Positive operators on extended second order cones

A positive operator on a cone is a linear operator that maps the cone to a subcone of itself. The extended second order cones were introduced by Németh and Zhang [17] as a working tool to solve mixed complementarity problems. Sznajder [23] determined the automorphism group and the Lyapunov (or bilinearity) ranks of these cones. Ferreira and Németh [9] reduced the problem of projecting onto the second order cone to a piecewise linear equation. Németh and Xiao [16] solved linear complementarity problems on the extended second order cone (motivated by portfolio optimization models) by reducing them to mixed complementarity problems with respect to the nonnegative orthant. As an extension of Sznajder's results, this paper aims to be a first work about finding necessary conditions and sufficient conditions for a linear operator to be a positive operator (which extends the notion of an automorphism) on an extended second order cone. Although, in the particular case of second order cones a necessary and sufficient condition is known, for extended second order cone such a condition is very difficult to find without restricting the structure of the linear operator. If the matrix of the linear operator is block-diagonal, we give such a necessary and sufficient condition.