Spaces of Analytic Functions and Their Applications

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 Poly?a-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.