On the Orbit of an m-Isometry

Bermúdez, Teresa; Marrero, Isabel; Martinón, Antonio

doi:10.1007/s00020-009-1700-3

Cited by 27 publications

(12 citation statements)

References 11 publications

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“…So, for example, in [14,18,19,26] different results about m-isometries are given. In [6,8,17] are proved some dynamic properties of m-isometries. Several authors have considered certain types of operators (composition, multiplication, shift) and have analyzed conditions under which these operators are m-isometries: [5,9,20,23,25].…”

Section: Introductionmentioning

confidence: 98%

Products of m-isometries

Bermúdez

Martinón

Noda

2013

Linear Algebra and its Applications

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Section: Introductionmentioning

confidence: 98%

Products of m-isometries

Bermúdez

Martinón

Noda

2013

Linear Algebra and its Applications

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“…Basic properties of m-isometries include the facts that every m-isometry is an (m + 1)-isometry, that m-isometries are bounded below and that their spectrum σ(T ) ⊆ K lies in the closed unit disc. The dynamics of m-isometric operators have been studied in [6] and [10]. If H is a complex Hilbert space, condition (1.1) can be rewritten as 2) and this formulation can be interpreted in an arbitrary Banach space.…”

Section: Introductionmentioning

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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“…He defines: Definition 1.3. An operator T is said to be n-supercyclic, n ≥ 1, if there is a subspace of dimension n in X with dense orbit.These operators have been studied in [1], [3] and [5] and [7]. Feldman gave some different classes of n-supercyclic operators and in particular:…”

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Strongly $n$-supercyclic operators

Ernst¹

2014

J. Operator Theory

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Abstract. In this paper, we are interested in the properties of a new class of operators, recently introduced by Shkarin, called strongly n-supercyclic operators. This notion is stronger than n-supercyclicity. We prove that such operators have interesting spectral properties and give examples and counter-examples answering some natural questions asked by Shkarin. IntroductionIn what follows X will denote completely separable Baire vector spaces over the field K = R, C and T will be a bounded linear operator on X. Since the last 1980's, density properties of orbits of operators have been of great interest for many mathematicians, particularly hypercyclic and cyclic operators for their link with the invariant subspace problem. Another reason explaining this interest is that they appear in many well-known classes of operators: weighted shifts, composition operators, translation operators,...is dense in X. The set of all hypercyclic vectors for T is denoted by HC(T ). The operator T is said to be hypercyclic if HC(T ) = ∅.One may remove linearity in this definition, then under the same assumptions, T is said to be universal. In the same way, in 1974, Hilden and Wallen [10] introduced the weaker notion of supercyclicity which does not deal with orbits of vectors any more but with orbits of lines. Definition 1.2. A vector x ∈ X is said supercyclic for T if its projective orbit {λT n x, n ∈ N, λ ∈ K} is dense in X. The set of all hypercyclic vectors for T is denoted by SC(T ). The operator T is called hypercyclic if SC(T ) = ∅.As we said before, these properties have been intensively studied and the reader can refer to [2] and [9] for a deep and complete survey. One of the main ingredient providing such operators is the so called Hypercyclicity Criterion given by Carol Kitai in 1982 [11].Theorem: Hypercyclicity Criterion. Let X be a separable Banach space and T ∈ L(X). T satisfies the Hypercyclicity Criterion if there exist a strictly increasing sequence (n k ) k∈N , two dense setsIf T satisfies the Hypercyclicity Criterion, then T is hypercyclic. Shkarin [14] proved that this criterion is only a sufficient condition for hypercyclicity providing counter-examples to the necessary condition. Actually, J. Bès and A. Peris [4] showed that any finite direct sum of an operator T with itself is hypercyclic if and only if T satisfies the Hypercyclicity Criterion. This characterisation will be of great use later. Similarly, H.N. Salas [12] gave a Supercyclicity Criterion which is only a sufficient condition too and verifies the same kind of characterisation as above.Theorem: Supercyclicity Criterion. Let X be a separable Banach space and T ∈ L(X). T satisfies the Supercyclicity Criterion if there exist a strictly increasing sequence (n k ) k∈N , two dense sets D 1 , D 2 ⊂ X in X and a sequence of maps S n k : D 2 → X such that: (a) T n k x S n k y → 0 for any x ∈ D 1 and y ∈ D 2 ; (b) T n k S n k y → y for any y ∈ D 2 . If T satisfies the Supercyclicity Criterion, then T is supercyclic.These results are at the very heart of th...

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On the Orbit of an m-Isometry

Cited by 27 publications

References 11 publications

Products of m-isometries

Products of m-isometries

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

Strongly $n$-supercyclic operators

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