An eigenvalue criterion for hypercyclicity due to the first author is improved. As a consequence, some new sufficient conditions for a sequence of infinite order linear differential operators to be hypercyclic on the space of holomorphic functions on certain domains of ${ℂ}^N$ are shown. Moreover, several necessary conditions are furnished. The equicontinuity of a family of operators as above is also studied, and it is characterized if the domain is ${ℂ}^N$. The results obtained extend or improve earlier work...

We study the ``smallness'' of the set of non-hypercyclic vectors for some classical hypercyclic operators.

Let T be a continuous linear operator on a Hausdorff topological vector space 𝓧 over the field ℂ. We show that if T is N-supercyclic, i.e., if 𝓧 has an N-dimensional subspace whose orbit under T is dense in 𝓧, then T* has at most N eigenvalues (counting geometric multiplicity). We then show that N-supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an N-dimensional subspace cannot be dense in an (N+1)-dimensional space. Finally, we show that a subnormal operator...

Let T be a continuous linear operator acting on a Banach space X. We examine whether certain fundamental results for hypercyclic operators are still valid in the Cesàro hypercyclicity setting. In particular, in connection with the somewhere dense orbit theorem of Bourdon and Feldman, we show that if for some vector x ∈ X the set Tx,T²/2 x,T³/3 x, ... is somewhere dense then for every 0 < ε < 1 the set (0,ε)Tx,T²/2 x,T³/3 x,... is dense in X. Inspired by a result of Feldman, we also prove...

We improve a result of Charpentier [Studia Math. 198 (2010)]. We prove that even on Fréchet spaces with a continuous norm, the existence of only one restrictively universal series implies the existence of a closed infinite-dimensional subspace of restrictively universal series.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\infty }$ be a sequence of positive numbers and $1\le p<\infty$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }\widehat{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\infty }{\left|\widehat{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\infty$. We investigate strict cyclicity of $H_p^{\infty }\left(\beta \right)$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^p\left(\beta \right)$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H_p^{\infty }\left(\beta \right)$. We also give a necessary condition for a composition operator to be bounded on $H^p\left(\beta \right)$ when $H_p^{\infty }\left(\beta \right)$ is strictly cyclic.

Two characterizations of the reductivity of a cyclic normal operator in Hilbert space are proved: the equality of the sets of cyclic and *-cyclic vectors, and the equality L²(μ) = P²(μ) for every measure μ equivalent to the scalar-valued spectral measure of the operator. A cyclic subnormal operator is reductive if and only if the first condition is satisfied. Several consequences are also presented.

For an unbounded operator S the question whether its subnormality can be built up from that of every $S_f$, the restriction of S to a cyclic space generated by f in the domain of S, is analyzed. Though the question at large has been left open some partial results are presented and a possible way to prove it is suggested as well.

We show that the Angle Criterion for testing supercyclic vectors depends in an essential way on the geometrical properties of the underlying space. In particular, we exhibit non-supercyclic vectors for the backward shift acting on c₀ that still satisfy such a criterion. Nevertheless, if ℬ is a locally uniformly convex Banach space, the Angle Criterion yields an equivalent condition for a vector to be supercyclic. Furthermore, we prove that local uniform convexity cannot be weakened to strict convexity....

An operator (linear and continuous) in a Fréchet space is hypercyclic if there exists a vector whose orbit under the operator is dense. If the scalar multiples of the elements in the orbit are dense, the operator is supercyclic. We give, for Fréchet space operators, a Supercyclicity Criterion reminiscent of the Hypercyclicity Criterion. We characterize the supercyclic bilateral weighted shifts in terms of their weight sequences. As a consequence, we show that a bilateral weighted shift is supercyclic...

We prove a Supercyclicity Criterion for a continuous linear mapping that is defined on the operator algebra of a separable Banach space ℬ. Our result extends a recent result on hypercyclicity on the operator algebra of a Hilbert space. This kind of result is a powerful tool to analyze the structure of supercyclic vectors of a supercyclic operator that is defined on ℬ. For instance, as a consequence of the main result, we give a very simple proof of the recently established fact that certain supercyclic...