A Hessenberg matrix \(D= (d_{i,j})^\infty_{i,j=0}\) is by definition a matrix of complex numbers satisfying \(d_{i,j}= 0\) whenever \(i> j+ 1\). That is, \(D\) is lower triangular, plus at most an upper diagonal string of nonzero elements. Jacobi matrices are Hessenberg, and one of the main aims of the paper under review is to draw a parallel between these two classes.
The authors study \(D\) as a densely defined (possibly unbounded) linear operator on \(l^2(\mathbb{N})\). The concept of a determinate Hessenberg matrix is developed on the basis of the example of Jacobi matrices associated to determinated moment sequences. This means, in rather imprecise terms, that the resolvent \((D- z)^{-1}U^+\) localized at the unilateral shift is Hilbert-Schmidt, see also [B. Beckermann, J. Comput. Appl. Math. 127, 17–65 (2001; Zbl 0977.30007)]. The finite \(n\)-section of the infinite matrix \(D\) is \(D_n = (d_{i,j})^n_{i,j= 0}\).
The main body of the paper is concerned with the convergence \((D_n- z)^{-1}\to (D- z)^{-1}\). This corresponds to the classical Padé approximation of Markov or Stieltjes functions, and the whole study is highly relevant for rational approximation theory. The main complication arises from the asymmetry (non-self-adjointness) of the operator represented by \(D\). However, the techniques are based on elementary operator theory (von-Neumann’s inequality, compactness, perturbation determinants). In this way, the whole note is made accessible to non-experts, for the benefit of a potential (and deserved) wide audience.