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Neutrinoless double-$\beta$ decay is of fundamental importance fordetermining the neutrino mass. Although double electron ($\beta^-\beta^-$)decay is the most promising mode, in very recent years interest in doublepositron ($\beta^+\beta^+$) decay, positron emitting electron capture($EC\beta^+$), and double electron capture ($ECEC$) has been renewed. Wepresent here results of a calculation of nuclear matrix elements forneutrinoless double-$\beta^+$ decay and positron emitting electron capturewithin the framework of the microscopic interacting boson model (IBM-2) for$^{58}$Ni, $^{64}$Zn, $^{78}$Kr, $^{96}$Ru, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba,and $^{136}$Ce decay. By combining these with a calculation of phase spacefactors we calculate expected half-lives.

Neutrinoless double positron decay and positron emitting electron capture in the interacting boson model