May 24, 2004

Dmitry Kalyuzhnyi-Verbovetzkii

Title: On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series

Abstract:

This talk is based on a joint work with Daniel Alpay. We are motivated by our investigation of the non-commutative counterpart of rational $ J$-unitary matrix-valued functions. The property of $ J$-unitarity, as well as many other properties of non-commutative formal power series, is formulated in terms of substitutions of matrices for the formal variables. Thus, we need to understand better the relationship between coefficients of formal power series and their values on matrices. The results which will be reported in this talk have an algebraic nature and are independent of the original motivation.

For a rational formal power series in $ N$ non-commuting indeterminates, with matrix coefficients, we establish the formula which relates the intersection of the null spaces of coefficients to the intersection of the null spaces of values of this series on $ N$-tuples of $ n\times n$ matrices, for $ n$ large enough. This formula, to a certain extent, gives a generalization of the known fact in the theory of rings which says that there are no rational identities valid for infinitely many matrix rings $ \mathbb{C}^{n_j\times n_j}, j=1,2,\ldots$. As an application, we formulate the criteria of observability, controllability, and minimality for Schützenberger-Fliess recognizable formal power series representations.