On weak positive supercyclicity

León-Saavedra, Fernando; Piqueras-Lerena, Antonio

doi:10.1007/s11856-008-1050-x

Cited by 10 publications

(8 citation statements)

References 16 publications

(17 reference statements)

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“…Let us recall that an operator T defined on a Banach space X is said to be weakly supercyclic if the orbit {λT n x : n ∈ N, λ ∈ C} is weakly dense in X. To show this, we need a weak version of the Positive Supercyclicity Theorem discovered in [18] (for some related results see [24]).…”

Section: Proposition 28mentioning

confidence: 99%

Orbits of Cesàro type operators

León-Saavedra

Lerena

Seoane‐Sepúlveda

2009

Mathematische Nachrichten

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A bounded linear operator T on a Banach space X is called hypercyclic if there exists a vector x ∈ X such that its orbit, {T n x}, is dense in X. In this paper we show hypercyclic properties of the orbits of the Cesàro operator defined on different spaces. For instance, we show that the Cesàro operator defined onMoreover, it is chaotic and it has supercyclic subspaces. On the other hand, the Cesàro operator defined on other spaces of functions behave differently. Motivated by this, we study weighted Cesàro operators and different degrees of hypercyclicity are obtained. The proofs are based on the classical Müntz-Szász theorem. We also propose problems and give new directions.

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Section: Proposition 28mentioning

confidence: 99%

Orbits of Cesàro type operators

León-Saavedra

Lerena

Seoane‐Sepúlveda

2009

Mathematische Nachrichten

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“…An important result concerning Γ-supercyclicity, for Γ = T, was due to León Saavedra and Müller [18], where they proved that a linear operator is T-supercyclic if and only if it is hypercyclic. The case when Γ = R + is also called positive supercyclicity (see [22]). The concept of Γ-supercyclicity was recently introduced by Charpentier, Ernst and Menet in [11].…”

Section: D-hypercyclic Weakly Hypercyclic and γ-Supercyclic Homogenementioning

confidence: 99%

Orbits of homogeneous polynomials on Banach spaces

Cardeccia

Muro

2020

Ergod. Th. Dynam. Sys.

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We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show, a simple and natural example of a homogeneous polynomial with an orbit that is at the same time d-dense (the orbit meets every ball of radius d), weakly dense and such that Γ · OrbP (x) is dense for every Γ ⊂ C that is either unbounded or that has 0 as an accumulation point. Moreover we generalize the construction to arbitrary infinite dimensional separable Fréchet spaces. To prove this we study Julia sets of homogeneous polynomials on Banach spaces.However, the behavior of the orbits induced by a homogeneous polynomial can be highly nontrivial and it is far from being understood. For example, in [4] Bernardes showed that the orbits may oscillate between infinity and the boundary of the limit ball. He also proved that every infinite dimensional and separable Banach space supports supercyclic homogeneous polynomials. More recently Peris, Kim and Song [16,17] proved that every separable Banach space of dimension greater than one supports numerically hypercyclic homogeneous polynomials. This means that there are vectors x ∈ S X , x * ∈ S X * for which its numerical orbit, N orb P (x, x * ) := {x * (P n (x)) : n ∈ N 0 } is dense in C.The following result shows that the Julia set of a homogeneous polynomial share some of the properties satisfied by the Julia set of a holomorphic function on the complex plane.Proposition 2.12 (Properties of J P ). Let P be a non linear homogeneous polynomial. The Julia set J P satisfies the following properties:Proof. i) This is clear since J P is the boundary of an open set.ii) First note that y ∈ A P if and only if P (y) ∈ A P . Thus, if x ∈ J P then P (x) is not in A P .On the other hand, there exists (a n ) n ⊆ A P such that a n → x. Since P (a n ) belongs also to A P and P (a n ) → P (x), we have that P (x) ∈ J P .iii) Just note that if x ∈ J P then λx ∈ J P for every |λ| = 1.iv) It suffices to show that A P n = A P . Clearly if P k (x) → 0 then (P n ) k (x) → 0 and therefore A P ⊆ A P n . The converse follows thanks to the existence of the limit ball (Proposition 2.1). If P nk (x) → 0 then Orb P n (x) meets eventually the limit ball of P and therefore P n (x) must tend to zero.In contrast to the one dimensional case, the Julia set J P may be empty. Indeed, if P ∈ P( 2 ℓ 2 ; ℓ 2 ) is defined as P (x) = (x 2 2 , x 2 3 , x 2 4 , . . .), then P n (x) → 0 for every x ∈ ℓ 2 . Thus A P = ℓ 2 and hence J P = R P = ∅.Recall that a set A is completely invariant under P if P (A) ⊆ A and P −1 (A) ⊆ A. The Julia set of a homogeneous polynomials need not to be completely invariant, as we will show in Example 3.19. On the other hand, under certain conditions on P , J P results completely invariant.have that P n (x) → 0 and hence P (x) ∈ A P . If R P = ∅ this implies that x ∈ J P . If R P = ∅ and x ∈ R P , there is some ǫ > 0 so that P n (y) → ∞ for every y ∈ B ǫ (x).Since P is open, P (B ǫ (x)) is an open neighborhood of...

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“…In Section 4 we will prove a weak version of the Positive Supercyclicity Theorem. These results are included in [28].…”

Section: Supercyclicity Is An Intermediate Property Between the Hypermentioning

confidence: 99%

“…After this paper was accepted, the authors improved Theorem 4.1 to the non-locally convex setting. In fact, Theorem 4.1 is true without the hypothesis on separability of the dual X ⋆ (see [28]). …”

mentioning

confidence: 99%

Positivity in the theory of supercyclic operators

León-Saavedra

Piqueras-Lerena

2007

Perspectives in Operator Theory

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Abstract. A bounded linear operator T defined on a Banach space X is said to be supercyclic if there exists a vector x ∈ X such that the projective orbit {λT n x : λ ∈ C, n ∈ N} is dense in X. The aim of this survey is to show the relationship between positivity and supercyclicity. This relationship comes from the so called Positive Supercyclicity Theorem. Throughout this exposition, interesting new directions and open problems will appear.

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On weak positive supercyclicity

Cited by 10 publications

References 16 publications

Orbits of Cesàro type operators

Orbits of Cesàro type operators

Orbits of homogeneous polynomials on Banach spaces

Positivity in the theory of supercyclic operators

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