Operator inequalities, reproducing kernels, and invariant subspaces

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AbstractRichter/SundbergRichter and Sundberg will continue their study of linear operators on Hilbert spaces modelled by multiplication operators on spaces of analytic functions, in both one and several variables. The study of such operators has a long history and has led to much progress in operator theory and complex analysis. For instance the unilateral shift, which is modelled by multiplication by z on the Hardy space of the unit disc, has been used to enhance our knowledge of contraction operators. Other classes of operators characterized by operator inequalities may be studied using operators modelled by multiplication by z on other Hilbert spaces of analytic functions. Such spaces have associated with them evaluation functionals called reproducing kernels, and there is an intimate connection between properties of such a kernel and properties of the multiplication operators on the corresponding space. An important classical example where this correspondence has been exploited is Nevanlinna-Pick interpolation. Recent work of several researchers has extended the applicability of the ideas behind Nevanlinna-Pick interpolation and introduced other types of inequalities on operators and reproducing kernels that seem to be very fruitful. Richter and Sundberg will investigate the connection between operator inequalities, reproducing kernels, and the structure of the lattice of invariant subspaces of multiplication operators. The proposed work involves ideas and problems from several areas of pure and applied mathematics. Operator Theory, which may be thought of as an infinite dimensional version of linear algebra, grew out of ideas used to study certain partial differential equations arising in physics in the 1800's, and became increasingly important with the advent of Quantum Mechanics in the twentieth century. Complex Analysis is a subject with a long and distinguished history, and remains a very active and broad area of research. These two areas have had a very fruitful interaction throughout this century, owing to the fact that some interesting and useful operators can be modelled by natural operations on spaces of analytic functions. At least since the 1960's it has been realized that a series of related results concerning certain of these operators are of importance in the study of Control Theory, an area of importance in electrical engineering and other practical applications. Among these results are the Beurling-Lax Theorem on invariant subspaces, the Nevanlinna-Pick Interpolation Theorem, and its close relative the Commutant Lifting Theorem. Work by a number of researchers since the 1980's has shown that the circle of ideas concerned with these results are applicable to a much wider class of objects than had previously been realized. This has resulted in a much improved understanding of the underlying mathematical systems