Level sets and Composition operators on the Dirichlet space

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal

Constructive approximation in de Branges-Rovnyak spaces

In most classical holomorphic function spaces on the unit disk, a function $f$ can be approximated in the norm of the space by its dilates f\_r(z):=f(rz)~(r \textless{} 1). We show that this is \emph{not} the case for the de Branges--Rovnyak spaces \cH(b). More precisely, we give an example of a non-extreme point $b$ of the unit ball of $H^\infty$ and a function f\in\cH(b) such that \lim\_{r\to1^-}\|f\_r\|\_{\cH(b)}=\infty. It is known that, if $b$ is a non-extreme point of the unit ball of $H^\infty$, then polynomials are dense in \cH(b). We give the first constructive proof of this fact

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 $E$.Comment: Journal of Geometric Analysis (2011

Dinuclear LnIII Complexes with 9‐Anthracenecarboxylate Showing Field‐Induced SMM and Visible/NIR Luminescence

The reaction of several Ln(NO3)3·6H2O salts with 9-anthracenecarboxylic acid (9-HAC) and 2,2′-bipyridine (bpy) in a mixture of CH3OH/H2O has allowed the isolation of the dinuclear compounds 1-6 with formula [Ln2(μ2-9-AC)4(9-AC)2(bpy)2] [LnIII = Nd (1), Eu (2), Gd (3), Tb (4), Er (5), and Yb (6)]. The molar magnetic susceptibility measurements of 1-6 in the 2-300 K temperature range indicate weak antiferromagnetic ex-. change for the isotropic GdIII compound 3. Compounds 1, 5, and 6 exhibit field-induced single-molecule magnet (SMM) behavior. The luminescence properties of compounds 1-6 in the solid state have been studied at different temperatures and show sensitization of the 4f-4f emission bands in the NIR range for compounds 1, 5, and 6