This work is an introduction to anisotropic spaces, which have an ω-weight of analytic functions and are generalizations of Lipshitz classes in the polydisc. We prove that these classes form an algebra and are invariant with respect to monomial multiplication. These operators are bounded in these (Lipshitz and Djrbashian) spaces. As an application, we show a theorem about the division by good-inner functions in the mentioned classes is proved.

In this dissertation we consider several problems in classical complex analysis and operator theory. In the first part we study basis properties of a system of complex exponentials with a given frequency sequence. We show that most of these basis properties can be characterized in terms of the invertibility properties of certain Toeplitz operators. We use this reformulation to give a metric description of the radius of l2-dependence. Using similar methods we solve the classical Beurling gap problem in the case of separated real sequences. In the second part we consider the classical Polýa-Levinson problem asking for a description of all real sequences with the property that every zero type entire function which is bounded on such a sequence must be a constant function. We first give a description in terms of injectivity of certain Toeplitz operators and then use this to give a metric description of all such sequences. In the last part we study the spectral changes of a partial isometry under unitary perturbations. We show that all the spectra can be described in terms of the characteristic function of the partial isometry that is being perturbed. Our main tool in the proofs is a Herglotz-type representation for generalized spectral measures. We furthermore use this representation to give a new proof of the classical Naimark's dilation theorem and to generalize Aleksandrov's disintegration theorem.

Cette thèse est consacrée à l'étude du problème de la cyclicité dans certains espaces de fonctions analytiques sur le disque unité. Nous nous intéressons aux espaces de type Bergman et aux espaces de type Korenblum. Dans la première partie, nous étudions les fonctions cycliques dans les espaces de type Korenblum en utilisant la notion des prémesures. Cette notion a été introduite et développée par B. Korenblum au début des années 1970s. En particulier, nous donnons une réponse positive à une conjecture énoncée par C. Deninger. Dans la deuxième partie, nous utilisons la méthode de la résolvante pour étudier la cyclicité des fonctions intérieures singulières associées aux mesures de Dirac dans les espaces de type Bergman à poids. / In this thesis, we study the cyclicity problem in some spaces of analytic functions on the open unit disc. We focus our attention on Korenblum type spaces and on weighted Bergman type spaces. First, we use the technique of premeasures, introduced and developed by Korenblum in the 1970-s and the 1980-s, to give a characterization of cyclic functions in the Korenblum type spaces. In particular, we give a positive answer to a conjecture by Deninger. Second, we use the so called resolvent transform method to study the cyclicity of the one point mass singular inner function in weighted Bergman type spaces, especially with weights depending on the distance to a subset of the unit circle.