This article explores the notions of $\mathcal {F}$-transitivity and topological $\mathcal {F}$-recurrence for backward shift operators on weighted $\ell ^p$-spaces and $c_0$-spaces on directed trees, where $\mathcal {F}$ represents a Furstenberg family of subsets of $\mathbb {N}_0$. In particular, we establish the equivalence between recurrence and hypercyclicity of these operators on unrooted directed trees. For rooted directed trees, a backward shift operator is hypercyclic if and only if it possesses an orbit of a bounded subset that is weakly dense.

We are interested in the optimal growth in terms of Lp-averages of hypercyclic and $\mathcal{U}$-frequently hypercyclic functions for some weighted Taylor shift operators acting on the space of analytic functions on the unit disc. We unify the results obtained by considering intermediate notions of upper frequent hypercyclicity between $\mathcal{U}$-frequent hypercyclicity and hypercyclicity.

We prove that a finite set of natural numbers J satisfies that $J\cup \{0\}$ is not Sidon if and only if for any operator T, the disjoint hypercyclicity of $\{T^j:j\in J\}$ implies that T is weakly mixing. As an application we show the existence of a non-weakly mixing operator T such that $T\oplus T^2\oplus\cdots \oplus T^n$ is hypercyclic for every n.

We study a class of left-invertible operators which we call weakly concave operators. It includes the class of concave operators and some subclasses of expansive strict $m$-isometries with $m > 2$. We prove a Wold-type decomposition for weakly concave operators. We also obtain a Berger–Shaw-type theorem for analytic finitely cyclic weakly concave operators. The proofs of these results rely heavily on a spectral dichotomy for left-invertible operators. It provides a fairly close relationship, written in terms of the reciprocal automorphism of the Riemann sphere, between the spectra of a left-invertible operator and any of its left inverses. We further place the class of weakly concave operators, as the term $\mathcal {A}_1$, in the chain $\mathcal {A}_0 \subseteq \mathcal {A}_1 \subseteq \ldots \subseteq \mathcal {A}_{\infty }$ of collections of left-invertible operators. We show that most of the aforementioned results can be proved for members of these classes. Subtleties arise depending on whether the index $k$ of the class $\mathcal {A}_k$ is finite or not. In particular, a Berger–Shaw-type theorem fails to be true for members of $\mathcal {A}_{\infty }$. This discrepancy is better revealed in the context of $C^*$- and $W^*$-algebras.

Given a sequence $\varrho =(r_n)_n\in [0,1)$ tending to $1$, we consider the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ of Abel universal functions consisting of holomorphic functions f in the open unit disk $\mathbb {D}$ such that for any compact set K included in the unit circle ${\mathbb {T}}$, different from ${\mathbb {T}}$, the set $\{z\mapsto f(r_n \cdot )\vert _K:n\in \mathbb {N}\}$ is dense in the space ${\mathcal {C}}(K)$ of continuous functions on K. It is known that the set ${\mathcal {U}}_A({\mathbb {D}},\varrho )$ is residual in $H(\mathbb {D})$. We prove that it does not coincide with any other classical sets of universal holomorphic functions. In particular, it is not even comparable in terms of inclusion to the set of holomorphic functions whose Taylor polynomials at $0$ are dense in ${\mathcal {C}}(K)$ for any compact set $K\subset {\mathbb {T}}$ different from ${\mathbb {T}}$. Moreover, we prove that the class of Abel universal functions is not invariant under the action of the differentiation operator. Finally, an Abel universal function can be viewed as a universal vector of the sequence of dilation operators $T_n:f\mapsto f(r_n \cdot )$ acting on $H(\mathbb {D})$. Thus, we study the dynamical properties of $(T_n)_n$ such as the multiuniversality and the (common) frequent universality. All the proofs are constructive.

The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$, the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$. It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$-spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]

We show that there exists an invertible frequently hypercyclic operator on $\ell ^1(\mathbb {N})$ whose inverse is not frequently hypercyclic.

For arbitrary closed countable subsets Z of the unit circle examples of topologically mixing operators on Hilbert spaces are given which have a densely spanning set of eigenvectors with unimodular eigenvalues restricted to Z. In particular, these operators cannot be ergodic in the Gaussian sense.

Bayart and Ruzsa [Difference sets and frequently hypercyclic weighted shifts. Ergod. Th. & Dynam. Sys.35 (2015), 691–709] have recently shown that every frequently hypercyclic weighted shift on $\ell ^p$ is chaotic. This contrasts with an earlier result of Bayart and Grivaux [Frequently hypercyclic operators. Trans. Amer. Math. Soc.358 (2006), 5083–5117], who constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . We first generalize the Bayart–Ruzsa theorem to all Banach sequence spaces in which the unit sequences form a boundedly complete unconditional basis. We then study the relationship between frequent hypercyclicity and chaos for weighted shifts on Fréchet sequence spaces, in particular, on Köthe sequence spaces, and then on the special class of power series spaces. We obtain, rather curiously, that every frequently hypercyclic weighted shift on $H(\mathbb {D})$ is chaotic, while $H(\mathbb {C})$ admits a non-chaotic frequently hypercyclic weighted shift.

For any $\alpha \in \mathbb {R},$ we consider the weighted Taylor shift operators $T_{\alpha }$ acting on the space of analytic functions in the unit disc given by $T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$

$$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$$

$T_\alpha $

$L^p$

$1\leq p\leq +\infty $

We study the dynamics induced by homogeneous polynomials on Banach spaces. It is known that no homogeneous polynomial defined on a Banach space can have a dense orbit. We show a simple and natural example of a homogeneous polynomial with an orbit that is at the same time $\unicode[STIX]{x1D6FF}$-dense (the orbit meets every ball of radius $\unicode[STIX]{x1D6FF}$), weakly dense and such that $\unicode[STIX]{x1D6E4}\cdot \text{Orb}_{P}(x)$ is dense for every $\unicode[STIX]{x1D6E4}\subset \mathbb{C}$ that either is unbounded or has 0 as an accumulation point. Moreover, we generalize the construction to arbitrary infinite-dimensional separable Banach spaces. To prove this, we study Julia sets of homogeneous polynomials on Banach spaces.

A well-known result in the area of dynamical systems asserts that any invertible hyperbolic operator on any Banach space is structurally stable. This result was originally obtained by Hartman in 1960 for operators on finite-dimensional spaces. The general case was independently obtained by Palis and Pugh around 1968. We will exhibit a class of examples of structurally stable operators that are not hyperbolic, thereby showing that the converse of the above-mentioned result is false in general. We will also prove that an invertible operator on a Banach space is hyperbolic if and only if it is expansive and has the shadowing property. Moreover, we will show that if a structurally stable operator is expansive, then it must be uniformly expansive. Finally, we will characterize the weighted shifts on the spaces $c_{0}(\mathbb{Z})$ and $\ell _{p}(\mathbb{Z})$ ($1\leq p<\infty$) that satisfy the shadowing property.

We improve a recent result by giving the optimal conclusion both to the frequent universality criterion and the frequent hypercyclicity criterion using the notion of $A$-densities, where $A$ refers to some weighted densities sharper than the natural lower density. Moreover, we construct an operator which is logarithmically frequently hypercyclic but not frequently hypercyclic.

It is not known whether the inverse of a frequently hypercyclic bilateral weighted shift on c0(ℤ) is again frequently hypercyclic. We show that the corresponding problem for upper frequent hypercyclicity has a positive answer. We characterise, more generally, when bilateral weighted shifts on Banach sequence spaces are (upper) frequently hypercyclic.

Let $1\le p<\infty $, and let $G$ be a discrete group. We give a sufficient and necessary condition for weighted translation operators on the Lebesgue space ${{\ell }^{p}}(G)$ to be densely disjoint hypercyclic. The characterization for the dual of a weighted translation to be densely disjoint hypercyclic is also obtained.

We prove the existence of a (in fact many) holomorphic function $f$ in ${{\mathbb{C}}^{d}}$ such that, for any $a\ne 0$, its translations $f(\cdot +na)$ are dense in $H({{\mathbb{C}}^{d}})$.

In this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.