Linear Dynamical Properties of Weighted Backward Shifts on Spaces of - - PowerPoint PPT Presentation

Linear Dynamical Properties of Weighted Backward Shifts on Spaces of Real Analytic Functions

Can Deha Karıksız (joint work with late Pawe l Doma´ nski)

¨ Ozye˘ gin University, Istanbul deha.kariksiz@ozyegin.edu.tr

03.07.2018

Outline

1 Introduction 2 Conditions on linear dynamical properties using dynamical

transference principles

3 Conditions on linear dynamical properties via eigenvalues

Introduction

Linear Dynamical Properties

An operator T on a topological vector space X is called hypercyclic if there is some x ∈ X such that the set {x, Tx, T 2x, · · · , T nx, · · · }, called the orbit of x under T, is dense in X.

Introduction

Linear Dynamical Properties

An operator T on a topological vector space X is called hypercyclic if there is some x ∈ X such that the set {x, Tx, T 2x, · · · , T nx, · · · }, called the orbit of x under T, is dense in X. topologically transitive if for any pair of nonempty open subsets U, V of X, there exists some n ∈ N such that T n(U) ∩ V = ∅.

Introduction

Linear Dynamical Properties

An operator T on a topological vector space X is called mixing if for any pair of nonempty open subsets U, V of X, there exists some N ∈ N such that for every n ≥ N we have T n(U) ∩ V = ∅.

Introduction

Linear Dynamical Properties

An operator T on a topological vector space X is called mixing if for any pair of nonempty open subsets U, V of X, there exists some N ∈ N such that for every n ≥ N we have T n(U) ∩ V = ∅. chaotic if T is topologically transitive and has a dense set of periodic points.

Introduction

Weighted Backward Shifts on Fr´ echet Sequence Spaces

Let X be a Fr´ echet sequence space with canonical unit sequences

• en. For a sequence of nonzero scalars ω = (ωn)n∈N, the operator

Bω : X → X defined by Bωen = ωnen−1, n ≥ 1, e0 = 0, is called a weighted backward shift on X.

Introduction

Weighted Backward Shifts on Fr´ echet Sequence Spaces

Let X be a Fr´ echet sequence space with canonical unit sequences

• en. For a sequence of nonzero scalars ω = (ωn)n∈N, the operator

Bω : X → X defined by Bωen = ωnen−1, n ≥ 1, e0 = 0, is called a weighted backward shift on X. Theorem Let Bω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which (en)n∈N is a basis. (i) Bω is hypercyclic ⇔ there is an increasing sequence (nk)k∈N

• f positive integers such that (nk

ν=1 ων)−1 enk → 0 in X as

k → ∞.

Introduction

Weighted Backward Shifts on Fr´ echet Sequence Spaces

Let X be a Fr´ echet sequence space with canonical unit sequences

• en. For a sequence of nonzero scalars ω = (ωn)n∈N, the operator

Bω : X → X defined by Bωen = ωnen−1, n ≥ 1, e0 = 0, is called a weighted backward shift on X. Theorem Let Bω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which (en)n∈N is a basis. (i) Bω is hypercyclic ⇔ there is an increasing sequence (nk)k∈N

• f positive integers such that (nk

ν=1 ων)−1 enk → 0 in X as

k → ∞. (ii) Bω is mixing ⇔ (n

ν=1 ων)−1 en → 0 in X as n → ∞.

Introduction

Weighted Backward Shifts on Fr´ echet Sequence Spaces

Let X be a Fr´ echet sequence space with canonical unit sequences

• en. For a sequence of nonzero scalars ω = (ωn)n∈N, the operator

Bω : X → X defined by Bωen = ωnen−1, n ≥ 1, e0 = 0, is called a weighted backward shift on X. Theorem Let Bω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which (en)n∈N is an unconditional basis. (i) Bω is hypercyclic ⇔ there is an increasing sequence (nk)k∈N

• f positive integers such that (nk

ν=1 ων)−1 enk → 0 in X as

k → ∞. (ii) Bω is mixing ⇔ (n

ν=1 ων)−1 en → 0 in X as n → ∞.

(iii) Bω is chaotic ⇔ ∞

n=1 (n ν=1 ων)−1 en converges in X.

Introduction

Spaces of Real Analytic Functions

Let A(Ω) denote the space of all complex-valued real analytic functions on an open set Ω in R.

Introduction

Spaces of Real Analytic Functions

Let A(Ω) denote the space of all complex-valued real analytic functions on an open set Ω in R. Equivalent Topologies on A(Ω) (Martineau 1966) Projective limit topology: A(Ω) = projN∈N H(KN) = projN∈N indn∈N H∞(UN,n), where (KN)N∈N is a compact increasing exhaustion of Ω, and (UN,n)n∈N are fundamental sequences of neighborhoods of KN for each N.

Introduction

Spaces of Real Analytic Functions

Let A(Ω) denote the space of all complex-valued real analytic functions on an open set Ω in R. Equivalent Topologies on A(Ω) (Martineau 1966) Projective limit topology: A(Ω) = projN∈N H(KN) = projN∈N indn∈N H∞(UN,n), where (KN)N∈N is a compact increasing exhaustion of Ω, and (UN,n)n∈N are fundamental sequences of neighborhoods of KN for each N. Inductive limit topology: A(Ω) = ind H(U) where the inductive limit is taken over all complex neighborhoods of Ω.

Introduction

Spaces of Real Analytic Functions

Main difficulties These locally convex topologies on A(Ω) are not metrizable, hence A(Ω) is not a Fr´ echet space. (Doma´ nski, Vogt 2000) A(Ω) has no Schauder basis.

Introduction

Weighted Backward Shifts on A(Ω)

Definition Given a sequence of nonzero scalars ω = (ωn)n∈N, a continuous linear operator Bω : A(Ω) → A(Ω), that sends the monomials xn to ωn−1xn−1 for all n ≥ 1, the unit function to the zero function, is called a weighted backward shift with the weight sequence ω.

Introduction

Weighted Backward Shifts on A(Ω)

Definition Given a sequence of nonzero scalars ω = (ωn)n∈N, a continuous linear operator Bω : A(Ω) → A(Ω), that sends the monomials xn to ωn−1xn−1 for all n ≥ 1, the unit function to the zero function, is called a weighted backward shift with the weight sequence ω. Problem: How to characterize well-defined weighted backward shifts on A(Ω)?

Introduction

Hadamard Multipliers on A(Ω)

A linear continuous operator M : A(Ω) → A(Ω) is called a (Hadamard) multiplier whenever every monomial is an eigenvector.

Introduction

Hadamard Multipliers on A(Ω)

A linear continuous operator M : A(Ω) → A(Ω) is called a (Hadamard) multiplier whenever every monomial is an eigenvector. Observation There is a one-to-one correspondence between the weighted backward shifts and the multipliers on A(Ω) whenever 0 ∈ Ω.

Introduction

Hadamard Multipliers on A(Ω)

A linear continuous operator M : A(Ω) → A(Ω) is called a (Hadamard) multiplier whenever every monomial is an eigenvector. Observation There is a one-to-one correspondence between the weighted backward shifts and the multipliers on A(Ω) whenever 0 ∈ Ω. In this case, we can use the representation theorems for Hadamard multipliers on A(Ω) (Doma´ nski, Langenbruch 2012). Proposition Let Ω ⊂ R with 0 ∈ Ω be an open set. Then, TFAE: (i) Bω is a w.b.s. with the weight sequence ω = (ωn). (ii) Bω maps a function ∞

n=0 fnzn around zero into a real

analytic function on Ω represented around zero by the series ∞

n=0 fnωn−1zn−1.

Conditions Using Dynamical Transference Principles

An operator T on X is called quasiconjugate to an operator S on Y via a continuous map φ : Y → X with dense range if T ◦ φ = φ ◦ S. Linear dynamical properties like hypercyclicity, mixing, and chaos are preserved under quasiconjugacy.

Conditions Using Dynamical Transference Principles

An operator T on X is called quasiconjugate to an operator S on Y via a continuous map φ : Y → X with dense range if T ◦ φ = φ ◦ S. Linear dynamical properties like hypercyclicity, mixing, and chaos are preserved under quasiconjugacy. By considering Bω as an operator acting on the space H(C) of entire functions, and the space H({0}) of germs of holomorphic functions at zero, we obtain the following two quasiconjugacies. H(C)

Bω

− − → H(C) ↓ ↓ A(Ω)

Bω

− − → A(Ω) A(Ω)

Bω

− − → A(Ω) ↓ ↓ H({0})

Bω

− − → H({0})

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H(C), we obtain the following sufficient conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms.

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H(C), we obtain the following sufficient conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) if there is a sequence (nk) such that for every R > 0, nk

• ν=1

ων−1 −1 Rnk → 0 then Bω is hypercyclic.

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H(C), we obtain the following sufficient conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) if there is a sequence (nk) such that for every R > 0, nk

• ν=1

ων−1 −1 Rnk → 0 then Bω is hypercyclic. (b) if for every R > 0,

• n
• ν=1

ων−1 −1 Rn → 0 then Bω is mixing and chaotic.

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H({0}) and a construction due to Bonet, we obtain the following necessary conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms.

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H({0}) and a construction due to Bonet, we obtain the following necessary conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) if Bω is topologically transitive then there is a sequence (nk) and there is R > 0 such that nk

• ν=1

ων−1 −1 Rnk → 0.

Conditions Using Dynamical Transference Principles

Using the quasiconjugacy involving H({0}) and a construction due to Bonet, we obtain the following necessary conditions. Proposition (Doma´ nski, K. 2018) Let Ω ⊂ R be an open set with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) if Bω is topologically transitive then there is a sequence (nk) and there is R > 0 such that nk

• ν=1

ων−1 −1 Rnk → 0. (b) if Bω is mixing then there is R > 0 such that

• n
• ν=1

ων−1 −1 Rn → 0.

Conditions Using Dynamical Transference Principles

Some Problems

Observation There are weight sequences satisfying the conditions nk

• ν=1

ων−1 −1 Rnk → 0

• r
• n
• ν=1

ων−1 −1 Rn → 0 for some R > 0, but not all R > 0.

Conditions Using Dynamical Transference Principles

Some Problems

Observation There are weight sequences satisfying the conditions nk

• ν=1

ων−1 −1 Rnk → 0

• r
• n
• ν=1

ων−1 −1 Rn → 0 for some R > 0, but not all R > 0. Example If we take the unweighted backward shift on A(Ω), that is, ω = (ωn) where ωn = 1 for all n ∈ N, then

• n
• ν=1

ων−1 −1 Rn = Rn → 0 only for 0 < R < 1.

Conditions using Eigenvalues

Godefroy-Shapiro criterion

Godefroy-Shapiro criterion Let T be an operator on a topological vector space X. If the subspaces X0 := span{x ∈ X : Tx = λx for some λ with |λ| < 1}, Y0 := span{x ∈ X : Tx = λx for some λ with |λ| > 1} are dense in X, then T is mixing. If, moreover, X is a complex space and the subspace Z0 := span{x ∈ X : Tx = eαπi for some rational α} is dense in X, then T is chaotic.

Conditions using Eigenvalues

Description of Eigenvalues of A(Ω)

Definition A set U is called the star of holomorphy of a germ f ∈ H({0}) if it is a maximal star-like set in C around zero such that f extends holomorphically to V . A set U is star-like if for any z ∈ U we have {tz : t ∈ [0, 1]} ⊂ U.

Conditions using Eigenvalues

Description of Eigenvalues of A(Ω)

Proposition Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. If sup

n

• n
• ν=1

ων−1 −1 Rn < ∞ for some R > 0, then (i) zero is an eigenvalue with eigenspace of dimension one,

Conditions using Eigenvalues

Description of Eigenvalues of A(Ω)

Proposition Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. If sup

n

• n
• ν=1

ων−1 −1 Rn < ∞ for some R > 0, then (i) zero is an eigenvalue with eigenspace of dimension one, (ii) λ = 0 is an eigenvalue of Bω if and only if λΩ is contained in the star of holomorphy of the universal eigenfunction Eω(z) := zn+1 +

∞

• j=n+2
• j−1
• k=n+1

ωk −1 zj.

Conditions using Eigenvalues

Description of Eigenvalues of A(Ω)

Proposition Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. If sup

n

• n
• ν=1

ων−1 −1 Rn < ∞ for some R > 0, then (i) zero is an eigenvalue with eigenspace of dimension one, (ii) λ = 0 is an eigenvalue of Bω if and only if λΩ is contained in the star of holomorphy of the universal eigenfunction Eω(z) := zn+1 +

∞

• j=n+2
• j−1
• k=n+1

ωk −1 zj. If λ = 0 is an eigenvalue of Bω, then its eigenspace is

• ne-dimensional, spanned by the function fλ(z) := Eω(λz).

Conditions using Eigenvalues

Application of Godefroy-Shapiro Criterion

Theorem (Doma´ nski, K. 2018) Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) If Bω has a universal eigenfunction such that its star of holomorphy contains at least one interval λΩ for some λ ∈ C with |λ| > 1, then Bω is mixing.

Conditions using Eigenvalues

Application of Godefroy-Shapiro Criterion

Theorem (Doma´ nski, K. 2018) Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) If Bω has a universal eigenfunction such that its star of holomorphy contains at least one interval λΩ for some λ ∈ C with |λ| > 1, then Bω is mixing. (b) If additionally to (a) the star of holomorphy of the universal eigenfunction contains a strip {z ∈ C : |z − tλ| < ε for some t ∈ R} for some ε > 0, then Bω is also hypercyclic.

Conditions using Eigenvalues

Application of Godefroy-Shapiro Criterion

Theorem (Doma´ nski, K. 2018) Let Ω ⊆ R be an interval with 0 ∈ Ω, and Bω : A(Ω) → A(Ω) be a weighted backward shift such that ω has no zero terms. (a) If Bω has a universal eigenfunction such that its star of holomorphy contains at least one interval λΩ for some λ ∈ C with |λ| > 1, then Bω is mixing. (b) If additionally to (a) the star of holomorphy of the universal eigenfunction contains a strip {z ∈ C : |z − tλ| < ε for some t ∈ R} for some ε > 0, then Bω is also hypercyclic. (c) If additionally to (a) the star of holomorphy of the universal eigenfunction contains a set of the form {teπai : t ∈ Ω, a ∈ (a1, a2)} for some −∞ < a1 < a2 < ∞ then Bω is also chaotic.

Conditions using Eigenvalues

Some Examples

Examples Let Ω be an interval containing zero. The following weighted backward shifts are chaotic, mixing, and hypercyclic. (i) the unweighted shift B1,

Conditions using Eigenvalues

Some Examples

Examples Let Ω be an interval containing zero. The following weighted backward shifts are chaotic, mixing, and hypercyclic. (i) the unweighted shift B1, (ii) the differentiation operator D : A(Ω) → A(Ω) which corresponds to the weighted shift with ωn = n + 1,

Conditions using Eigenvalues

Some Examples

Examples Let Ω be an interval containing zero. The following weighted backward shifts are chaotic, mixing, and hypercyclic. (i) the unweighted shift B1, (ii) the differentiation operator D : A(Ω) → A(Ω) which corresponds to the weighted shift with ωn = n + 1, (iii) the weighted backward shift Qk : A(Ω) → A(Ω), k > 0, Qk(f )(z) = 1

0 f ′(zt)tk dt which corresponds to the weighted

backward shift with ωn =

n+1 n+1+k .

References I

• P. Doma´

nski, C.D. Karıksız, Eigenvalues and dynamical properties of weighted backward shifts on the space of real analytic functions, Studia Math. 242 (2018), 57–78.

• P. Doma´

nski, M. Langenbruch, Representation of multipliers

• n spaces of real analytic functions, Analysis 32 (2012),

137–162.

• P. Doma´

nski, D. Vogt, The space of real-analytic functions has no basis, Studia Math. 142 (2) (2000), 187–200.

• A. Martineau, Sur la topologie des espaces de fonctions

holomorphes, Math. Ann. 163 (1966), 62–88.

The End