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1.
We prove a criterion that guarantees that in a given class of operators the set of hypercyclic ones is residual. We also prove the existence of quasinilpotent Volterra composition operators, Vj{V_\varphi} , such that both Vj{V_\varphi} and Vj*{V_\varphi^\star} are supercyclic and both I + Vj{I + V_\varphi} and I + Vj*{I + V_\varphi^\star} are hypercyclic.  相似文献   

2.
Let G be a locally compact group and let 1 ≤ p < 1. Recently, Chen et al. characterized hypercyclic, supercyclic and chaotic weighted translations on locally compact groups and their homogeneous spaces. There has been an increasing interest in studying the disjoint hypercyclicity acting on various spaces of holomorphic functions. In this note, we will study disjoint hypercyclic and disjoint supercyclic powers of weighted translation operators on the Lebesgue space L p(G) in terms of the weights. Sufficient and necessary conditions for disjoint hypercyclic and disjoint supercyclic powers of weighted translations generated by aperiodic elements on groups will be given.  相似文献   

3.
Using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for T for which the a-Browder's theorem or the a-Weyl's theorem holds. Also, the relation between hypercyclic operators (or supercyclic operators) and the operators which satisfy Weyl type theorem is discussed.  相似文献   

4.
We study hypercyclicity and supercyclicity of weighted shifts on ℓ, with respect to the weak * topology. We show that there exist bilateral shifts that are weak * hypercyclic but fail to be weak * sequentially hypercyclic. In the unilateral case, a shift T is weak * hypercyclic if and only if it is weak * sequentially hypercyclic, and this is equivalent to T being either norm, weak, or weak-sequentially hypercyclic on c0 or ℓp (1 ≤ p < ∞). We also show that the set of weak * hypercyclic vectors of any unilateral or bilateral shift on ℓ is norm nowhere dense. Finally, we show that ℓ supports an isometry that is weak * sequentially supercyclic.  相似文献   

5.
A bounded linear operator T acting on a Banach space B is called weakly hypercyclic if there exists xB such that the orbit is weakly dense in B and T is called weakly supercyclic if there is xB for which the projective orbit is weakly dense in B. If weak density is replaced by weak sequential density, then T is said to be weakly sequentially hypercyclic or supercyclic, respectively. It is shown that on a separable Hilbert space there are weakly supercyclic operators which are not weakly sequentially supercyclic. This is achieved by constructing a Borel probability measure μ on the unit circle for which the Fourier coefficients vanish at infinity and the multiplication operator Mf(z)=zf(z) acting on L2(μ) is weakly supercyclic. It is not weakly sequentially supercyclic, since the projective orbit under M of each element in L2(μ) is weakly sequentially closed. This answers a question posed by Bayart and Matheron. It is proved that the bilateral shift on ?p(Z), 1?p<∞, is weakly supercyclic if and only if 2<p<∞ and that any weakly supercyclic weighted bilateral shift on ?p(Z) for 1?p?2 is norm supercyclic. It is also shown that any weakly hypercyclic weighted bilateral shift on ?p(Z) for 1?p<2 is norm hypercyclic, which answers a question of Chan and Sanders.  相似文献   

6.
Let T B(X) be a hypercyclic operator and a complex number of modulus 1. Then T is hypercyclic and has the same set of hypercyclic vectors as T. A version of this results gives for a wide class of supercyclic operators that x X is supercyclic for T if and only if the set {tTn x : t > 0, n = 0, 1, ...} is dense in X. This gives answers to several questions studied in the literature.  相似文献   

7.
We study hypercyclicity properties of functions of Banach space operators. Generalizations of the results of Herzog–Schmoeger and Bermudez–Miller are obtained. As a corollary we also show that each non-trivial operator commuting with a generalized backward shift is supercyclic. This gives a positive answer to a conjecture of Godefroy and Shapiro. Furthermore, we show that the norm-closures of the set of all hypercyclic (mixing, chaotic, frequently hypercyclic, respectively) operators on a Hilbert space coincide. This implies that the set of all hypercyclic operators that do not satisfy the hypercyclicity criterion is rather small—of first category (in the norm-closure of hypercyclic operators).  相似文献   

8.
Herrero conjectured in 1991 that every multi-hypercyclic (respectively, multi-supercyclic) operator on a Hilbert space is in fact hypercyclic (respectively, supercyclic). In this article we settle this conjecture in the affirmative even for continuous linear operators defined on arbitrary locally convex spaces. More precisely, we show that, if is a continuous linear operator on a locally convex space E such that there is a finite collection of orbits of T satisfying that each element in E can be arbitrarily approximated by a vector of one of these orbits, then there is a single orbit dense in E. We also prove the corresponding result for a weaker notion of approximation, called supercyclicity . Received October 18, 1999 / Published online February 5, 2001  相似文献   

9.
A pair of commuting operators, (A,B), on a Hilbert space is said to be hypercyclic if there exists a vector such that {A n B k x : n, k ≥ 0} is dense in . If f, gH (G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M * f , M * g ) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, gH (G) such that the pair (M * f , M * g ) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.  相似文献   

10.
Necessary and sufficient conditions for hypercyclic/supercyclic Banach space operators T to satisfy are proved.  相似文献   

11.
We show that if V α (α > 0) is the Riemann-Liouville fractional integration operator and T is an invertible operator on L 2(0, 1) which commutes with V , then TV α is not supercyclic on L 2(0, 1); in particular, many Volterra convolution operators are not supercyclic. The technique is based on an argument used by Gallardo-Gutiérrez and Montes-Rodríguez to show that V is not supercyclic.  相似文献   

12.
Complementing the existing literature in d-hypercyclicity, we characterize disjoint supercyclicity for a finite family of weighted shift operators. Using this characterization, we answer Question 2 in a recent paper by Bès, Martin and Peris in the negative by constructing examples of disjoint supercyclic weighted shifts whose direct sum operator is hypercyclic, but the same shifts operators fail to be disjoint hypercyclic. We also show the Disjoint Blow-Up/Collapse Property and the Strong Disjoint Blow-Up/Collapse Property for disjoint supercyclicity are equivalent when dealing with a finite family with two or more weighted shifts. However, those weighted shifts operators will never satisfy the Disjoint Supercyclicity Criterion. This provides a sharp distinction between disjoint supercyclicity and supercyclicity for a single operator. We provide a partial answer to disjoint supercyclic version of Question 3 in a recent paper by Salas by showing that we can always select an additional operator to add to an family of d-supercyclic weighted shift operators while maintaining the d-supercyclicity. We also show that, in general, this additional operator cannot be another weighted shift.  相似文献   

13.
A unitary operator V and a rank 2 operator R acting on a Hilbert space H{\mathcal{H}} are constructed such that V + R is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.  相似文献   

14.
We show that an operator on a separable complex Banach space with sufficiently many eigenvectors associated to eigenvalues of modulus 1 is hypercyclic. We apply this result to construct hypercyclic operators with prescribed Kσ unimodular point spectrum. We show how eigenvectors associated to unimodular eigenvalues can be used to exhibit common hypercyclic vectors for uncountable families of operators, and prove that the family of composition operators C? on H2(D), where ? is a disk automorphism having +1 as attractive fixed point, has a residual set of common hypercyclic vectors.  相似文献   

15.
We consider analytic self‐maps φ on $\mathbf {D}$ and prove that the composition operator Cφ acting on $H_{v}^0$ is hypercyclic if φ is an automorphism or a hyperbolic non‐automorphic symbol with no fixed point. We give examples of weights v and parabolic non‐automorphisms φ on $\mathbf {D}$ which yield non‐hypercyclic composition operators Cφ on $H_{v}^0$.  相似文献   

16.
We study the dynamic behaviour of weighted composition operators on the space of holomorphic functions on a simply connected plane domain and endowed with the compact open topology. Any such operator is weakly supercyclic if and only if it is topologically mixing and, when the weight is bounded, if and only if the operator has a hypercyclic subspace. We also provide conditions on the symbols that ensure the operator to be Devaney-chaotic and to have a frequently hypercyclic subspace.  相似文献   

17.
Inspired by a recent result that a weakly hypercyclic operator may fail to be norm hypercyclic, we show there exists a weakly supercyclic operator that fails to be norm supercyclic. Moreover, despite a classical result of Hilden and Wallen that every unilateral weighted backward shift is supercyclic, we show such an operator may have a weakly supercyclic vector that is not a norm supercyclic vector. In addition to these results, we extend a result of Kitai by showing a hyponormal operator cannot be weakly hypercyclic.  相似文献   

18.
In this paper we extend the notion of a locally hypercyclic operator to that of a locally hypercyclic tuple of operators. We then show that the class of hypercyclic tuples of operators forms a proper subclass to that of locally hypercyclic tuples of operators. What is rather remarkable is that in every finite dimensional vector space over R or C, a pair of commuting matrices exists which forms a locally hypercyclic, non-hypercyclic tuple. This comes in direct contrast to the case of hypercyclic tuples where the minimal number of matrices required for hypercyclicity is related to the dimension of the vector space. In this direction we prove that the minimal number of diagonal matrices required to form a hypercyclic tuple on Rn is n+1, thus complementing a recent result due to Feldman.  相似文献   

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