Related papers: On the Brown--Shields conjecture for cyclicity in …
A radially weighted Besov space $H$ is a space of holomorphic functions on the unit ball $\mathbb{B}_d \subseteq \mathbb{C}^d$ whose $N$-th radial derivative is square integrable with respect to a given admissible radial measure. We write…
Every function in the Dirichlet space on the unit disc has an inner/outer factorization. We study which inner functions occur in this way. For Blaschke products, this is the well known question of which subsets of the disc are zero sets for…
For spaces of analytic functions defined on an open set in $\mathbb{C}^n$ that satisfy certain nice properties, we show that operators that preserve shift-cyclic functions are necessarily weighted composition operators. Examples of spaces…
We prove the property that a function is cyclic (resp., non-cyclic) is not preserved by norm convergence in Dirichlet-type spaces $D_\alpha$, and show how other significant quantities for cyclicity do remain preserved under the limit of…
We study the cyclicity of polynomials in Poletsky-Stessin weighted Bergman spaces on various domains in $\mathbb{C}^2$, including the unit ball, the bidisk, and the complex ellipsoid. To this end, we introduce a natural extension of the…
We first generalize the results of Le\'on and M\"uller [Studia Math. 175(1) 2006] on hypercyclic subspaces to sequences of operators on Fr\'echet spaces with a continuous norm. Then we study the particular case of iterates of an operator T…
In this paper, we study the cyclicity of the shift operator $S$ acting on a Banach space $\X$ of analytic functions on the open unit disc $\D$. We develop a general framework where a method based on a corona theorem can be used to show that…
The aim of these notes is to discuss the completeness of the dilated systems in a most general framework of an arbitrary sequence lattice $X$, including weighted $\ell^p$ spaces. In particular, general multiplicative and completely…
We study the cyclicity in weighted $\ell^p(\mathbb{Z})$ spaces. For $p \geq 1$ and $\beta \geq 0$, let $\ell^p\_\beta(\mathbb{Z})$ be the space of sequences $u=(u\_n)\_{n\in \mathbb{Z}}$ such that $(u\_n |n|^{\beta})\in \ell^p(\mathbb{Z})…
It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard $d$-shift acting on the full Fock space. Upon settling for a softer relation than unitary…
We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(\mu)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $\mu$ is a measure supported in the closed…
We prove that smooth $C^\infty$ functions are dense in weighted fractional Sobolev spaces on an arbitrary open set, under some mild conditions on the weight. We also obtain a~similar result in non-weighted spaces defined by some kernel…
We consider weak-star closed invariant subspaces of the shift operator in the classical Bloch space. We prove that any bounded analytic function decomposes into two factors, one which is cyclic and another one generating a proper shift…
We extend results of B\'en\'eteau, Condori, Liaw, Seco, and the author concerning cyclic vectors in Dirichlet-type spaces to the setting of the unit ball, identifying some classes of cyclic and non-cyclic functions, and noting the necessity…
The classical Dirichlet space is a complete Pick space, hence by a theorem of Agler and McCarthy, there exists an embedding $b$ of the unit disc into a $d$-dimensional ball such that composition with $b$ realizes the Dirichlet space as a…
In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional…
For a (not necessarily locally convex) topological vector space $\mathcal{X}$ of holomorphic functions in one complex variable, we show that the shift invariant subspace generated by a set of polynomials is $\mathcal{X}$ if and only if…
We introduce new quantitative measures for cyclicity in radially weighted Besov spaces, including the Drury-Arveson space, by defining cyclicity indices based on potential theory and capacity. Extensions to non-commutative settings are…
This article consists of two connected parts. In the first part, we study the shift invariant subspaces in certain $\mathcal{P}^2(\mu)$-spaces, which are the closures of analytic polynomials in the Lebesgue spaces $\mathcal{L}^2(\mu)$…
We consider the space of real-valued continuously differentiable functions on a compact subset of a euclidean space. We characterize the completeness of this space and prove that the space of restrictions of continuously differentiable…