Conical open mapping theorems and regularity

Bauschke, Heinz H. and Borwein, Jonathan M. (1999) Conical open mapping theorems and regularity. In: Proceedings of the Centre for Mathematics and its Applications (Australian National University) , March 1998, National Symposium on Functional Analysis, Optimization and Applications.

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Suppose $T$ is a continuous linear operator between two Hilbert spaces $X$ and $Y$ and let $K$ be a closed convex nonempty cone in $X$. We investigate the possible existence of $\delta > 0$ such that $\delta B_Y \cap T(K) \subseteq T(B_X \cap K)$, where $B_X,B_Y$ denote the closed unit balls in $X$ and $Y$ respectively. This property, which we call openness relative to $K$, is a generalization of the classical openness of linear operators. We relate relative openness to Jameson's property~(G), to the strong conical hull intersection property, to bounded linear regularity, and to metric regularity. Our results allow a simple construction of two closed convex cones that have the strong conical hull intersection property but fail to be boundedly linearly regular.