n-Weakly hypercyclic and n-weakly supercyclic operators

Feldman, Nathan S.

doi:10.1016/j.jfa.2012.07.006

Cited by 11 publications

(21 citation statements)

References 28 publications

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“…In fact, with only a small modification of his proof, we see that his weak angle criterion actually proves that the given vector is not a 2-weakly supercyclic vector, see Feldman [10] for the details.…”

Section: Proposition 3 If T Is 1-weakly Hypercyclic Thenmentioning

confidence: 90%

“…The following result is an elementary consequence of the Weak Angle Criterion, see Feldman [10] for the details.…”

Section: Theorem 2 (Weak Angle Criterion) Suppose That T Is a Boundedmentioning

confidence: 94%

“…Item (1) below follows from the Weak Ratio Criterion and item (2) follows easily from the definition of 2-weak supercyclicity, see Feldman [10] for the details.…”

Section: Corollary 1 (Weak Ratio Criterion) If T Is a Bounded Linear mentioning

confidence: 99%

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N-weakly supercyclic matrices

Feldman

2011

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We define an operator to n-weakly hypercyclic if it has an orbit that has a dense projection onto every n-dimensional subspace. Similarly, an operator is n-weakly supercyclic if it has a scaled orbit that has a dense projection onto every n-dimensional subspace. In this paper, we show the following results: (i) There are no n-weakly hypercyclic matrices on R n or C n . (ii) There are no 2-weakly supercyclic matrices on C n for n ≥ 2. (iii) There are no 3-weakly supercyclic matrices on R n for n ≥ 3; and (iv) there are 2-weakly supercyclic matrices on R n if and only if n is even. Finally, we show that there is an onto isometry on 2 R (N) that is 2-weakly supercyclic, but not 3-weakly supercyclic and also give some examples involving tuples of matrices. We conclude with some questions.

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Section: Proposition 3 If T Is 1-weakly Hypercyclic Thenmentioning

confidence: 90%

“…The following result is an elementary consequence of the Weak Angle Criterion, see Feldman [10] for the details.…”

Section: Theorem 2 (Weak Angle Criterion) Suppose That T Is a Boundedmentioning

confidence: 94%

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N-weakly supercyclic matrices

Feldman

2011

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“…Let ξ " e 2πi{5 and for α, β P T, let upα, βq " pαξ, αξ 2 , αξ 3 , αξ 4 , α, βξ 2 , βξ 4 , βξ, βξ 3 , βq P T 10 . Then S upα,βq " ST α,β , where T α,β : C 2 Ñ C 2 , T α pz, wq " pαz, βwq and…”

Section: Infinite Sums and Integralsmentioning

confidence: 99%

On numerically hypercyclic operators

Shkarin

2013

Preprint

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According to Kim, Peris and Song, a continuous linear operator T on a complex Banach space X is called numerically hypercyclic if the numerical orbit tf pT n xq : n P Nu is dense in C for some x P X and f P X ˚satisfying }x} " }f } " f pxq " 1. They have characterized numerically hypercyclic weighted shifts and provided an example of a numerically hypercyclic operator on C 2 .We answer two questions of Kim, Peris and Song. Namely, we construct a numerically hypercyclic operator, whose square is not numerically hypercyclic as well as an operator which is not numerically hypercyclic but has two numerical orbits whose union is dense in C. We characterize numerically hypercyclic operators on C 2 as well as the operators similar to a numerically hypercyclic one and those operators whose conjugacy class consists entirely of numerically hypercyclic operators. We describe in spectral terms the operator norm closure of the set of numerically hypercyclic operators on a reflexive Banach space. Finally, we provide criteria for numeric hypercyclicity and decide upon the numerical hypercyclicity of operators from various classes.

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“…Similarly, x is called a weakly hypercyclic vector for T if O(T, x) is dense in X with respect to the weak topology on X. Recently, Feldman [11] has introduced and studied the concept of an n-weakly hypercyclic vector. Namely, for n ∈ N, x is called an n-weakly hypercyclic vector for T if for every continuous surjective linear map S : X → K n , the set S(O(T, x)) is dense in K n .…”

Section: Introductionmentioning

confidence: 99%

Orbits of coanalytic Toeplitz operators and weak hypercyclicity

Shkarin

2012

Preprint

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We prove a new criterion of weak hypercyclicity of a bounded linear operator on a Banach space. Applying this criterion, we solve few open questions. Namely, we show that if G is a region of C bounded by a smooth Jordan curve Γ such that G does not meet the unit ball but Γ intersects the unit circle in a non-trivial arc, then M * is a weakly hypercyclic operator on H 2 (G), where M is the multiplication by the argument operator M f (z) = zf (z). We also prove that if g is a non-constant function from the Hardy space H ∞ (D) on the unit disk D such that g(D) ∩ D = ∅ and the set {z ∈ C : |z| = 1, |g(z)| = 1} is a subset of the unit circle T of positive Lebesgue measure, then the coanalytic Toeplitz operator T * g on the Hardy space H 2 (D) is weakly hypercyclic. On the contrary, if g(D) ∩ D = ∅, |g| > 1 almost everywhere on T and log(|g| − 1) ∈ L 1 (T), then T * g is not 1-weakly hypercyclic and hence is not weakly hypercyclic (a bounded linear operator T on a complex Banach space X is called n-weakly hypercyclic if there is x ∈ X such that for every surjective continuous linear operator S : X → C n , the set {S(T m x) : m ∈ N} is dense in C n ). The last result is based upon lower estimates of the norms of the members of orbits of a coanalytic Toeplitz operator. Finally, we show that there is a 1-weakly hypercyclic operator on a Hilbert space, whose square is non-cyclic and prove that a Banach space operator is weakly hypercyclic if and only if it is n-weakly hypercyclic for every n ∈ N.

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n-Weakly hypercyclic and n-weakly supercyclic operators

Cited by 11 publications

References 28 publications

N-weakly supercyclic matrices

N-weakly supercyclic matrices

On numerically hypercyclic operators

Orbits of coanalytic Toeplitz operators and weak hypercyclicity

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