Let V V denote an infinite dimensional Banach space over the complex field, B [ V ] B[V] the bounded linear operators on V V and F F a closed subspace of V V . An element of T F = { T | T ∈ B [ V ] , T ( F ) ⊆ F } {\mathcal {T}_F} = \{ T|T \in B[V],T(F) \subseteq F\} is called a conservative operator. Some sufficient conditions for T ∈ T F T \in {\mathcal {T}_F} to be in the boundary, B \mathcal {B} , of the maximal group, M \mathcal {M} , of invertible elements are determined. For example, if T ∈ T F T \in {\mathcal {T}_F} , is such that (i) V V is the topological direct sum of R ( T ) \mathcal {R}(T) and N ( T ) ≠ { θ } N(T) \ne \{ \theta \} , (ii) T T is an automorphism on R ( T ) ∩ F \mathcal {R}(T) \cap F , then T ∈ B T \in \mathcal {B} . Also, the complement of the closure of M \mathcal {M} is discussed. This is an extension of another paper by the same authors [6].