Spectral computation and free probability theory

Research project R 2 Spectral computation and free probability theory Franz LEHNER 27.11.2000 Free probability was introduced by D. Voiculescu as an abstract approach to haxmonic analysis on the free group. Central to the theory is the concept of free convolution, which solves spectral problems of convolution operators supported on the generators of the free product of discrete groups. Free convolution was found also independently by random walk theorists, who are interested in spectral data of the transition operators of random walks. Moreover, Voiculescu has shown that free convolution also describes the asymptotic behaviour of sums of random matrices as the dimension grows to infinity. Voiculescu has also developed a theory of operator valued free probability, based on the amalga-mated free product of algebras with conditional expectations. This to gether with a matrix trick yields to a theoretical approach to compute the spectrum of arbitrary finitely supported convolution operators on the free group. Our project is mainly concerned with matrix valued non-selfadjoint spectral problems on the free product of discrete groups, which have not yet been treated extensively in the literature. Our aims are threefold. - Concrete methods of computation of spectra and spectral radii of matrix-valued convolution operators on the free group. We have developed so fax an approach based on Haagerup inequality and explicit resolvent calculus to compute the contours of spectra of (scalar-valued) convolution operators supported on the generators, which can be extended to the matrix valued case. - Investigation of the behaviour of orthogonal polynomials under free convolution. These can be used to construct asymptotic eigenvectors which give information about the spectrum of the associated Jacobi operator. Then try to find asymptotic eigenvectors for non-selfadjoint free sums of operators, in order to compute the their complete spectrum. - As a complementary means for spectral computation, find criteria for the convergence of the spectral measure of random matrices to the Brown measure of the weak limit as the dimension grows to infinity. mension grows to infinity.