From the introduction: Let \(0 \leq m \leq n-1\) and \(R_{ij}\) be a given set of \(\mu_ i \times \mu_ j\) matrices for \(i,j = 1,\dots,n\) and \(| i - j | \leq m\). H. Dym and I. Gohberg [Linear Alg. Appl. 36, 1-24 (1981; Zbl 0461.15002)], established necessary and sufficient conditions for the existence and uniqueness of an invertible block matrix \(F = (F_{ij})_{i,j = 1, \dots, n}\) such that \(F_{ij} = R_{ij}\) for \(| i - j | \leq m\) and \(F^{-1}\) has a band triangular factorization and so \((F^{-1})_{ij} = 0\) for \(| i - j | > m\). The aim of this paper is to generalize these results in two directions. First, we would like to consider operator matrices i.e. we allow \(R_{ij}\) to be (linear bounded) operators acting between the Hilbert spaces \(H_ j\) and \(H_ j\). Secondly, we will allow the set of indices of the given \(R_{ij}\) to be more general then banded ones. In fact, we will consider index sets which have an associated graph which is chordal.