Block Triangular Matrices in Banach Space: Minimal Completions and Factorability

Abstract

This paper considers the extension of four issues from applied matrix theory to Banach space operators under a finite partition: block LPU factorability with L, U invertible; block LU factorability with L, U not necessarily invertible; rank factorization; and, above all, the minimal rank completion problem of block triangular type. This extension requires the replacement of rank considerations by range and kernel inclusions.

LPU factorability appears as a natural condition under which the other issues can be fully analyzed. In practice it reduces to a finite sequence of complementability conditions. When the completion problem has LPU factorable data, the minimal completions of the data are factorable operators and admit a complete description; otherwise, non-factorable minimal completions may exist due to a well-known Banach space anomaly (Embry’s theorem).