The role of the spectrum in the cyclic behavior of composition operators

Gallardo-Gutiérrez, Eva A.; Montes-Rodrı́guez, Alfonso

doi:10.1090/memo/0791

Cited by 46 publications

(40 citation statements)

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“…Analogue extensions were done to Seidel and Walsh's [27] hypercyclicity result on non-Euclidean translations acting on the space H (D) of holomorphic functions on the unit disc: Bourdon and Shapiro [12] and Gallardo-Gutiérrez and Montes-Rodríguez [18] showed the existence of slow-growth universal functions by showing the hypercyclicity of these non-Euclidean translations on the Hardy space and on weighted Dirichlet spaces. With respect to the second theme, it is well known that every separable, infinite-dimensional Fréchet space X supports a hypercyclic operator T , thanks to the works of Ansari [1], BernalGonzález [5], and Bonet and Peris [11].…”

Section: Introductionmentioning

confidence: 93%

Disjoint mixing operators

Bès¹,

Martin²,

Peris³

et al. 2012

Journal of Functional Analysis

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Chan and Shapiro showed that each (non-trivial) translation operator f (z) T λ → f (z + λ) acting on the Fréchet space of entire functions endowed with the topology of locally uniform convergence supports a universal function of exponential type zero. We show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. We also show that every separable infinite-dimensional Fréchet space supports an arbitrarily large finite and commuting disjoint mixing collection of operators. When this space is a Banach space, it supports an arbitrarily large finite disjoint mixing collection of C 0 -semigroups. We also provide an easy proof of the result of Salas that every infinitedimensional Banach space supports arbitrarily large tuples of dual d-hypercyclic operators, and construct an example of a mixing Hilbert space operator T so that (T , T 2 ) is not d-mixing.

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Section: Introductionmentioning

confidence: 93%

Disjoint mixing operators

Bès¹,

Martin²,

Peris³

et al. 2012

Journal of Functional Analysis

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“…See, for example [5] and [10]. A bounded operator T acting on the Hilbert space H is called cyclic if there is a vector x ∈ H whose orbit under T , Orb(T, x) := {T n x : n ∈ N}, has dense linear span.…”

Section: Each Function K W (W ∈ D) Is Known As a Reproducing Kernel Omentioning

confidence: 99%

Composition operators on spaces of entire functions

Chacón

Giménez

2007

Proc. Amer. Math. Soc.

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“…where y ∈ R \ {0} (see, for instance, [10] for a similar computation when a = 1). For a Möbius map ψ(z) = z−α 1−αz we have…”

Section: The Next Results Identifies Invertible Weighted Composition Omentioning

confidence: 99%

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

2015

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Abstract. Boundedness of weighted composition operators Wu,ϕ acting on the classical Dirichlet space D as W h,ϕ f = h (f •ϕ) is studied in terms of the multiplier space associated to the symbol ϕ, i.e., M(ϕ) = {u ∈ D : Wu,ϕ is bounded on D}. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of Wu,ϕ in D whenever ϕ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko [13] to the context of the Dirichlet space.

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The role of the spectrum in the cyclic behavior of composition operators

Cited by 46 publications

References 0 publications

Disjoint mixing operators

Disjoint mixing operators

Composition operators on spaces of entire functions

Weighted composition operators on the Dirichlet space: boundedness and spectral properties

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