Bounded, conservative, linear operators and the maximal group

Abstract:

Let V denote a Banach space over the reals, $B[V]$ the bounded linear operators on V,f a linear functional defined on a complete subspace, (f), of V. A conservative operator is an element of the set ${\mathcal {T}_f} = \{ T|T \in B[V], T((f)) \subseteq (f)\}$. In this setting this paper extends some of the results of a recent paper by Rhoades [Triangular summability methods and the boundary of the maximal group, Math. Z. 105 (1968), 284-290]. In this setting necessary and sufficient conditions are proven for $T \in {\mathcal {T}_f}$ to be in the maximal group of invertible elements, $\mathcal {M}$. Sufficient conditions are proven for $T \in {\mathcal {T}_f}$ to be in the boundary, $\mathcal {B}$, of $\mathcal {M}$. It is proven that $\mathcal {B}$ is a multiplicative semigroup and if (f) is nontrivial, then $\mathcal {B}$ is nonconvex. Two questions raised in the paper by Rhoades were answered.