On the Salas Theorem and Hypercyclicity of f(T)

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).     

Pathological hypercyclic operators

We exhibit a hypercyclic operator whose square is not hypercyclic. Our operator is necessarily unbounded since a result of S. Ansari asserts that powers of a hypercyclic bounded operator are also hypercyclic. We also exhibit an unbounded Hilbert space operator whose non-zero vectors are hypercyclic. Received: 19 March 2005; revised: 18 July 2005     

Invariant measures for frequently hypercyclic operators

We investigate frequently hypercyclic and chaotic linear operators from a measure-theoretic point of view. Among other things, we show that any frequently hypercyclic operator T acting on a reflexive Banach space admits an invariant probability measure with full support, which may be required to vanish on the set of all periodic vectors for T  ; that there exist frequently hypercyclic operators on the sequence space _c0$c_0$ admitting no ergodic measure with full support; and that if an operator admits an ergodic measure with full support, then it has a comeager set of distributionally irregular vectors. We also give some necessary and sufficient conditions (which are satisfied by all the known chaotic operators) for an operator T to admit an invariant measure supported on the set of its hypercyclic vectors and belonging to the closed convex hull of its periodic measures. Finally, we give a Baire category proof of the fact that any operator with a perfectly spanning set of unimodular eigenvectors admits an ergodic measure with full support.     

The norm of a composition operator with linear symbol acting on the Dirichlet space

We obtain a representation for the norm of a composition operator on the Dirichlet space induced by a map of the form φ(z)=az+b. We compare this result to an upper bound for ‖Cφ‖ that is valid whenever φ is univalent. Our work relies heavily on an adjoint formula recently discovered by Gallardo-Gutiérrez and Montes-Rodríguez.     

On Products of Hypercyclic Semigroups

If (S(t)) is a hypercyclic (discrete or continuous) semigroup of linear operators, it is known that (S(t) ⊗ S(t)) is hypercyclic, if and only if (S(t)) satisfies the so-called hypercyclicity criterion (HCC). We give a strengthened version of the hypercyclicity criterion, which we call recurrent hypercyclicity criterion (RHCC). It is a necessary and sufficient condition on a semigroup (S(t)), such that the product with any semigroup (T(t)) satisfying HCC is again hypercyclic. The RHCC is inherited by products (obviously) and by subsemigroups. Any chaotic semigroup satisfies the RHCC.     

Common Hypercyclic Vectors and the Hypercyclicity Criterion

An operator on a separable, infinite dimensional Banach space satisfies the Hypercyclicity Criterion if and only if the associated left multiplication operator is hypercyclic; see [14], [16], [29]. By examining paths of operators where each operator along the path satisfies the criterion, we provide necessary and sufficient conditions for a path of left multiplication operators to have an SOT-dense set of common hypercyclic vectors. As a corollary, we establish a natural sufficient condition for a path of operators to have a common hypercyclic subspace.     

Recurrence properties of hypercyclic operators

We generalize the notions of hypercyclic operators, \(\mathfrak {U}\)-frequently hypercyclic operators and frequently hypercyclic operators by introducing a new concept in linear dynamics, namely \(\mathcal {A}\)-hypercyclicity. We then state an \(\mathcal {A}\)-hypercyclicity criterion, inspired by the hypercyclicity criterion and the frequent hypercyclicity criterion, and we show that this criterion characterizes the \(\mathcal {A}\)-hypercyclicity for weighted shifts. We also investigate which density properties can the sets \({N(x, U)=\{n\in \mathbb {N}\ ; \ T^nx\in U\}}\) have for a given hypercyclic operator, and we study the new notion of reiteratively hypercyclic operators.     

Disjointness in hypercyclicity

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.     

The Density of the Hypercyclic Operators in the Strong Operator Topology

In this note we give a simple proof of the fact that the set of all hypercyclic operators on a separable Hilbert space is dense in the strong operator topology.     

Perturbations of hypercyclic vectors

We show that a linear operator can have an orbit that comes within a bounded distance of every point, yet is not dense. We also prove that such an operator must be hypercyclic. This gives a more general form of the hypercyclicity criterion. We also show that a sufficiently small perturbation of a hypercyclic vector is still hypercyclic.     

Dynamics of the differentiation operator on weighted spaces of entire functions

The continuity of the differentiation operator on weighted Banach spaces of entire functions with sup-norm has been characterized recently by Harutyunyan and Lusky. We give necessary and sufficient conditions to ensure that the differentiation operator on these weighted Banach spaces of entire functions is hypercyclic or chaotic, when it is continuous. This research was partially supported by MEC and FEDER Project MTM2007-62643.     

<Emphasis Type="Italic">q</Emphasis>-Frequent hypercyclicity in spaces of operators

We provide conditions for a linear map of the form \(C_{R,T}(S)=RST\) to be q-frequently hypercyclic on algebras of operators on separable Banach spaces. In particular, if R is a bounded operator satisfying the q-frequent hypercyclicity criterion, then the map \(C_{R}(S)=RSR^*\) is shown to be q-frequently hypercyclic on the space \(\mathcal {K}(H)\) of all compact operators and the real topological vector space \(\mathcal {S}(H)\) of all self-adjoint operators on a separable Hilbert space H. Further we provide a condition for \(C_{R,T}\) to be q-frequently hypercyclic on the Schatten von Neumann classes \(S_p(H)\). We also characterize frequent hypercyclicity of \(C_{M^*_\varphi ,M_\psi }\) on the trace-class of the Hardy space, where the symbol \(M_\varphi \) denotes the multiplication operator associated to \(\varphi \).     

On the minimal number of matrices which form a locally hypercyclic, non-hypercyclic tuple

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.     

Positive operators on Krein spaces

A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literature. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace. Dedicated to the memory of M.G. Krein (1907–1989)     

Common hypercyclic vectors for the conjugate class of a hypercyclic operator

Given a separable, infinite dimensional Hilbert space, it was recently shown by the authors that there is a path of chaotic operators, which is dense in the operator algebra with the strong operator topology, and along which every operator has the exact same dense Gδ set of hypercyclic vectors. In the present work, we show that the conjugate set of any hypercyclic operator on a separable, infinite dimensional Banach space always contains a path of operators which is dense with the strong operator topology, and yet the set of common hypercyclic vectors for the entire path is a dense Gδ set. As a corollary, the hypercyclic operators on such a Banach space form a connected subset of the operator algebra with the strong operator topology.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

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.     

Subspace hypercyclicity

A bounded linear operator T on Hilbert space is subspace-hypercyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace in a relatively dense set. We construct examples to show that subspace-hypercyclicity is interesting, including a nontrivial subspace-hypercyclic operator that is not hypercyclic. There is a Kitai-like criterion that implies subspace-hypercyclicity and although the spectrum of a subspace-hypercyclic operator must intersect the unit circle, not every component of the spectrum will do so. We show that, like hypercyclicity, subspace-hypercyclicity is a strictly infinite-dimensional phenomenon. Additionally, compact or hyponormal operators can never be subspace-hypercyclic.     

REAL AND COMPLEX OPERATOR IDEALS

Abstract

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or 2-summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one-to-one correspondence between “real properties” and “complex properties” defining an operator ideal? In other words, does there exist for every real operator ideal a uniquely determined corresponding complex ideal and vice versa?

Unfortunately, we are not abel to give a final answer. Nevertheless, some preliminary results are obtained. In particular, we construct for every real operator ideal a corresponding complex operator ideal and for every complex operator ideal a corresponding real one. However, we conjecture that there exists a complex operator ideal which can not be obtained from a real one by this construction.

The following approach is based on the observation that every complex Banach space can be viewed as a real Banach space with an isometry acting on it like the scalar multiplication by the imaginary unit i.     

A remark on complete positivity of elementary operators

We show that many of the features of the theory of hypercyclic and supercyclic operators extend to that of finitely hypercyclic/supercyclic operators. In particular, subnormal operators, Banach space isometries, and thereforeC ₁ contractions are not finitely supercyclic.