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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.
Lawrence A. Harris 《Journal of Mathematical Analysis and Applications》2010,368(1):374-381
Our object is to present an independent proof of the extension of V.A. Markov's theorem to Gâteaux derivatives of arbitrary order for continuous polynomials on any real normed linear space. The statement of this theorem differs little from the classical case for the real line except that absolute values are replaced by norms. Our proof depends only on elementary computations and explicit formulas and gives a new proof of the classical theorem as a special case. Our approach makes no use of the classical polynomial inequalities usually associated with Markov's theorem. Instead, the essential ingredients are a Lagrange interpolation formula for the Chebyshev nodes and a Christoffel-Darboux identity for the corresponding bivariate Lagrange polynomials. We use these tools to extend a single variable inequality of Rogosinski to the case of two real variables. The general Markov theorem is an easy consequence of this. 相似文献
3.
Stanislav Shkarin 《Journal of Functional Analysis》2010,258(1):132-160
We treat the question of existence of common hypercyclic vectors for families of continuous linear operators. It is shown that for any continuous linear operator T on a complex Fréchet space X and a set Λ⊆R+×C which is not of zero three-dimensional Lebesgue measure, the family has no common hypercyclic vectors. This allows to answer negatively questions raised by Godefroy and Shapiro and by Aron. We also prove a sufficient condition for a family of scalar multiples of a given operator on a complex Fréchet space to have a common hypercyclic vector. It allows to show that if and φ∈H∞(D) is non-constant, then the family has a common hypercyclic vector, where Mφ:H2(D)→H2(D), Mφf=φf, and , providing an affirmative answer to a question by Bayart and Grivaux. Finally, extending a result of Costakis and Sambarino, we prove that the family has a common hypercyclic vector, where Tbf(z)=f(z−b) acts on the Fréchet space H(C) of entire functions on one complex variable. 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.
We introduce a notion of disjointness for finitely many hypercyclic operators acting on a common space, notion that is weaker than Furstenberg's disjointness of fluid flows. We provide a criterion to construct disjoint hypercyclic operators, that generalizes some well-known connections between the Hypercyclicity Criterion, hereditary hypercyclicity and topological mixing to the setting of disjointness in hypercyclicity. We provide examples of disjoint hypercyclic operators for powers of weighted shifts on a Hilbert space and for differentiation operators on the space of entire functions on the complex plane. 相似文献
6.
Jochen Wengenroth 《Proceedings of the American Mathematical Society》2003,131(6):1759-1761
We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].
7.
Geraldo Botelho Daniel Pellegrino 《Journal of Mathematical Analysis and Applications》2006,321(1):50-58
If X is a Banach space with a normalized unconditional Schauder basis (xn), we define whenever and obtain estimates for μX,(xn) when every continuous m-homogeneous polynomial from X into Y is absolutely (q,1) summing. Our results provide new information on coincidence situations between the space of absolutely summing m-homogeneous polynomials and the whole space of continuous m-homogeneous polynomials. In particular, when m=1, we obtain new contributions to the linear theory of absolutely summing operators. 相似文献
8.
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
9.
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
10.
11.
The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive
forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is
a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators
can be carried over to this case.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
12.
In this paper, we study the matrix multiplication operators on Banach function spaces and discuss their applications in semigroups
for solving the abstract Cauchy problem. 相似文献
13.
We provide in this paper a direct and constructive proof of the following fact: for a Banach space there are bounded linear operators having hypercyclic vectors if and only if is separable and dim. This is a special case of a recent result, which in turn solves a problem proposed by S. Rolewicz.
14.
Rich Stankewitz 《Proceedings of the American Mathematical Society》1999,127(10):2889-2898
Let be a semigroup of rational functions of degree at least two, under composition of functions. Suppose that contains two polynomials with non-equal Julia sets. We prove that the smallest closed subset of the Riemann sphere which contains at least three points and is completely invariant under each element of , is the sphere itself.
15.
Geraldo Botelho Daniel M. Pellegrino 《Proceedings of the American Mathematical Society》2006,134(6):1743-1751
It is well known that 2-homogeneous polynomials on -spaces are 2-dominated. Motivated by the fact that related coincidence results are possible only for polynomials defined on symmetrically regular spaces, we investigate the situation in several classes of symmetrically regular spaces. We prove a number of non-coincidence results which makes us suspect that there is no infinite dimensional Banach space such that every scalar-valued homogeneous polynomial on is -dominated for every .
16.
Leonardo Pellegrini 《Journal of Mathematical Analysis and Applications》2007,332(1):272-278
In this work we present some conditions of equivalence for the existence of a monomial basis in spaces of homogeneous polynomials on Banach spaces. 相似文献
17.
Pá draig Kirwan Raymond A. Ryan 《Proceedings of the American Mathematical Society》1998,126(4):1023-1029
We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.
18.
J.M. Calabuig E.A. Sánchez Pérez 《Journal of Mathematical Analysis and Applications》2010,364(1):88-136
In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces. 相似文献
19.
M.A. El-Gebeily K. Al Shammari 《Journal of Mathematical Analysis and Applications》2009,358(2):345-354
We establish some existence results for the nonlinear problem Au=f in a reflexive Banach space V, without and with upper and lower solutions. We then consider the application of the quasilinearization method to the above mentioned problem. Under fairly general assumptions on the nonlinear operator A and the Banach space V, we show that this problem has a solution that can be obtained as the strong limit of two quadratically convergent monotone sequences of solutions of certain related linear equations. 相似文献
20.
We study the boundedness and the compactness of composition operators on some Banach function spaces such as absolutely continuous Banach function spaces on a -finite measure space, Lorentz function spaces on a -finite measure space and rearrangement invariant spaces on a resonant measure space. In addition, we study some properties of the spectra of a composition operator on the general Banach function spaces.