## Proceedings of the Centre for Mathematics and its Applications

- Proc. Centre Math. Appl.
- National Symposium on Functional Analysis, Optimization and Applications. John Giles and Brett Ninness, eds. Proceedings of the Centre for Mathematics and its Applications, v. 36. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1999), 1 - 9

### Conical open mapping theorems and regularity

Heinz H. Bauschke and Jonathan M. Borwein

#### 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

**Permanent link to this document**

https://projecteuclid.org/
euclid.pcma/1416323140

**Zentralblatt MATH identifier**

1193.90167

#### Citation

Bauschke, Heinz H.; Borwein, Jonathan M. Conical open mapping theorems and regularity. National Symposium on Functional Analysis, Optimization and Applications, 1--9, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1999. https://projecteuclid.org/euclid.pcma/1416323140