the Conjugate Gradient (CG) on the $A = M^HM$ Hermitian operator. Although there are cases where BICGSTAB on the nonsymmetric operator can be twice as fast [@Frommer_bicgstabdQCD; @Fischer:1996th], in other cases, such as in domain wall fermions, CG is not only the fastest but also the most robust method[^4]. In this paper, we focus on CG for Hermitian systems for two reasons. First, CG is characterized by optimal convergence. Second, the eigenvectors of $A$ are also eigenvectors of $\gamma_5 M$. Hence, after an initial CG phase, computed eigenvectors can be used to deflate BICGSTAB on $\gamma_5 M$ for the rest of the right hand sides.

Solving a linear system with many right hand sides is an outstanding problem. Traditional approaches solve each system one by one. Unknowingly, such methods regenerate search directions within previously explored subspaces, thus wasting iterations in successive right hand sides. Sharing information between systems has long been recognized as the key idea [@CLanczos_bcg]. This is typically performed through seed methods [@VS_EG_95; @TFC_WLW_94; @Saad_87a] or through block methods [@Golub_Underwood_77; @DOLeary_80a; @Guennouni_Jbilou_Sadok]. For Hermitian matrices, a selective sharing of only the useful part of information between systems can be achieved through invariant subspaces. Such ideas have been tried 