# The dual Z-property for the Lorentz cone

## TL;DR

This paper investigates the dual cone of linear maps with the Z-property specifically for the Lorentz cone, extending the understanding of cone-complementarity in this context.

## Contribution

It provides a solution for characterizing the dual cone of Z-property maps with respect to the Lorentz cone, a specific case not previously addressed.

## Key findings

- Derived the dual cone of Z-property maps for the Lorentz cone.
- Extended the theory of cone-complementarity to Lorentz cones.
- Clarified the structure of Z-property maps in this setting.

## Abstract

The Z-property of a linear map with respect to a cone is an extension of the notion of Z-matrices. In a recent paper of Orlitzky (see Corollary 6.2 in M. Orlitzky. Positive and $\mathbf{Z}$-operators on closed convex cones, Electron. J Linear Algebra, 444--458, 2018) the characterisation of cone-complementarity is given in terms of the dual of the cone of linear maps satisfying the Z-property. Therefore, it is meaningful to consider the problem of finding the dual cone of the cone of linear maps which have the Z-property with respect to a cone. This short note will solve this problem in the particular case when the Z-property is considered with respect to the Lorentz cone.