A STUDY OF DE BRANGES - ROVNYAK AND DIRICHLET SPACES

This doctoral thesis is devoted to the study of certain spaces of analytic functions on the unit disk $\mathbb{D}$ of the complex plain, $\mathbb{D}:=\{z\in\mathbb{C}\colon |z|<1\}$. This work lies in the intersection between complex analysis, operator theory and harmonic analysis. In particular, we are interested in the de Branges-Rovnyak spaces $H(b)$, that is a class of spaces that arise from the Hardy space $H^2(\mathbb{D})$, and in the harmonically weighted Dirichlet spaces $\mathcal{D}_\mu$, an important generalization of the classical Dirichlet space $\mathcal{D}$. Special attention is given to a specific subclass of $H(b)$ spaces, the so-called \emph{model spaces} $K_u$. The first chapter contains the relevant background definitions and results. The remaining three chapters contain the original results, and they stem from three different papers or preprints. In Chapter 2, we study a special class of operators on the spaces $H(b)$, the \emph{difference quotient operators} $Q_\zeta^b$. The idea is to study from an operator-theoretic point of view the fundamental quantities \[\frac{f(z)-f(\zeta)}{z-\zeta},\] for functions $f\in H(b)$ and boundary points $\zeta\in\mathbb{T}:=\partial\mathbb{D}$. A priori, the values $f(\zeta)$ are not defined for analytic functions on the unit disk $\mathbb{D}$. However, as we carefully motivate in the thesis, the operator $Q_\zeta^b$ can be properly defined for special points $\zeta$. We conduct a spectral analysis of such operators and we deduce two estimates for the operator norm $\|Q_\zeta^b\|$, that allow us to completely characterize the boundary points for which these operators can be bounded. Also, this analysis leads to a complete characterization of a property of the function $b$ that defines the space $H(b)$: when $b$ is \emph{inner}, we prove that $b$ is \emph{one-component} if and only if the norm $\|Q_\zeta^b\|$ is uniformly bounded by the modulus of the first derivative $|b'(\zeta)|.$ In Chapter 3, we investigate the relations between $H(b)$ spaces and $\mathcal{D}_\mu$ spaces. This is connected to the content of Chapter 2, as the operator norm $\|Q_\zeta^b\|$ plays a role in the membership in Dirichlet spaces. In the last three decades, it has been shown that for special functions $b$ and measures $\mu$ we may have an equality of sets $H(b)=\mathcal{D}_\mu$, with equality or equivalence of norms. This is very interesting, since the two classes of spaces arise from very different contexts and are apparently unrelated. This problem is also fairly difficult, as the structure of both spaces is rather mysterious, except for very special cases. We obtain necessary and sufficient conditions for the embedding $H(b)\hookrightarrow \mathcal{D}_\mu$. These conditions also produce a complete characterization in some special cases. Using different techniques, we study in detail another special class of embeddings $H(b)\hookrightarrow\mathcal{D}_\mu$, namely, when the function $b$ is non-extreme and $\mu$ is a sum of finitely many Dirac deltas. We completely characterize the conditions for these embeddings, and we manage to complete this result with a characterization of the full identity $H(b)=\mathcal{D}_\mu$. In the last part of the thesis, we study in detail a problem of polynomial approximation in Dirichlet spaces, using the tools of the Hadamard multipliers on such spaces. Given the analytic function $f(z)=\sum_{k=0}^n a_kz^k$ in an appropriate weighted Dirichlet space, we study the generalized Ces\`{a}ro means \[ (\sigma_n^{\alpha} f)(z)=\binom{n+\alpha}{\alpha}^{-1}\sum_{k=0}^n\binom{n-k+\alpha}{\alpha}a_kz^k, \] where $\alpha$ is a parameter in the interval $[0,1]$. We point out that, for $\alpha=0$, we recover the $n$-th partial Taylor sum, while for $\alpha=1$, we obtain the standard Cesàro mean. We investigate the convergence $\sigma_n^\alpha f \to f$ as $n\to\infty$. In particular, we are interested in the asymptotical behavior of the norm $\|\sigma_n^\alpha\|$, as $n\to\infty$, and its dependence on the parameter $\alpha\in[0,1]$.