## Proceedings of the Centre for Mathematics and its Applications

### Conical open mapping theorems and regularity

#### Abstract

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 $\delta B_y \bigcap T(K) \subseteq T(B_x \bigcap K)$, where $Bx, By$ 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.

#### Article information

Dates
First available in Project Euclid: 18 November 2014