Positive operators on extended second order cones

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