<i>m</i>‐isometries on Banach spaces

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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“…For partial results see [14]. We recall the notion of an (m, p)-isometry in general Banach spaces, introduced by Bayart in [5].…”

Section: Proposition 83 If a Hyponormal Operator T ∶ H → H Is Recurmentioning

confidence: 99%

Recurrent Linear Operators

Costakis

Manoussos

Parissis

2013

Complex Anal. Oper. Theory

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“…In this section, we cite some basic results concerning (m, p)-isometric operators proven by Bayart in [5]. There, it is assumed that p ≥ 1, but on inspection it is clear that that this restriction is unnecessary and one can allow p ∈ (0, ∞).…”

Section: Preliminariesmentioning

confidence: 99%

“…Bayart proves in [5,Theorem 3.4] that an isometry on a complex infinite-dimensional Banach space is not N -supercyclic, for any N ≥ 1. Hence, Theorem 5.2 implies:…”

Section: (M ∞)-And (M P)-isometriesmentioning

confidence: 99%

On the Second Parameter of an (m, p)-Isometry

Hoffmann

Mackey

Searcóid

2011

Integr. Equ. Oper. Theory

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A bounded linear operator T on a Banach space X is called an (m, p)-isometry if it satisfies the equationIn this paper we study the structure which underlies the second parameter of (m, p)-isometric operators. We concentrate on determining when an (m, p)-isometry is a (µ, q)-isometry for some pair (µ, q). We also extend the definition of (m, p)-isometry, to include p = ∞ and study basic properties of these (m, ∞)-isometries.

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“…Also, sufficient conditions under which an m-isometric operator is not N -supercyclic are given by Bermúdez et al(see [6]). Bayart [4] completed the result by showing that m-isometric operators cannot be N -supercyclic. Sufficient conditions under which an A-m-isometry is not supercyclic or N -supercyclic are given by Rabaoui and Saddi in [13].…”

Section: Let H Denote An Infinite Dimensional Hilbert Space and B(h) mentioning

confidence: 96%