Classical analysis, Operator theory, Geometry of Banach spaces, their interplay and their applications - ANALYSIS AND OPERATORS

The research project is organized along three strongly related directions, Classical function theory, Operator theory and Geometry of Banach spaces. The main objectives concern Bergman spaces, harmonic approximation, removable and thin sets and capacities as related to the Cauchy integral, Hankel/Toeplitz operators and function models, convex geometry and operator spaces. Particular attention will be devoted to the invariant subspace problem for Hilbert spaces, and to applications of Classical Analysis, Operator Theory and Convex Geometry to integral and differential equations and control theory. The whole subject is based on the interplay between results and methods arising from the three main directions of research of the network. Break troughs performed in recent years by network participants concerning factorisation and interpolating sequences in Bergman and related spaces, the Cauchy integral in relation with capacities. Levin's problem, applications of the theory of almost analytic functions to weighted shifts provide an array of new tools suitable to reach the research objectives of the projected network, relying on the complementarily between network teams. The counter example to Halmos'10th. problem, constructed recently by using tools from the three main directions of research of the network, illustrates the power of methods based on the interplay between these areas.