Localized Frames and Compactness

We introduce the concept of weak-localization for generalized frames and use this concept to define a class of weakly localized operators. This class contains many important operators, including: Short Time Fourier Transform multipliers, Calderon-Toeplitz operators, Toeplitz operators on various functions spaces, Anti-Wick operators, and many others. In this paper, we study the boundedness and compactness of weakly localized operators. In particular, we provide a characterization of compactness for weakly localized operators in terms of the behavior of their Berezin transform

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 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

The Essential Norm of Operators on Aαp(Bn)A^p_\alpha(\mathbb{B}_n)

In this paper we characterize the compact operators on $A^p_\alpha(\mathbb{B}_n)$ when $1-1$. The main result shows that an operator on $A^p_\alpha(\mathbb{B}_n)$ is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.Comment: v1: 32 pages; v2: 32 pages, typos corrected; v3: 32 pages, typos corrected, presentation improved based on referee comment