Lower upper factorizations of operators with middle terms (Q1105173)

The authors consider factorizations of the form \(I- K=(I+K_+)(D+F)(I+K_+)\) where \(I_-\), \(K_+\), and D are lower, upper, and diagonal operators relative to a maximal chain \({\mathcal P}\) of orthoprojections in a separable Hilbert space. The main result (Theorem 2.1) asserts the following: Let K be a Hilbert-Schmidt operator and consider factorizations of the type \(I-K=(I+X_-)(D+F)(I+X_+)\) where \(X_+\) and \(X_-\) are Volterra operators satisfying \(X_+P=PX_+P\), \(PX_-=PX_-P\) (P\(\in {\mathcal P})\), D is invertible, \(DP=PD\) for each \(P\in {\mathcal P}\) and F is a finite-dimensional operator. Then min rank (F)\(=\max \{\dim \ker (I- PKP):o\in {\mathcal P}\}\) where the minimum is taken over all the factorizations in which \(X_+\) and \(X_-\) are Hilbert-Schmidt. The proof makes use of the factorization theorem [\textit{I. C. Gohberg} and \textit{M. G. Krein}, Theory and Applications of Volterra Operators in Hilbert Space, Moscow (1967; Zbl 0168.120), English transl.: Trans. Math. Monographs 24 (1970; Zbl 0194.438) theorem 6.2, p. 181.]