Factorization of diagonally dominant operators on \(\ell _ 1\)
This enjoyable paper is concerned with the LU factorization of bounded operators on the Banach space \(\ell_ 1\) of absolutely summable bi- infinite sequences. The authors prove that every bounded, diagonally (column) dominant, boundedly invertible, linear operator B on \(\ell_ 1\) can be factored as \(B=\) LU, where L, U, \(U^{-1}\), and \(L^{-1}\) are bounded operators on \(\ell_ 1\), L and \(L^{-1}\) are unit lower triangular, while U and \(U^{-1}\) are upper triangular with respect to the canonical basis for \(\ell_ 1\). Let \(P_ n\) be the linear projection on \(\ell_ 1\) given by \(P_ ne_ i=e_ i\) if \(| i| \leq n\), \(P_ ne_ 1=0\) if \(| i| >n\). For a given operator B on \(\ell_ 1\), let \(B_ n=P_ nBP_ n\). If \(B=LU\) is as above, and \(B_ n=L_ nU_ n\) is the corresponding factorization of \(B_ n\), then the authors prove that \(L_ n\) and \(U_ n\) converge to L and U, respectively, in the strong operator topology.
scientific article; zbMATH DE number 3996452