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Let      X X       be a separable Banach space. It is shown that every diagonally dominant invertible operator on      L 1 ( [ 0 , 1 ] , X ) {L_1}([0,\,1],\,X)       can be factored uniquely as a product of an invertible upper triangular operator and an invertible unit lower triangular operator.

Factorization of diagonally dominant operators on 𝐿₁([0,1],𝑋)