Interpolating Matrices

This Thesis work aims to extend the notion of interpolating sequences to objects having a non-trivial algebraic structure, such as a sequence of square matrices A (of eventually unbounded dimensions). First, we will consider the case in which the spectra of A lie in the unit disc, a necessary assumption if we want to apply to each matrix of the sequence a bounded holomorphic function on the unit disc. Since analytic functions preserve algebraic properties of square matrices, such as their eigenspaces, we can\u27t extend the definition of interpolating sequences to matrices by just considering targets which are bounded in the operator norm. We will overcome this first obstruction by identifying a target sequence with a bounded sequence of bounded analytic functions. We will then propose a way to separate matrices that agrees with the pseudo-hyperbolic distance in the unit disc whenever matrices reduce to points, and we will show that such separation condition corresponds to separated finite dimensional model spaces in the Hardy space. Our first main result will then extend Carleson interpolation Theorem to the sequence A.We will also extend some known results on interpolating sequences for bounded analytic functions in the polydisc to sequences of d-tuples of commuting matrices with joint spectra in the polydisc. The main difference with the one variable scenario, as for the scalar case, is that the separation conditions that allows one to describe interpolating sequences in this multi-variable setting are stated in terms of the whole class of admissible reproducing kernels Hilbert spaces, since the Hardy space alone is not able to encode interpolation properties, for d\u3e1. Moreover, we will try to get a better geometric understanding of the well known partial results on interpolating sequences (of scalars) in the polydisc, by considering random sequences which are almost surely separated. Finally, we will consider interpolating sequences of non-commuting d-tuples of matrices. This will be done by referring to well developed theory of non-commutative Pick kernels