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We provide a new proof of Volbergs Theorem characterizing thin interpolating sequences as those for which the Gram matrix associated to the normalized reproducing kernels is a compact perturbation of the identity. In the same paper, Volberg characterized sequences for which the Gram matrix is a compact perturbation of a unitary as well as those for which the Gram matrix is a Schatten-$2$ class perturbation of a unitary operator. We extend this characterization from $2$ to $p$, where $2 le p le infty$.
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We study thin interpolating sequences ${lambda_n}$ and their relationship to interpolation in the Hardy space $H^2$ and the model spaces $K_Theta = H^2 ominus Theta H^2$, where $Theta$ is an inner function. Our results, phrased in terms of the functions that do the interpolation as well as Carleson measures, show that under the assumption that $Theta(lambda_n) to 0$ the interpolation properties in $H^2$ are essentially the same as those in $K_Theta$.
We look at thin interpolating sequences and the role they play in uniform algebras, Hardy spaces, and model spaces.
We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^{sigma}left(mathbb{B}_{d}right)$, in the range $0 < sigmaleq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $mathrm{H}^2(mathbb{D}^d)$ and its multiplier algebra $mathrm{H}^infty(mathbb{D}^d)$: in the polydisc, we obtain a sufficient and a necessary condition for a sequence to be $mathrm{H}^infty(mathbb{D}^d)$-interpolating almost surely. Those two conditions do not coincide, due to the fact that the deterministic starting point is less descriptive of interpolating sequences than its counterpart for the unit ball. On the other hand, we give the $0-1$ law for random interpolating sequences for $mathrm{H}^2(mathbb{D}^d)$.
We study Sobolev estimates for solutions of the inhomogenous Cauchy-Riemann equations on annuli in $cx^n$, by constructing exact sequences relating the Dolbeault cohomology of the annulus with respect to Sobolev spaces of forms with those of the envelope and the hole. We also obtain solutions with prescibed support and estimates in Sobolev spaces using our method.
We consider two-variable model spaces associated to rational inner functions $Theta$ on the bidisk, which always possess canonical $z_2$-invariant subspaces $mathcal{S}_2.$ A particularly interesting compression of the shift is the compression of multiplication by $z_1$ to $mathcal{S}_2$, namely $ S^1_{Theta}:= P_{mathcal{S}_2} M_{z_1} |_{mathcal{S}_2}$. We show that these compressed shifts are unitarily equivalent to matrix-valued Toeplitz operators with well-behaved symbols and characterize their numerical ranges and radii. We later specialize to particularly simple rational inner functions and study the geometry of the associated numerical ranges, find formulas for the boundaries, answer the zero inclusion question, and determine whether the numerical ranges are ever circular.
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