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The Hartree-Fock-Bogolyubov (HFB) plus proton-neutron quasiparticle randomphase approximation (pnQRPA) approach based on Skyrme interaction is applied tostudy the nuclear $\beta^+$/EC decay for nuclei near the proton magic numbers$Z=$20, 28, and 50. With properly selected Skyrme interactions that include thetensor terms, and the $T=0$ pairing interaction, the experimental $\beta^+$/ECdecay half-lives of these nuclei can be systematically reproduced quite well.It is shown that the tensor and $T=0$ pairing interactions play different rolesin different nuclei. The specific effect is relevant to the configurations thatcontribute to the decay. The attractive and repulsive properties of tensorinteraction for the GT state is also studied. The present results indicate thatit works repulsively on the GT states dominated by the configurations from $\pij_\gtrless$ to $\nu j_\gtrless$ orbits, while working attractively for thosemainly composed of the configurations from $\pi j_\gtrless$ to $\nu j_\lessgtr$orbits.

Effects of tensor and $T=0$ pairing interactions on nuclear $\beta^+$/EC decay