Exponential dichotomy for noninvertible linear difference equations: block triangular systems

In this paper, block upper triangular systems of linear difference equations are considered, in which the coefficient matrices are not assumed invertible. The relationship between the exponential dichotomy properties of such a system and its associated block diagonal system is studied. The reason it is important to study triangular systems is that any system of linear difference equations is kinematically similar to an upper triangular system. In the bounded invertible case, it is known that for equations on the intervals J = Z(+) or Z(-), a block upper triangular system has an exponential dichotomy if and only if the associated block diagonal system has one. However, when J = Z, only the sufficiency holds. The sufficiency extends to the noninvertible case, provided the off-diagonal matrices are bounded. However, the necessity does not hold even when J = Z(+) or Z(-). Nevertheless, if certain conditions are added, then the necessity does hold and it is also shown that these conditions are needed since it turns out that if both the triangular and diagonal systems have dichotomies, then these extra conditions must hold.