LARGE BERGMAN · Toeplitz and related operators in large Bergman spaces

Toeplitz and related operators in large Bergman spaces

The topic and research objectives of this project belong to mathematical analysis, more precisely, to the theory of linear operators in Banach and Hilbert spaces of analytic functions. In general, the techniques of operator theory form a standard approach and toolkit for solving partial and ordinary differential equations arising in physics, chemistry, biology, economy and other branches of science and its applications. For example, spectra of linear operators are directly connected with central physical concepts: in particular they can be identified with the frequencies of propagating (electromagnetic, acoustic, elastic, quantum etc.) waves in various types of media or materials. Central mathematical questions for linear operators concern their boundedness, compactness, invertibility and Fredholm and spectral properties. For linear operators, the property of invertibility amounts to the question of the solvability of an equation containing the operator, and boundedness is equivalent to the continuity of the operator, which is relevant for the stability of the solution under small errors and perturbations for initial data contained in an equation. The project focuses on the properties of special classes of linear operators in Bergman spaces of analytic functions. For example, Toeplitz operators are defined with the help of the so called Bergman kernel, which is the integral kernel of a natural orthogonal projection between the involved Hilbert spaces. The kernel depends on the domains and weights and can be explicitly presented in very simple cases only. Due to its applications and connections to other problems and areas in analysis, the Bergman kernel is in itself a much studied object. It is also closely connected or even an essential part of the definition of other operators studied in this project. Central, standard techniques for the study of these operators concern characterizations of the properties of the operators in terms of Carleson measures, averaging functions and Berezin transforms. The predecessor of the study of operator theory in Bergman spaces is the corresponding research in Hardy spaces of analytic functions. The theory of Hardy spaces is a central subject in modern analysis with many deep problems and results. It has close connections to many phenomena in harmonic, complex and functional analysis and even in partial differential equations. Most of the central theory of Hardy spaces and operators on them were well understood already by the 1970's, but their natural counterparts and analogous problems for Bergman spaces were generally viewed as essentially more difficult, and only some isolated progress was achieved. The terminology goes back to the classical book by Stefan Bergman, which contained the first systematic treatment of the Hilbert space of square integrable analytic functions with respect to Lebesgue area measure on the unit disc, which is nowadays known as the standard or classical Bergman space; the reproducing kernel used by Bergman became known as the Bergman kernel function. Operator theory in Bergman spaces was initiated in 1980's with new contributions by several authors, but the main breakthroughs came only in the 1990's, when a number of difficult problems were solved in a several important contributions. Also, a direct application of Toeplitz operators in analytic function spaces was established via the so called deformation quantization, which is a special case of the canonical quantization in theoretical physics. The Bergman spaces under consideration are equipped with weighted norms vanishing rapidly at the boundary of the underlying complex domain. Operator theory on such spaces have been studied actively mainly only during the last 10 years, although some techniques go back to Hörmander's studies on several complex variables in 1960's. In contrast, spaces with slower, so called doubling, weights, and operators on them, have been investigated actively already since 1980's. Such studies heavily use the classical hyperbolic geometry of the underlying domain, but in the case of exponentially decreasing weigths, the connection with the hyperbolic geometry is broken, and nontrivial changes are needed. These have been introduced by many authors in the literature during the last years. In the present project we have adapted and developed these techniques and obtained a number of new results on central questions in theory. In particular the project has focused on improving the existing techniques involving Carleson measures, averaging functions, Berezin transforms, Kahane-Khinchine inequalities, and others.

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