  Nuclei in the vicinity of driplines have been receiving a lot of attention in
nuclear structure studies. In the nuclei, the continuum coupling is crucial in
reproducing weakly-bound and unbound phenomena. To calculate observables of the
nuclei as open quantum systems, we have developed valence-space effective
operators in the complex-energy Berggren basis using many-body perturbation
theory. We focus on the Gamow-Teller $\beta$ decay in the {\it sd} shell. The
two- plus three-nucleon force from the chiral effective field theory (EFT),
named EM1.8/2.0, has been used. The Gamow shell model which takes the continuum
coupling into account can properly reproduce experimental observations of
weakly-bound and unbound states. The $\beta$-decay isospin asymmetry between
the dripline nucleus $^{22}\rm Si$ and its mirror partner $^{22}\rm O$ is
reproduced, in which the $s_{1/2}$ continuum plays a key role. Significant
Thomas-Ehrman shift is seen through mirror energy differences between the
mirror daughters $^{22}\rm Al$ and $^{22}\rm F$, in which the continuum effect
plays an important role.
