We survey results on linear operators leaving invariant different kinds of generalized numerical ranges and numerical radii.

There has been considerable interest in studying inclusion regions for numerical ranges. It is in fact very useful in knowing inclusion regions for W (A). For example, it is well known (see [4, Chapter 1]) that W (A) ⊆ IR if and only if A = A∗; W (A) ⊆ [0,∞) if and only if A is positive semidefinite; andW (A) ⊆ (0,∞) if and only if A is positive definite. Moreover, Ando [1] (see also [2]) showe...

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)...

Miranda-Thompson majorization is a group-induced cone ordering on $mathbb{R}^{n}$ induced by the group of generalized permutation with determinants equal to 1. In this paper, we generalize Miranda-Thompson majorization on the matrices. For $X$, $Yin M_{m,n}$, $X$ is said to be  Miranda-Thompson majorized by $Y$ (denoted by $Xprec_{mt}Y$) if there exists some $Din rm{Conv(G)}$ s...

In this paper, we describe linear maps between complex Banach algebras that preserve products equal to fixed elements. This generalizes some important special cases where the elements are zero or identity element. First show if such map preserves a finite-rank operator, then it must also product. several instances, is enough product preserving be scalar multiple of an algebra homomorphism. Seco...

We note that the well-known result of Von Neumann \cite{von} is not valid for all doubly substochastic operators on discrete Lebesgue spaces $\ell^p(I)$, $p\in[1,\infty)$. This fact lead us to distinguish two classes these operators. Precisely, class increasable $\ell^p(I)$ isolated with property an analogue in this true. The submajorization relation $\prec_s$ positive cone $\ell^p(I)^+$, when ...