Strong continuity of semigroups of composition operators on Morre

Transcription

1 Semigroups on Strong continuity of semigroups of composition operators on Noel Merchán Universidad de Málaga, Spain Joint work with Petros Galanopoulos and Aristomenis G. Siskakis New Developments in Complex Analysis and Function Theory Heraklion, Greece. 2-6 July 2018

2 Index Notation and definitions Semigroups on 1 Notation and definitions 2 3 Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation 4 5 Semigroups on

3 Semigroups on Spaces of analytic functions in the unit disc D = {z C : z < 1}, the unit disc. Hol(D) is the space of all analytic functions in D. Automorphisms on D We consider It is known that Aut(D) = {ϕ : D D : ϕ is conformal}. Aut(D) = {λσ a : λ = 1, a D} where σ a : D D is the Möbius map σ a (z) = z a 1 az.

4 Semigroups on Hardy spaces If 0 < r < 1 and f Hol(D), we set ( 1 M p (r, f ) = 2π 2π M (r, f ) = sup f (z). z =r 0 f (re it ) p dt) 1/p, 0 < p <, If 0 < p, we consider the Hardy spaces H p, { } H p = f Hol(D) : f def H p = sup 0<r<1 M p (r, f ) <.

5 Semigroups on Bergman spaces If 0 < p <, we consider the Bergman spaces A p, { } A p = f Hol(D) : f (z) p da(z) <. D BMOA BMOA = { ( f H 1 : f e iθ) } BMO. H BMOA H p. 0<p<

6 Semigroups on Bloch space B = {f Hol(D) : sup(1 z 2 ) f (z) < }. z D H BMOA B.

7 Semigroups on For 0 < λ < 1 we define the Morrey space L 2,λ as { } L 2,λ = f H 2 : sup (1 a 2 ) 1 λ 2 f σ a f (a) H 2 < a D. We also define for 0 < λ < 1 the little L 2,λ 0 as { } L 2,λ 0 = f L 2,λ : lim (1 a 2 ) 1 λ 2 f σ a f (a) H 2 = 0. a 1 For 0 < λ < 1 BMOA = L 2,1 L 2,λ L 2,0 = H 2.

8 Semigroups on f (z) = z 2n B \ L 2,λ 0 < λ < 1. n=0 f (z) = (1 z) 1 λ 2 L 2,λ \ B 0 < λ < 1. Growth in For 0 < λ < 1 there exists a constant C such that if f L 2,λ then C f (z) z D. (1 z ) 1 λ 2 It follows that L 2,λ B 3 λ 2 0 < λ < 1.

9 Semigroups on α-bloch spaces If α > 0 we can consider the spaces B α = {f Hol(D) : sup(1 z 2 ) α f (z) < }. z D B = B 1 B α 1 B α 2, 1 α 1 α 2.

10 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Semigroups of analytic functions A semigroup (ϕ t ) for t 0 consists of analytic functions on D with ϕ t (D) D which satisfies the following: ϕ 0 is the identity in D. ϕ t+s = ϕ t ϕ s, for all t, s 0. ϕ t ϕ 0, as t 0, uniformly on compact subsets of D. Each semigroup (ϕ t ) gives rise to a semigroup (C t ) consisting on composition operators on Hol(D), C t (f ) = f ϕ t, f Hol(D).

11 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation We are going to be interested in the restriction of (C t ) to certain linear subspaces of Hol(D). Definition Given a Banach space X Hol(D) and a semigroup (ϕ t ), we say that (ϕ t ) generates a semigroup of operators on X if (C t ) is a well defined strongly continuous semigroup of bounded operators in X. This means that for every f X, we have C t (f ) X for all t 0 and lim t 0 + C t(f ) f X = 0.

12 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Definition For a semigroup (ϕ t ) we define the infinitesimal generator G of (ϕ t ) as ϕ t (z) z G(z) = lim, z D. t 0 + t This convergence holds uniformly on compact subsets of D so G Hol(D). Moreover G (ϕ t (z)) = ϕ t(z) t = G(z) ϕ t(z), z D, t 0. z

13 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Examples of semigroups Some examples of semigroups are: ϕ t (z) = z, t 0 G(z) = 0 (Trivial semigroup). ϕ t (z) = e t z, t 0 G(z) = z. ϕ t (z) = e it z, t 0 G(z) = iz.

14 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Representation of the infinitesimal generator G has a unique representation G(z) = (bz 1)(z b)p(z), z D, where b D and P Hol(D) with Re P(z) 0 for all z D. If G 0, (b, P) is uniquely determined from (ϕ t ). The point b is called Denjoy-Wolff point of the semigroup.

15 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Denjoy-Wolff point in the disc Studying the semigroup in the case b D can be reduced by renormalization to the case b = 0. Then ϕ t (z) = h 1 ( e ct h(z) ), where h : D h(d) = Ω is a univalent function with Ω a spirallike domain, h(0) = 0, Re c 0 and ωe ct Ω for each ω Ω, t 0.

16 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation Denjoy-Wolff point in the boundary If b D it may be reduced to b = 1. Then ϕ t (z) = h 1 (h(z) + ct), where h : D h(d) = Ω is a univalent function with Ω a close-to-convex domain, h(0) = 0, Re c 0 and ω + ct Ω for each ω Ω, t 0.

17 Semigroups on Strong continuity Infinitesimal generator Examples of semigroups Denjoy-Wolff point Motivation This connection between composition operators (C t ) and semigroups (ϕ t ) opens the possibility of studying properties of the semigroup of operators (C t ) in terms of the theory of functions.

18 Semigroups on Some results Every semigroup (ϕ t ) generates a semigroup of operators on the Hardy spaces H p (1 p < ), the Bergman spaces A p (1 p < ), the Dirichlet space, and on the spaces VMOA and little Bloch. No non-trivial semigroup generates a semigroup of operators in spaces H, BMOA, B or L 2,λ, 0 < λ < 1. There are plenty of semigroups (but not all) which generate semigroups of operators in the disc algebra A.

19 Semigroups on Theorem Let be 0 < λ < 1 and (ϕ t ) a semigroup of analytic functions. Then there exists a closed subspace Y L 2,λ such that (ϕ t ) generates a semigroup of operators on Y and such that any other subspace of L 2,λ with this property is contained in Y. We write that space Y as [ϕ t, L 2,λ ].

20 Semigroups on Theorem Let be 0 < λ < 1 and (ϕ t ) a semigroup of analytic functions. Let G the infinitesimal generator of (ϕ t ) then [ϕ t, L 2,λ ] = {f L 2,λ : Gf L 2,λ }.

21 Semigroups on Theorem For 0 < λ < 1, every semigroup (ϕ t ) generates a semigroup of operators on L 2,λ 0. L 2,λ 0 [ϕ t, L 2,λ ] L 2,λ 0 < λ < 1. The inclusion L 2,λ 0 [ϕ t, L 2,λ ] can be proper. For the semigroup ϕ t (z) = e t z + 1 e t, t 0, z D the function f (z) = (1 z) 1 λ 2 L 2,λ \ L 2,λ 0 satisfies f ϕ t f L 2,λ = e t 1 λ 2 (1 z) 1 λ 2 (1 z) 1 λ 2 L 2,λ ) = C (e t 1 λ

22 Semigroups on At this point it is natural to ask about conditions in (ϕ t ) such that L 2,λ 0 = [ϕ t, L 2,λ ] or L 2,λ = [ϕ t, L 2,λ ].

23 Semigroups on Conditions for L 2,λ 0 = [ϕ t, L 2,λ ] Theorem Let (ϕ t ) be a semigroup with infinitesimal generator G and 0 < λ < 1. Assume that for some 0 < α < 1/2, (1 z ) α G(z) = O(1) as z 1. Then L 2,λ 0 = [ϕ t, L 2,λ ].

24 Semigroups on We can prove that as a consequence of a stronger theorem. Theorem Let (ϕ t ) be a semigroup with infinitesimal generator G and 0 < λ < 1. Assume that 1 1 z lim da(z) = 0. I 0 I G(z) 2 Then L 2,λ 0 = [ϕ t, L 2,λ ]. S(I)

25 Semigroups on Theorem Let (ϕ t ) be a semigroup with infinitesimal generator G and Denjoy-Wolff point b D. If L 2,λ 0 = [ϕ t, L 2,λ ] then lim z 1 (1 z ) 3 λ 2 G(z) = 0.

26 Semigroups on Problem Are there semigroups such that L 2,λ = [ϕ t, L 2,λ ]? Theorem (BCD-MMPS) There are no non-trivial semigroups such that [ϕ t, B] = B. Proof: It is strongly used that every B α is a Grothendieck space with the Dunford-Pettis property. We do not know if L 2,λ, 0 < λ < 1 satisfy this property. BMOA does not have it.

27 Semigroups on This question has remained open for BMOA until Theorem (Anderson, Jovovic, Smith. 2017) Suppose H X B. Then there are no non-trivial semigroups such that [ϕ t, X] = X. Theorem Suppose 0 < λ < 1 and L 2,λ X B 3 λ 2. Then there are no non-trivial semigroups such that [ϕ t, X] = X. Corollary For 0 < λ 1 there are no non-trivial semigroups such that [ϕ t, L 2,λ ] = L 2,λ.

28 Semigroups on Proof Given any non-trivial semigroup (ϕ t ) and 0 < λ < 1 we just need a function f L 2,λ such that 1 lim inf f ϕ t f 3 λ. t 0 B 2 If the Denjoy-Wolff point of (ϕ t ) is b = 0 then ϕ t (z) = h 1 ( e ct h(z) ). When Re c = 0 the (ϕ t ) are rotations of the disc.

29 Semigroups on Proof The function f (z) = (1 z) 1 λ 2 L 2,λ and lim f (r) (1 r) 3 λ 2 = 1 λ > 0. r 1 2 lim f (re iθ ) (1 r) 3 λ 2 = 0 for θ 0. r 1 ( ) So if ϕ t (z) = ze iat for real a 0 for t 0, 2π a f ϕ t f 3 λ sup f (ϕ t (r))ϕ B 2 t(r) f (r) (1 r) 3 λ 2 1 λ 0<r<1 2.

30 Semigroups on Proof If Re c > 0, (ϕ t ) does not consist of automorphisms. Since Ω is spirallike about 0, we can choose ω 0 Ω such that ω 0 = inf{ ω : ω Ω}. Since h is univalent there is a γ 0 D such that lim r 1 h(rγ 0) exists and is equal to ω 0. Thus, lim ϕ t(rγ 0 ) = h 1 (e ct ω 0 ) D, t > 0. r 1 Since ϕ t U H D B 3 λ 2 0 t 0 we have lim r 1 ϕ t(rγ 0 ) (1 r) 3 λ 2 = 0.

31 Semigroups on Proof Letting f (z) = (1 γ 0 z) (1 λ)/2, we have Thus for all t 0 lim f (rγ 0 ) (1 r) 3 λ 2 = 1 λ > 0. r 1 2 f ϕ t f 3 λ lim sup f (ϕ t (rγ 0 ))ϕ B 2 t(rγ 0 ) f (rγ 0 ) (1 r) 3 λ 2 r 1 1 λ 2. If b = 1 we use also f (z) = (1 γ 0 z) (1 λ)/2.

32 Semigroups on THANK YOU! ευχαριστ ώ πoλύ