Universal functions for composition operators with non-automorphic symbol

Grosse‐Erdmann, Karl‐G.; Mortini, Raymond

doi:10.1007/s11854-009-0013-4

Cited by 50 publications

(58 citation statements)

References 23 publications

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“…However, if φ has no fixed point in D it follows as in the proof of Theorem 6.10 that for every compact set K ⊂ D there exists a positive integer n o such that φ [n] (K) ∩ K = ∅ for all n ≥ n o . This implies that C φ is hereditarily hypercyclic; see for instance [37,Theorem 3.2]. In this case C φ cannot be rigid.…”

Section: Corollary 611 Let φ ∈ Lfm(d) the Composition Operator C φmentioning

confidence: 99%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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Abstract. We study the notion of recurrence and some of its variations for linear operators acting on Banach spaces. We characterize recurrence for several classes of linear operators such as weighted shifts, composition operators and multiplication operators on classical Banach spaces. We show that on separable complex Hilbert spaces the study of recurrent operators reduces, in many cases, to the study of unitary operators. Finally, we study the notion of product recurrence and state some relevant open questions. The most studied notion in linear dynamics is that of hypercyclicity: a bounded linear operator T acting on a separable Banach space is hypercyclic if there exists a vector whose orbit under T is dense in the space. On the other hand, a very central notion in topological dynamics is that of recurrence. This notion goes back to Poincaré and Birkhoff and it refers to the existence of points in the space for which parts of their orbits under a continuous map "return" to themselves. The purpose of this note is the study of the notion of recurrence, together with its variations, in the context of linear dynamics. Some examples and characterizations of recurrence for special In an effort to characterize recurrent linear operators one many times falls back to the notion of hypercyclicity. This is for example the case when we study the recurrence properties of backwards shifts, say on ℓ 2 (Z). The reason behind is that, according to a result of Seceleanu, [51], the orbits of these operators satisfy a zero-one law: if the orbit of a weighted backward shift contains a non-zero limit point then the corresponding shift is actually hypercyclic. Thus a weighted backward shift on ℓ 2 (Z) is recurrent if and only if it is hypercyclic. The same equivalence is true, albeit for different reasons, for the adjoint of a multiplication operators on the Hardy space H 2 (D). These connections to hypercyclicity, already observed in [17], come up naturally and thus motivate a further search on whether the properties of recurrent operators resemble the properties of hypercyclic ones, in general. It turns out that, indeed, there are many structural similarities between the set of hypercyclic vectors and the set of recurrent vectors in the sense that they exhibit the same invariances. Furthermore, the spectral properties of hypercyclic and recurrent operators are somewhat similar, although this vague statement should be interpreted with some care. However, these similarities cannot be pushed too much as there are obviously many classes of operators which are recurrent without being hypercyclic. One can find such examples among composition operators on the Hardy space H 2 (D). However, the primordial example is given just by considering unimodular multiples of the identity operator. A more general class for which one needs to address the recurrence properties independently of hypercyclicity is that of unitary operators on Hilbert spaces.The discussion above hopefully justifies why we will shortly recall a full set of definitions rel...

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Section: Corollary 611 Let φ ∈ Lfm(d) the Composition Operator C φmentioning

confidence: 99%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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“…A related interesting question is when C ϕ is mean ergodic [3,9,17,22]. We complement important results on universality which were obtained in [5,6,12]. These papers do not discuss when C ϕ is power bounded or mean ergodic on H (U ).…”

Section: Introduction and Notationmentioning

confidence: 75%

A note on mean ergodic composition operators on spaces of holomorphic functions

Bonet

Domański

2011

RACSAM

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“…In the special case q n = 1 for all n ∈ N, Theorem 6 corresponds to Theorem 5.1 (1-3) of [31]. As a direct consequence of Theorem 6, we have…”

Section: The Alpha- Beta-and Gamma-duals Of the Space μ(λ B; P)mentioning

confidence: 81%

Some new paranormed sequence spaces and core theorems

Karaisa

Başar

2014

AIP Conference Proceedings

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Let μ ∈ { ∞ , c, c 0 }. In this study, by using lambda matrix Λ and difference matrix B, we introduce the new nonabsolute type paranormed sequence space μ(λ , B; p) and prove that μ(λ , B; p) and μ(p) linearly isomorphic. Further, we give some inclusion relations concerning the space μ(λ , B; p). Afterwards, we determine the alpha-, beta-and gamma-duals of the space μ(λ , p; B). We also give the characterization of the classes (μ(λ , B; p) : ν) and (ν : μ(λ , B; p)), where ν is any given sequence space. Moreover, we introduce the Λ-core of a complex valued sequence and determine the necessary and sufficient conditions on a matrix A for which Λ − core(Ax) ⊆ K − core(x) and Λ − core(Ax) ⊆ st − core(x) for all x ∈ ∞ .

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Universal functions for composition operators with non-automorphic symbol

Cited by 50 publications

References 23 publications

Recurrent Linear Operators

Recurrent Linear Operators

A note on mean ergodic composition operators on spaces of holomorphic functions

Some new paranormed sequence spaces and core theorems

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