Dynamics of composition operators on spaces of real analytic functions

Dynamics of composition operators on spaces of real analytic functions

Jos´e Bonet

Instituto Universitario de Matem´atica Pura y Aplicada IUMPA Universidad Polit´ecnica de Valencia

Operator Theory and Related Topics, Lille (France), June 2010

Joint work with P. Doma´nski, Pozna´n (Poland)

Statement of the Problem

Problem

Study the behavior of iterates of a composition operator on the space of real analytic functionsA(Ω) .

We are mainly interested in the case when all the orbits are bounded.

Composition operator

C_ϕ(f) :=f ◦ϕ

ϕ:U→U holomorphic,U ⊂C^d open set Cϕ:H(U)→H(U) ϕ: Ω→Ω real analytic, Ω⊂R^d open set

C_ϕ:A(Ω) →A(Ω) Iterates:

(Cϕ)ⁿ:=Cϕ◦Cϕ◦ · · · ◦Cϕ

| {z }

n times

=Cϕⁿ

Universal functions for composition operators have been investigated by Bernal, Bonilla, Godefroy, Grosse-Erdmann, Le´on, Luh, Montes, Mortini, Shapiro and others.

Definitions

X is a Hausdorff locally convex space (lcs).

L(X)is the space of all continuous linear operators onX. Power bounded operators

An operatorT ∈ L(X) is said to bepower bounded if{T^m}^∞_m=1 is an equicontinuous subset ofL(X).

IfX is a Fr´echet space, an operatorT is power bounded if and only if the orbits{T^m(x)}^∞_m=1of all the elements x∈X underT are bounded.

This is a consequence of the uniform boundedness principle.

Hypercyclic operators

An operatorT ∈ L(X) is calledhypercyclic if there isx∈X such that its orbit is dense inX.

Definitions

Mean ergodic operators

An operatorT ∈ L(X) is said to bemean ergodic if the limits Px := lim

n→∞

1 n

n

X

m=1

T^mx, x∈X, (1)

exist inX.

A power bounded operatorT is mean ergodic precisely when

X =Ker(I−T)⊕Im(I−T), (2) whereI is the identity operator,Im(I−T) denotes the range of (I−T) and the bar denotes the “closure inX”.

Motivation

Trivial observation

λ∈C, |λ| ≤1.

λ_[n]:= 1

n(λ+λ²+...+λⁿ) λ= 1⇒λ_[n] = 1,n= 1,2,3, ...

λ6= 1⇒ |λ_[n]| ≤ 2

n|1−λ| →0 asn→ ∞.

Motivation

von Neumann, 1931

LetHbe a Hilbert space and letT ∈ L(H) be unitary. Then there is a projectionP onH such thatT_[n]:= ¹_nPn

m=1T^mconverges to Pin the strong operator topology.

Lorch, 1939

LetX be a reflexive Banach space. IfT ∈ L(X) is a power bounded operator, then there is a projectionP onX such thatT[n] converges toP in the strong operator topology; i.e.T is mean ergodic.

Hille, 1945

There exist mean ergodic operatorsT on L₁([0,1]) which are not power bounded.

Motivation

Problem attributed to Sucheston, 1976

LetX be a Banach space such that every power bounded operator T ∈ L(X) is mean ergodic. Does it follow thatX is reflexive?

YES if X is a Banach lattice (Emelyanov, 1997).

YES if X is a Banach space with basis (Fonf, Lin, Wojtaszczyk, J.

Funct. Anal. 2001). This was a major breakthrough.

There are non reflexive Banach spaces in which all contractions are mean ergodic (Fonf, Lin, Wojtaszczyk, Preprint 2009).

Yosida’s mean ergodic Theorem

Barrelled locally convex spaces Ls(X), Lb(X)

Yosida, 1960

LetX be a barrelled lcs. The operatorT ∈ L(X) is mean ergodic if and only if it lim_n→∞¹_nTⁿ= 0, inLs(X) and

{T_[n]x}^∞_n=1 is relativelyσ(X,X⁰)–compact,∀x∈X . (3) SettingP:=τs-lim_n→∞T_[n], the operatorP is a projection which commutes withT and satisfiesIm(P) =Ker(I−T) and

Ker(P) =Im(I−T).

Yosida’s mean ergodic Theorem

If{T_[n]}^∞_n=1 happens to be convergent in L_b(X), thenT is called uniformly mean ergodic.

Corollary

IfX is a reflexive lcs, then every power bounded operator onX is mean ergodic.

IfX is a Montel space, then every power bounded operator onX is uniformly mean ergodic.

Results of Albanese, Bonet, Ricker, 2008

Main Results

Let X be a complete barrelled lcs with a Schauder basis. ThenX is reflexive if and only if every power bounded operator onX is mean ergodic.

Let X be a complete barrelled lcs with a Schauder basis. ThenX is Montel if and only if every power bounded operator onX is uniformly mean ergodic.

Let X be a sequentially complete lcs which contains an isomorphic copy of the Banach space c₀. Then there exists a power bounded operator on X which is not mean ergodic.

Results of Albanese, Bonet, Ricker, 2008

Results about Schauder basis

A complete, barrelled lcs with a basis is reflexive if and only if every basis is shrinking if and only if every basis is boundedly complete.

This extends a result of Zippin for Banach spaces and answers positively a problem of Kalton from 1970.

Every non-reflexive Fr´echet space contains a non-reflexive, closed subspace with a basis.

This is an extension of a result of A. Pelczynski for Banach spaces.

Bonet, de Pagter and Ricker (Preprint 2010)

A Fr´echet latticeX is reflexive if and only if every power bounded operator onX is mean ergodic.

Composition operators on spaces of analytic functions

Theorem

LetU⊆C^d be a connected domain of holomorphy. The following assertions are equivalent:

(a) C_ϕ:H(U)→H(U) is power bounded;

(b) Cϕ:H(U)→H(U) is uniformly mean ergodic;

(c) Cϕ:H(U)→H(U) is mean ergodic;

(d) The map ϕhasstable orbitson U:

∀K bU ∃LbU such thatϕⁿ(K)⊆Lfor everyn∈N;

(e) There is a fundamental family of (connected) compact sets (Lj) inU such that ϕ(Lj)⊆Lj for everyj∈N.

Composition operators on spaces of analytic functions

The result is valid for Stein manifoldsU.

The equivalence (a)⇔(b)⇔(c) is true for arbitrary open connected U ⊆C^d.

IfU is hyperbolic, the conditions of the theorem are equivalent to the fact that all the orbits ofϕare compact.

IfU is complete hyperbolic, the conditions of the theorem are equivalent to the fact that there is a compact orbit ofϕ.

Composition operators on spaces of analytic functions

Theorem

LetU be a connected open subset ofCand letϕ:U→U be a holomorphic self-map. IfCϕis power bounded, then eitherϕis an automorphism ofU or all orbits ofϕtends to a constantu∈U such that uis a fixed point ofϕ.

If additionallyU=D, then in the automorphism caseϕhas also a fixed pointuand

ϕ=ϕ⁻¹_u ◦r_θ◦ϕ_u, θ∈[0,2), where

ϕu(z) := u−z

1−¯uz, rθ(z) =e^iθπz

Theorem continued Moreover,

(i) Ifϕis not an automorphism then the projectionP associated toC_ϕ is given: P(f)(z) =f(u).

(ii) Ifϕis an automorphism andθ=^p_q is rational then P(f)(z) = ¹_qPq−1

j=0 f(ϕ⁻¹_u (rjp q

(ϕ_u(z)))).

(iii) Ifϕis an automorphism andθis irrational then P(f)(z) = ¹₂R2

0f(ϕ⁻¹_u (rθ(ϕu(z))))dθ.

Composition operators on spaces of real analytic functions

Ω⊆R^d open connected set.

The space of real analytic functionsA(Ω) is equipped with the unique locally convex topology such that for any U⊆C^d open, R^d∩U = Ω, the restriction mapR:H(U)−→A(Ω) is continuous and for any compact setK ⊆Ω the restriction map

r :A(Ω) −→H(K) is continuous. In fact,

A(Ω) = proj_N∈N H(K_N) = proj_N∈N ind_n∈NH^∞(U_N,n).

A(Ω) is complete, separable, barrelled and Montel.

Doma´nski, Vogt, 2000. The spaceA(Ω) has no Schauder basis.

Composition operators on spaces of real analytic functions

Theorem

Letϕ: Ω→Ω, Ω⊆R^d open connected, be a real analytic map. Then the following assertions are equivalent:

(a) Cϕ:A(Ω) →A(Ω) is power bounded;

(b) Cϕ:A(Ω) →A(Ω) is uniformly mean ergodic;

(c) Cϕ:A(Ω) →A(Ω) is mean ergodic;

(d) ∀K bΩ ∃LbΩ ∀U complex neighbourhood ofL ∃V complex neighbourhood ofK:

∀n∈N ϕⁿ is defined onV andϕⁿ(V)⊆U;

(e) For every complex neighbourhoodU of Ω there is a complex (open!) neighbourhoodV ⊆U of Ω such thatϕextends as a holomorphic function toV andϕ(V)⊆V.

Composition operators on spaces of real analytic functions

Corollary

Letϕ: Ω→Ω be a real analytic map, Ω⊆R^d open connected set.

IfC_ϕ:A(Ω) →A(Ω) is power bounded, then for every complex neighbourhoodU of Ω there is a complex neighbourhoodV ⊆U of Ω and there is a fundamental system of compact sets (L_j) inV such that ϕ(L_j)⊆L_j.

In particular,ϕ(V)⊆V andC_ϕ:H(V)→H(V) is power bounded and (uniformly) mean ergodic.

Composition operators on spaces of real analytic functions

Corollary

Let Ω be a real analytic connected manifold and letϕ: Ω→Ω be a real analytic map. IfC_ϕ is hypercyclic, then

(1) ϕis injective, its derivativeϕ⁰(z) at arbitrary point z∈Ω is never a singular linear map and ϕruns away, i.e., for every compact set K ⊆Ω there isn∈Nsuch that ϕⁿ(K)∩K =∅.

(2) There is a complex neighbourhood U of Ω such that for every complex neighbourhoodV ⊆U of Ω the mapϕdoes not extend as a holomorphic self map on V.

Composition operators on spaces of real analytic functions

Example (Why does the complex case not solve the real analytic case?):

Even ifϕ:]−1,1[→]−1,1[ maps all bounded sets in ]−1,1[ in one compact set it does not follow thatC_ϕ:A(]−1,1[)→A(]−1,1[) is power bounded.

ϕ_α(z) := 2 α·πln

1−iz 1 +iz

.

ϕ_α maps the unit disc onto the vertical strip with Rez ∈ −_α¹,_α¹ . If ^π·α₄ <1, the zero point is an attractive fixed point on the imaginary line for ϕ⁻¹_α . This implies thatϕ⁻ⁿ_α (i)→0 asn→ ∞.

Since on this sequenceϕⁿ⁺¹_α is not defined, there is no common neighbourhood of zero such that all ϕⁿ_α are defined for alln∈N(α, α <1, ^π·α₄ <1). ThusCϕ:A(]−1,1[)→A(]−1,1[) is not power bounded.

Composition operators on spaces of real analytic functions

Theorem

Leta,b∈R¯ and letϕ:]a,b[→]a,b[ be real analytic. The following are equivalent:

(a) Cϕ:A(]a,b[)−→A(]a,b[) is power bounded;

(b) there exists a complex neighbourhood U of ]a,b[ such that ϕ(U)⊆U,C\U contains at least two points, and ϕhas a (real) fixed pointu, or equivalently, there is a fundamental family of such neighbourhoods of ]a,b[;

(c) ϕis one of the following forms:

1 ϕ= id ;

2 ϕ²= id ;

3 Asn→ ∞the sequenceϕⁿtends to a constant function≡u∈]a,b[

inA(]a,b[).

Composition operators on spaces of real analytic functions

Theorem continued

Ifu is the fixed point ofϕthen the above cases in (c) correspond to:

1 ϕ⁰(u) = 1;

2 ϕ⁰(u) =−1;

3 |ϕ⁰(u)|<1.

Moreover,C_ϕ is uniformly mean ergodic and the projection P:= lim_N∈NN¹ PN

n=1Cϕⁿ is of the following form:

1 P= id ;

2 P(f) =^f+f₂^◦ϕ, imP={f :f =f ◦ϕ}, kerP={f :f =−f ◦ϕ};

3 P(f) =f(u), imP= the set of constant functions, kerP={f :f(u) = 0}.

Composition operators on spaces of real analytic functions

Proposition

Letϕbe a self map on the unit disc Dwhich is holomorphic on a neighbourhood ofDand satisfiesϕ(]−1,1[)⊂]−1,1[.

Assume thatϕis injective and its derivative does not vanish on ]−1,1[.

Ifϕruns away on ]−1,1[, then Cϕ is hypercyclic onA(]−1,1[).

Summary

Characterization of holomorphic ϕsuch thatC_ϕpower bounded — form of projectionP. Characterization of hypercyclicity already known, at least for one complex variable, by the work of Bernal, Montes, Grosse-Erdamnn, Mortini and others.

Characterization of real analytic ϕsuch that C_ϕ power bounded — form of projectionP. Some consequences about hypercyclic composition operators.

The power bounded case is the only one where real analytic iterates of a selfmap can be fully understood through iterates of a

holomorphic selfmap.

References

1 A. A. Albanese, J. Bonet, W. J. Ricker, Mean Ergodic Operators in Fr´echet Spaces, Anal. Acad. Math. Sci. Fenn. Math. 34 (2009), 401-436.

2 J. Bonet, P. Doma´nski,Mean ergodic composition operators on spaces of holomorphic functions, RACSAM, to appear in 2011.

3 J. Bonet, P. Doma´nski,Power bounded composition operators on spaces of analytic functions, Preprint, 2010.