ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 39A70, 47B39 Zygmunt, Marcin J. Matrix Moment Problem Let ∑ = (σ(i, j))_{i,j=1}^{N} be a positive matrix of borel measures on the real line ℝ. We say that ∑ has all moments finite if

m_{n}(i, j) = ∫_{ℝ} x^{n}dσ(i, j)(x)

are finite. In this case we define a sequence of matrix moments M_{n} = (m_{n}(i, j))_{i,j=1}^{N}. The matrix moment problem asks how to construct a matrix measure ∑ from a given sequence of matrix moments and if such a measure is unique. It is closely connected with the analysis of self-adjoint three-term difference operators acting on sequences of matrices. The uniqueness of solutions of the matrix moment problem is related to the dimensions of the kernel of Γ(Z) as an operator on ℂ^{N}, where

Γ(z) = lim_{n+∞} (∑_{k=0}^{n} P_{k}^{∗}(z)P_{k}(z))^{−1}

and P_{n}(z) are the matrix polynomials orthogonal with respect to the following “quasi-scalar” product:

(F, G) = ∫_{ℝ} F (x)d ∑(x)G^{∗}(x) = ( ∑_{k,l=1}^{N} ∫_{ℝ} f(i, k)(x)\overline{G(j, l)(x)}dσ(k, l)(x)) .

All matrix measures ∑ for which the matrix polynomials form a dense subset in L^{2}(ℝ, ℂ^{n}; ∑) (they are so-called N-extremal measures) can be identified with unitary matrices acting on a subspace of ℂ^{N}.