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 2 x , · · · , T n x , · · · } , 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 2 x , · · · , T n x , · · · } , 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 e n . For a sequence of nonzero scalars ω = ( ω n ) n ∈ N , the operator B ω : X → X defined by B ω e n = ω n e n − 1 , n ≥ 1 , e 0 = 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 e n . For a sequence of nonzero scalars ω = ( ω n ) n ∈ N , the operator B ω : X → X defined by B ω e n = ω n e n − 1 , n ≥ 1 , e 0 = 0 , is called a weighted backward shift on X . Theorem Let B ω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which ( e n ) n ∈ N is a basis. (i) B ω is hypercyclic ⇔ there is an increasing sequence ( n k ) k ∈ N ν =1 ω ν ) − 1 e n k → 0 in X as of positive integers such that ( � n k k → ∞ .

Introduction Weighted Backward Shifts on Fr´ echet Sequence Spaces Let X be a Fr´ echet sequence space with canonical unit sequences e n . For a sequence of nonzero scalars ω = ( ω n ) n ∈ N , the operator B ω : X → X defined by B ω e n = ω n e n − 1 , n ≥ 1 , e 0 = 0 , is called a weighted backward shift on X . Theorem Let B ω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which ( e n ) n ∈ N is a basis. (i) B ω is hypercyclic ⇔ there is an increasing sequence ( n k ) k ∈ N ν =1 ω ν ) − 1 e n k → 0 in X as of positive integers such that ( � n k k → ∞ . ν =1 ω ν ) − 1 e n → 0 in X as n → ∞ . (ii) B ω is mixing ⇔ ( � n

Introduction Weighted Backward Shifts on Fr´ echet Sequence Spaces Let X be a Fr´ echet sequence space with canonical unit sequences e n . For a sequence of nonzero scalars ω = ( ω n ) n ∈ N , the operator B ω : X → X defined by B ω e n = ω n e n − 1 , n ≥ 1 , e 0 = 0 , is called a weighted backward shift on X . Theorem Let B ω : X → X be a weighted backward shift acting on a Fr´ echet sequence space X in which ( e n ) n ∈ N is an unconditional basis. (i) B ω is hypercyclic ⇔ there is an increasing sequence ( n k ) k ∈ N ν =1 ω ν ) − 1 e n k → 0 in X as of positive integers such that ( � n k k → ∞ . ν =1 ω ν ) − 1 e n → 0 in X as n → ∞ . (ii) B ω is mixing ⇔ ( � n ν =1 ω ν ) − 1 e n converges in X. n =1 ( � n (iii) B ω is chaotic ⇔ � ∞

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) = proj N ∈ N H ( K N ) = proj N ∈ N ind n ∈ N H ∞ ( U N , n ) , where ( K N ) N ∈ N is a compact increasing exhaustion of Ω, and ( U N , n ) n ∈ N are fundamental sequences of neighborhoods of K N 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) = proj N ∈ N H ( K N ) = proj N ∈ N ind n ∈ N H ∞ ( U N , n ) , where ( K N ) N ∈ N is a compact increasing exhaustion of Ω, and ( U N , n ) n ∈ N are fundamental sequences of neighborhoods of K N 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 = ( ω n ) n ∈ N , a continuous linear operator B ω : A (Ω) → A (Ω) , that sends the monomials x n to ω n − 1 x n − 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 = ( ω n ) n ∈ N , a continuous linear operator B ω : A (Ω) → A (Ω) , that sends the monomials x n to ω n − 1 x n − 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) B ω is a w.b.s. with the weight sequence ω = ( ω n ) . n =0 f n z n around zero into a real (ii) B ω maps a function � ∞ analytic function on Ω represented around zero by the series � ∞ n =0 f n ω n − 1 z n − 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 B ω 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. B ω B ω H ( C ) − − → H ( C ) A (Ω) − − → A (Ω) ↓ ↓ ↓ ↓ B ω B ω A (Ω) − − → A (Ω) H ( { 0 } ) − − → 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 B ω : 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 B ω : A (Ω) → A (Ω) be a weighted backward shift such that ω has no zero terms. (a) if there is a sequence ( n k ) such that for every R > 0 , � n k � − 1 � R n k → 0 then B ω is hypercyclic. ω ν − 1 ν =1