Let Mn be the semigroup of n× n complex matrices under the usual multiplication, and let S be different subgroups or semigroups in Mn including the (special) unitary group, (special) general linear group, the semigroups of matrices with bounded ranks. Suppose Λk(A) is the rank-k numerical range and rk(A) is the rank-k numerical radius of A ∈ Mn. Multiplicative maps φ : S → Mn satisfying rk(φ(A)...

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP = λP for some rank k orthogonal projection P . In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1 ≤ r < k it is shown that Λk(A) ⊆ Λk−r(A + F ) for any op...

For any n-by-n complex matrix A and any k, 1 ≤ k ≤ n, let Λk(A) = {λ ∈ C : X∗AX = λIk for some n-by-k X satisfying X∗X = Ik} be its rank-k numerical range. It is shown that if A is an n-by-n contraction, then Λk(A) = ∩{Λk(U) : U is an (n + dA)-by-(n + dA) unitary dilation of A}, where dA = rank (In − A∗A). This extends and refines previous results of Choi and Li on constrained unitary dilations...

‎Let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $A$ be an $n$-square quaternion matrix‎. ‎In this paper‎, ‎some results on the $k-$numerical range of $A$ are investigated‎. ‎Moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $A$ are introduced‎, ‎and some of their algebraic properties are studied‎.

Abstract. It is known that the i-th order laminated microstructures can be resolved by the k-th order rank-one convex envelopes with k ≥ i. So the requirement of establishing an efficient numerical scheme for the computation of the finite order rank-one convex envelopes arises. In this paper, we develop an iterative scheme for such a purpose. The 1-st order rank-one convex envelope R1f is appro...