For a rank-$1$ matrix $A={𝐚𝐛}^t$, we define the perimeter of $A$ as the number of nonzero entries in both $𝐚$ and $𝐛$. We characterize the linear operators which preserve the rank and perimeter of rank-$1$ matrices over semifields. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices over semifields if and only if it has the form $T\left(A\right)=UAV$, or $T\left(A\right)=U{A}^{t}V$ with some invertible matrices U and V.

We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator $T$ preserves the rank and perimeter of rank-$1$ matrices if and only if it has the form $T\left(A\right)=P(A\circ B)Q$, or $T\left(A\right)=P(A^t\circ B)Q$ with appropriate permutation matrices $P$ and $Q$ and positive integer matrix $B$, where $\circ$ denotes Hadamard product.

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k\in V$ by $A₁*\cdots *A_k=A_{i₁}\cdots A_{iₘ}$. This includes the usual product $A₁*\cdots *A_k=A₁\cdots A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying ${\mu }^m=1$ such that ϕ: V → V has the form A...