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).
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Cohen, N. (2019). Block Triangular Matrices in Banach Space: Minimal Completions and Factorability. In: Bolotnikov, V., ter Horst, S., Ran, A., Vinnikov, V. (eds) Interpolation and Realization Theory with Applications to Control Theory. Operator Theory: Advances and Applications, vol 272. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-11614-9_4
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DOI: https://doi.org/10.1007/978-3-030-11614-9_4
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-11613-2
Online ISBN: 978-3-030-11614-9
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