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1.
Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces 总被引:1,自引:0,他引:1
By a recent result of M. De La Rosa and C. Read, there exist hypercyclic Banach space operators which do not satisfy the Hypercyclicity Criterion. In the present paper, we prove that such operators can be constructed on a large class of Banach spaces, including or . 相似文献
2.
Daniel Carando Verónica Dimant Santiago Muro 《Journal of Mathematical Analysis and Applications》2007,336(2):1324-1340
A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class. 相似文献
3.
We provide an alternative proof to those by Shkarin and by Bayart and Matheron that the operator D of complex differentiation supports a hypercyclic algebra on the space of entire functions. In particular we obtain hypercyclic algebras for many convolution operators not induced by polynomials, such as , , or , where . In contrast, weighted composition operators on function algebras of analytic functions on a plane domain fail to support supercyclic algebras. 相似文献
4.
Antonio Bonilla Pedro J. Miana 《Proceedings of the American Mathematical Society》2008,136(2):519-528
Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Banach space admits a topologically mixing uniformly continuous cosine family.
5.
Frédéric Bayart 《Journal of Functional Analysis》2019,276(11):3441-3467
We prove the existence of algebras of hypercyclic vectors in three cases: convolution operators, composition operators, and backward shift operators. 相似文献
6.
Henrik Petersson 《Journal of Approximation Theory》2006,138(2):168-183
A sequence of continuous linear operators is said to be hypercyclic if there exists a vector , called hypercyclic for , such that {Tnx:n0} is dense. A continuous linear operator, acting on some suitable function space, is PDE-preserving for a given set of convolution operators, when it map every kernel set for these operators invariantly. We establish hypercyclic sequences of PDE-preserving operators on , and study closed infinite-dimensional subspaces of, except for zero, hypercyclic vectors for these sequences. 相似文献
7.
8.
Henrik Petersson 《Journal of Mathematical Analysis and Applications》2006,319(2):764-782
A continuous linear operator is hypercyclic if there is an x∈X such that the orbit {Tnx} is dense, and such a vector x is said to be hypercyclic for T. Recent progress show that it is possible to characterize Banach space operators that have a hypercyclic subspace, i.e., an infinite dimensional closed subspace H⊆X of, except for zero, hypercyclic vectors. The following is known to hold: A Banach space operator T has a hypercyclic subspace if there is a sequence (ni) and an infinite dimensional closed subspace E⊆X such that T is hereditarily hypercyclic for (ni) and Tni→0 pointwise on E. In this note we extend this result to the setting of Fréchet spaces that admit a continuous norm, and study some applications for important function spaces. As an application we also prove that any infinite dimensional separable Fréchet space with a continuous norm admits an operator with a hypercyclic subspace. 相似文献
9.
We show that any countable family of operators of the form P(B), where P is a non-constant polynomial and B is the backward shift operator on ω, the countably infinite product of lines, has a common hypercyclic subspace. 相似文献
10.
Gustavo Fernández André Arbex Hallack 《Journal of Mathematical Analysis and Applications》2005,309(1):52-55
In this paper we correct a proof by Aron and Markose in [R. Aron, D. Markose, On universal functions, J. Korean Math. Soc. 41 (2004) 65-76] for the hypercyclicity of the operator given by , in the case b≠0. 相似文献
11.
Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, Köthe (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems. 相似文献
12.
Nathan S. Feldman 《Journal of Mathematical Analysis and Applications》2008,346(1):82-98
In this paper we prove that there are hypercyclic (n+1)-tuples of diagonal matrices on Cn and that there are no hypercyclic n-tuples of diagonalizable matrices on Cn. We use the last result to show that there are no hypercyclic subnormal tuples in infinite dimensions. We then show that on real Hilbert spaces there are tuples with somewhere dense orbits that are not dense, but we also give sufficient conditions on a tuple to insure that a somewhere dense orbit, on a real or complex space, must be dense. 相似文献
13.
Paul S. Bourdon Joel H. Shapiro 《Transactions of the American Mathematical Society》2000,352(11):5293-5316
The backward shift on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: ``Which operators that commute with inherit its hypercyclicity?' We show that the problem reduces to the study of operators of the form where is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely is allowed to approach the unit circle as .
14.
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
15.
Stanislav Shkarin 《Journal of Mathematical Analysis and Applications》2011,382(2):516-522
We show that for every supercyclic strongly continuous operator semigroup {Tt}t?0 acting on a complex F-space, every Tt with t>0 is supercyclic. Moreover, the set of supercyclic vectors of each Tt with t>0 is exactly the set of supercyclic vectors of the entire semigroup. 相似文献
16.
17.
George Costakis Demetris Hadjiloucas 《Proceedings of the American Mathematical Society》2008,136(3):937-946
In this article we look at skew-products of multiples of the backward shift and examine conditions under which the skew-product is topologically transitive or hypercyclic in the second coordinate. We also give an application of the theory to iterated function systems of multiples of backward shift operators.
18.
We characterize disjoint hypercyclicity and disjoint supercyclicity of finitely many linear fractional composition operators acting on spaces of holomorphic functions on the unit disc, answering a question of Bernal-González. We also study mixing and disjoint mixing behavior of projective limits of endomorphisms of a projective spectrum. In particular, we show that a linear fractional composition operator is mixing on the projective limit of the Sv spaces strictly containing the Dirichlet space if and only if the operator is mixing on the Hardy space. 相似文献
19.
Héctor N. Salas 《Journal of Mathematical Analysis and Applications》2011,374(1):106-117
Let E be a separable Fréchet space. The operators T1,…,Tm are disjoint hypercyclic if there exists x∈E such that the orbit of (x,…,x) under (T1,…,Tm) is dense in E×?×E. We show that every separable Banach space E admits an m-tuple of bounded linear operators which are disjoint hypercyclic. If, in addition, its dual E∗ is separable, then they can be constructed such that are also disjoint hypercyclic. 相似文献
20.
In this note, we show that every infinite-dimensional separable Fréchet space admitting a continuous norm supports an operator for which there is an infinite-dimensional closed subspace consisting, except for zero, of hypercyclic vectors. The family of such operators is even dense in the space of bounded operators when endowed with the strong operator topology. This completes the earlier work of several authors.