Frames and bases in holomorphic function spaces – FRAB

We are going to consider several fundamental problems related to bases and frames in Banach spaces of analytic functions. A frame in a Hilbert space is a system of vectors such that taking the scalar products of a given vector with those produces "coordinates", the sum of squared moduli of which will be comparable to the squared norm of the given vector. Any finite union of Riesz basic sequences (the images of orthonormal sequences under bounded invertible maps) will produce a frame in the closed subspace it spans. Recently, the so called Feichtinger conjecture claiming that every frame is the union of Riesz basic sequences attracted attention of many analysts. In particular, in order to intensify the research in this direction a selective meeting was organized by the American Institute of Mathematics (Stanford University, Palo Alto, autumn 2006). Numerous results published by the participants (P.Casazza, V.Paulsen, Ch.Akemann and others), show many relations between the Feichtinger conjecture and other famous problems in Operator theory and Harmonic Analysis. For instance, the Feichtinger conjecture is equivalent to a conjecture generalizing the Bourgain-Tzafriri Restricted Invertibility Theorem, and to the "Paving Conjecture" of Kadison-Singer. The latter states that given any linear contraction whose diagonal matrix elements are zero with respect to a given basis, there is a partition of the basis in a universal number of pieces such that the compression of the operator to the span of each piece of the partition (or the minor matrix generated by the piece, if one prefers) has norm bounded above by one half. The aim of this project is to push forward the Feichtinger-Kadison-Singer problem in the framework of spaces of holomorphic functions. The above mentioned conceptual breakthroughs and a close collaboration of specialists in different areas of analysis, involved in the project, permit us to expect substantial progress in our research program that includes (1) studying the Feichtinger problem for systems of reproducing kernels in different spaces of analytic functions, including the Fock spaces and the de Branges spaces; here we deal with known and new sampling and interpolation problems; (2) estimating the unconditional basis constants for different types of bases, with applications (3) to the spectral theory of Toeplitz operators, to the evolution semigroups, to the systems theory; (4) to the Reproducing Kernel Thesis for Toeplitz and Hankel operators. In the project we use a variety of methods and results arising from real and complex analysis, harmonic analysis, and operator theory. In particular, these methods include the Bergman space techniques, d-bar equations, approximation on the complex plane, potential theory, the Hilbert transform, spectral theory, and functional calculi. Our research groups centered at Marseille (coordinator group), Bordeaux, and Lyon will take part in all main tasks of the project. Furthermore, we expect to collaborate in our research both with foreign colleagues (USA, Spain, Norway, Sweden, Canada, England) and with specialists interested in the applications of frames techniques to Linear Systems theory (INRIA, Sophia-Antipolis).