions hold. (1) If R is a semi-embedding and X is separable then R is a G δ -embedding. (2) If R is a G δ -embedding, then R | Z : Z → Y and RS are G δ -embeddings,for any closed subspace Z ⊂ X and any isomorphism S : W → X . (3) If X = c and R is a G δ -embedding, then R is bounded from below,i.e. there exists C > such that k Rx k > C k x k , x ∈ X. The proofs of (1) and (3) can be found in [9, Prop. 1.8 and Prop. 2.2].The other assertion is clear by definition.We are now able to prove our main result.
Theorem 2.9.
Let S be a strongly continuous semigroup on a Banach space X with generator A . Then the following assertions are equivalent (1) A − is L ∞ -admissible, (2) F av ( S ) = D ( A ) and S satisfies C -maximal-regularity, (3) A is a bounded operator.Proof: The implication (3) ⇒ (1) is clear as admissibility of A − = A istrivial when A is bounded. Furthermore, (1) ⇒ (2) follows by Theorem 2.6and since L ∞ -admissibility implies C -admissibility of A − which is equiva-lent to S satisfying C -maximal-regularity, Proposition 2.2.Hence, it remains to show (2) ⇒ (3). Suppose that F av ( S ) = D ( A ) and that S satisfies C -maximal-regularity. Suppose that A is unbounded. Thus wemay consider the Baillon space Z as given in Proposition 2.5. Let ˜ Z be thesmallest closed S -invariant subspace containing Z . Since ˜ Z = { S ( t ) z : z ∈ Z, t ≥ } , it is easy to see that ˜ Z is separable, since Z is separable. Since by assumption D ( A ) = F av ( S ) , we conclude that D ( A | ˜ Z ) = F av ( S | ˜ Z ) and thus, by [38,Theorem 3.2.9], that R ( λ, A | ˜ Z ) K is norm-closed in X , where K = { x ∈ ˜ Z : ||| x ||| } . Hence, R ( λ, A | ˜ Z ) is a semi-embedding on the separable space ˜ Z . Therefore, Lemma 2.8(1) yields that R ( λ, A ) | ˜ Z = R ( λ, A | ˜ Z ) : ˜ Z → X is a G δ -embedding from ( ˜ Z, ||| · ||| ) to X . Since the property of being a G δ -embedding is inherited by restrictions on closed subspaces and invariantunder isomorphisms, Lemma 2.8(2), we infer that also R ( λ, A ) | Z : Z → X isa G δ -embedding, where Z is equipped with the norm k·k , which is equivalent to ||| · ||| . Since Z is isomorphic to c , we conclude by Bourgain–Rosenthal,Lemma 2.8(3) and 2.8(2), that R ( λ, A ) | Z is bounded from below. Hence,there exists a constant C > such that k R ( λ, A ) x k > C k x k , x ∈ Z. This, however, contradicts the property of a Baillon space that R ( λ, A ) z n tends to as n → ∞ , Proposition 2.5(4), where ( z n ) n ∈ N is the sequencespanning Z , for which it holds that inf n k z n k > , Proposition 2.5(3).In the study of adjoint semigroups (on non-reflexive spaces), the no-tion of sun-reflexivity—linking the space and the semigroup—is classical.Recall that X is sun-reflexive (or “ ⊙ -reflexive”) for a given strongly con-tinuous semigroup S on X if the isometry j defined in (2.2) maps X onto X ⊙⊙ = ( X ⊙ ) ⊙ . Obviously, if X is reflexive, then X is sun-reflexive withrespect to any strongly continuous semigroup S on X . Also note that byde Pagter’s result [14], sun-reflexivity can be reformulated by the conditionthat the resolvent is weakly compact. In the following we show that if X is non-reflexive, but sun-reflexive with respect to S , then A − cannot be L ∞ -admissible. Corollary 2.10.
Let S be a strongly continuous semigroup such that X issun-reflexive. If A − is L ∞ -admissible, then A is bounded and X is reflexive.Proof: By Theorem 2.9, we only have to show that X is reflexive.This, however, is clear since S is uniformly continuous which implies that X ⊙⊙ = X ∗∗ and that j defined in (2.2) equals the canonical embedding of X in X ∗∗ .We point out that Corollary 2.10 can also be proved without referringto Theorem 2.9. Indeed, by Theorem 2.6, we know that F av ( S ) = D ( A ) .Since X is sun-reflexive, this fact together with [38, Theorem 6.2.14] impliesthat X must have the Radon-Nikodym property. Now A must be boundedbecause otherwise Baillon’s theorem shows that X contains c which con-tradicts that X has the Radon-Nikodym property. That X is reflexive nowfollows in the same way as in the other proof of the corollary.3. Comments on the discrete-time case
The study of discrete-time versions of maximal regularity was initiatedby Blunck [7, 8] and, with a focus on the extremal cases ℓ ∞ and ℓ , furtherstudied by Kalton and Portal in [34] and [27]. For a power-bounded operator T ∈ L ( X ) , we consider the solutions to x n = T x n − + Bu n , n ∈ N , (3.1) x = 0 , where ( u n ) n ∈ N is in ℓ ∞ ( N , X ) , the space of X -valued bounded sequences. Definition 3.1 (discrete-time admissibility) . Let T ∈ L ( X ) be power-bounded, and p ∈ [1 , ∞ ] . We say that B ∈ L ( U, X ) is an ℓ p -admissible REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 13 control operator for T if there exists κ > such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 T n − k Bu k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) κ k u k ℓ p ( { ,...,n } ,U ) , ∀ n ∈ N , u ∈ ℓ p ( N , U ) . Furthermore, C ∈ L ( X, Y ) is called an ℓ p -admissible observation operator for T if there exists ˜ κ > such that (cid:13)(cid:13)(cid:13)(cid:0) CT k x (cid:1) k ∈ N (cid:13)(cid:13)(cid:13) ℓ p ( N ,X ) ˜ κ k x k , x ∈ X. We will always explicitly state whether an operator is ℓ p -admissible asobservation or control operator in order to avoid confusion. This is contrastto the notation Section 2, where this was clear from the context. Remark 3.2. • Note that by a uniform boundedness principle argu-ment, if p ∈ [1 , ∞ ) , an operator B ∈ L ( U, X ) is an ℓ p -admissiblecontrol operator if and only if the limit lim n →∞ P nk =1 T n − k Bu k ex-ists and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) lim n →∞ n X k =1 T n − k Bu k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) κ k u k ℓ p , u ∈ ℓ p ( N , X ) . For p = ∞ , an analogous statement holds with ℓ ∞ replaced by c . • Comparing the definitions of discrete-time and continuous-time ad-missibility leads to the following observation. Whereas our defini-tion for continuous-time deals with fixed (and finite) times τ > ,the property in the discrete case is connected to uniform estimatesin n . We point out that, without loss of generality, we could haverestricted ourselves to infinite-time admissible operators in Section2 as well.On the other hand, the operator T is said to satisfy ℓ p -maximal regularity for p ∈ [1 , ∞ ] if for B = I the solution ( x n ) n ∈ N to (3.1) satisfies ( x n − x n − ) n ∈ N ∈ ℓ p ( N , X ) , ∀ ( u n ) n ∈ N ∈ ℓ p ( N , X ) . Note that this is equivalent to the existence of some constant κ > suchthat (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 T n − k ( I − T ) u k ! n ∈ N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ p ( N ,X ) κ k u k ℓ p ( N ,X ) , u ∈ ℓ p ( N , X ) . (3.2)The following result was implicitly shown in [27] without using the notionof admissibility. Proposition 3.3.
Let T ∈ L ( X ) be power-bounded. Then the followingassertions are equivalent. (1) T satisfies ℓ ∞ -maximal regularity, (2) T ∗ satisfies ℓ -maximal regularity, (3) I − T is an ℓ ∞ -admissible control operator, (4) I − T ∗ is an ℓ -admissible observation operator. Proof:
The equivalences (1) ⇔ (2) ⇔ (4) follow from [27, Proposition 2.3and Theorem 3.1]. The proof of (1) ⇔ (3) is clear from the definitions.The duality in Proposition 3.3 between maximal-regularity and admissi-bility with respect to ℓ and ℓ ∞ may come as a surprise when compared tothe continuous-time situation, where such a result does not hold. The rea-son for this rests on the fact that there is no difference between c -maximalregularity (replacing ℓ ∞ by c in (3.2)) and ℓ ∞ -maximal regularity.In [27, Theorem 3.5] Kalton and Portal show a version of Baillon’s resultfor discrete-time. More precisely, they prove that if X does not contain c and T satisfies ℓ ∞ -maximal regularity, then X = X + X for T -invariantclosed subspaces X and X such that T | X = I X and r ( T | X ) < , where r ( · ) refers to the spectral radius. The following example demonstrates thatthe assumption on X cannot be dropped and an analogous result as Theo-rem 2.9 does not hold in the discrete-time case. Example 3.4.
Let X = c ( N ) and let ( e n ) n ∈ N refer to the canonical basis.It is easy to see that the operator T y = P ∞ m =1 (1 − m ) y m e m , defined for y = P ∞ n =1 y n e n , satisfies ℓ ∞ -maximal regularity. Indeed, to see this, let x k = P ∞ m =1 x k,m e m ∈ X and consider, for n ∈ N , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 T n − k ( I − T ) x k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n X k =1 ∞ X m =1 (cid:0) − m (cid:1) n − k m x k,m e m (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = sup m m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 (cid:0) − m (cid:1) n − k x k,m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) sup k =1 ,..,n k x k k . On the other hand, it is obvious that there is no decomposition X into T -invariant closed subspaces X and X such that T | X = I and such that thespectral radius of T | X is less than .4. An alternative proof for Theorem 2.9
We sketch an alternative argument which shows Theorem 2.9 whichavoids the notion of G δ -embeddings and the fact that F av ( S ) = D ( A ) under the additional assumption that the dual semigroup S ∗ is stronglycontinuous. The key is the following lemma which can be proved by care-fully studying Baillon’s space. Lemma 4.1.
Let S be a strongly continuous semigroup with unbounded gen-erator A . Suppose that A − is L ∞ -admissible and let ( z n ) n ∈ N be the elementsspanning the Baillon space Z from Proposition 2.5. Then P ∞ n =1 z n convergesto an element z ∈ X in the norm k R ( λ, A ) · k , i.e. lim N →∞ R ( λ, A ) N X n =1 z n = R ( λ, A ) z (4.1) for λ ∈ R ( λ, A ) . REFINEMENT OF BAILLON’S THEOREM ON MAXIMAL REGULARITY 15
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