Linear maps on L(ₚⁿ,ₚᵐ), (p∈ \1,∞\) preserving parallel pairs

Two vectors $x,y$ of a Banach space are said to form a parallel (resp. triangle equality attaining or TEA) pair if $\|x+λy\|=\|x\|+\|y\|$ holds for some scalar $λ$ with $|λ|=1$ (resp. $λ=1$). For p∈ \1,∞\, and m,n≥ 2, we study the linear maps T: L(ₚⁿ, ₚᵐ) → L(ₚⁿ,ₚᵐ) that preserve parallel (resp. TEA) pairs, that is, those linear maps T for which $T(A),T(B)$ form a parallel (resp. TEA) pair whenever $A,B$ form a parallel (resp. TEA) pair of L(ₚⁿ,ₚᵐ). We prove that if T is non-zero, then the following are equivalent: (1) T preserves TEA pairs. (2) T preserves parallel pairs and rank$(T)>1$. (3) T preserves parallel pairs and T is invertible. (4) T is a scalar multiple of an isometry.