Linear Maps Preserving Rank-additivity and Rank-sum-minimal on Tensor Products of Matrix Spaces

The problems of characterizing maps that preserve certain invariant on given sets are called the preserving problems, which have become one of the core research areas in matrix theory. If for any A1 ⊕···⊕ Ak’B1 ⊕···⊕ Bk ∈ M n1 ⊕···⊕ Mnk, a linear map, Φ : Mn1 ⊕···⊕ Mnk → Mn1 ⊕···⊕ Mnk , as R (A1 ⊕···⊕ Ak + B1 ⊕···⊕ Bk) = R (A1 ⊕···⊕ Ak) + R (B1 ⊕···⊕ Bk) established, there is R (Φ (A1 ⊕···⊕ Ak + B1 ⊕···⊕ BK)) = R (Φ (A1 ⊕···⊕ Ak)) + R (Φ(B1 ⊕···⊕ BK)) we say that Φ preserves the rank-additivity. If for any A1 ⊕···⊕ Ak′B1 ⊕···⊕ Bk ∈ Mn1 ⊕···⊕ Mnk, and a linear map, Φ : Mn1 ⊕···⊕ Mnk → Mn1 ⊕···⊕ Mnk , as established, there is R(A1 ⊕···⊕ Ak + B1 ⊕···⊕ Bk) = |R(A1( ⊕···⊕ Ak) — R (B1 ⊕···⊕ Bk) we say that Φ rank-sum-miminal. In this paper, we characterize the form of linear mapping Φ.