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Karim Kellay

Publications and source records attributed to Karim Kellay.

31 records · Page 2Linked to original sources
arXiv2015

Dirichlet spaces with superharmonic weights and de Branges-Rovnyak spaces

We consider Dirichlet spaces with superharmonic weights. This class contains both the harmonic weights and the power weights. Our main result is a characterization of the Dirichlet spaces with superharmonic weights that can be identified as de Branges-Rovnyak spaces. As an application, we obtain the dilation inequality \[ {\cal D}_ω(f_r)\le \frac{2r}{1+r}{\cal D}_ω(f) \qquad(0\le r<1), \] where ${\cal D}_ω$ denotes the Dirichlet integral with superharmonic weight $ω$, and $f_r(z):=f(rz)$ is the $r$-dilation of the holomorphic function $f$.

math.CV↗
arXiv2015

Geometry of reproducing kernels in model spaces near the boundary

We study two geometric properties of reproducing kernels in model spaces $K\_θ$where $θ$ is an inner function in the disc: overcompleteness and existence of uniformly minimalsystems of reproducing kernels which do not contain Riesz basic sequences. Both of these properties are related to the notion of the Ahern--Clark point. It is shown that "uniformly minimal non-Riesz"$ $ sequences of reproducing kernelsexist near each Ahern--Clark point which is not an analyticity point for $θ$, whileovercompleteness may occur only near the Ahern--Clark points of infinite orderand is equivalent to a "zero localization property". In this context the notion ofquasi-analyticity appears naturally, and as a by-product of our results we give conditions in thespirit of Ahern--Clark for the restriction of a model space to a radius to be a class ofquasi-analyticity.

math.CV↗
arXiv2014

A dynamical system approach to Heisenberg Uniqueness Pairs

Let $Λ$ be a set of lines in $\mathbb{R}^2$ that intersect at the origin. For $Γ\subset\mathbb{R}^2$ a smooth curve, we denote by $\mathcal{A}\mathcal{C}(Γ)$ the subset of finite measures on $Γ$ that are absolutely continuous with respect to arc length on $Γ$. For such a $μ$, $\widehatμ$ denotes the Fourier transform of $μ$. Following Hendenmalm and Montes-Rodríguez, we will say that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair if $μ\in\mathcal{A}\mathcal{C}(Γ)$ is such that $\widehatμ=0$ on $Λ$, then $μ=0$. The aim of this paper is to provide new tools to establish this property. To do so, we will reformulate the fact that $\widehatμ$ vanishes on $Λ$ in terms of an invariance property of $μ$ induced by $Λ$. This leads us to a dynamical system on $Γ$ generated by $Λ$. The investigation of this dynamical system allows us to establish that $(Γ,Λ)$ is a Heisenberg Uniqueness Pair. This way we both unify proofs of known cases (circle, parabola, hyperbola) and obtain many new examples. This method also allows to have a better geometric intuition on why $(Γ,Λ)$ is a Heisenberg Uniqueness Pair.

math.CA↗
arXiv2014

Sampling, interpolation and Riesz bases in small Fock spaces

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets--Ingham 1/4 theorem for Paley--Wiener spaces. Contrarily to the situation in Paley--Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

math.CV↗
arXiv2011

On zero sets in the Dirichlet space

We study the zeros sets of functions in the Dirichlet space. Using Carleson formula for Dirichlet integral, we obtain some new families of zero sets. We also show that any closed subset of $E \subset \TT$ with logarithmic capacity zero is the accumulation points of the zeros of a function in the Dirichlet space. The zeros satisfy a growth restriction which depends on $E$.

math.CA↗
arXiv2010

Cantor sets and cyclicity in weighted Dirichlet spaces

We treat the problem of characterizing the cyclic vectors in the weighted Dirichlet spaces, extending some of our earlier results in the classical Dirichlet space. The absence of a Carleson-type formula for weighted Dirichlet integrals necessitates the introduction of new techniques.

math.CV↗
arXiv2008

On the Brown--Shields conjecture for cyclicity in the Dirichlet space

Let $\cD$ be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We establish a new sufficient condition for a function $f\in\cD$ to be {\em cyclic}, i.e. for $\{pf: p\text{a polynomial}\}$ to be dense in $\cD$. This allows us to prove a special case of the conjecture of Brown and Shields that a function is cyclic in $\cD$ iff it is outer and its zero set (defined appropriately) is of capacity zero.

math.CV↗
arXiv2008

Compact operators that commute with a contraction

Let $T$ be a $C_0$--contraction on a separable Hilbert space. We assume that $I_H-T^*T$ is compact. For a function $f$ holomorphic in the unit disk $\DD$ and continuous on $\bar\DD$, we show that $f(T)$ is compact if and only if $f$ vanishes on $σ(T)\cap \TT$, where $σ(T)$ is the spectrum of $T$ and $\TT$ the unit circle. If $f$ is just a bounded holomorphic function on $\DD$ we prove that $f(T)$ is compact if and only if $\lim_{n\to \infty} T^nf(T) =0$.

math.FA↗