Complementary Triangular Forms for Infinite Matrices

Abstract

If A and Z are complex, square finite matrices, and if one of them is diagonable, then there exists an invertible matrix S, such that S ^-1 AS is upper triangular, and S ^-1 ZS is lower triangular. This paper presents analogues of this result for pairs of bounded operators, acting on the separable Hilbert space l ₂(Z ⁺). The main result states that there exist two diagonable operators A and Z acting on l ₂(Z ⁺), that are not simultaneously similar respectivily to an upper triangular operator and a lower triangular operator. The example is based on the existence of a unitary operator on l ₂(Z ⁺), that does not admit lower-upper factorization, even after independently permuting rows and columns. On the other hand, for pairs of bounded operators, where one of the operators is of finite rank, positive results are obtained.