Bounded, conservative, linear operators and the maximal group. II

Abstract:

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