An interrelationship between the numerical range of matrix polynomials and its factorization, with linear or nonlinear factors, is investigated here. Also, a bound on the number of components of numerical range is given.
The numerical range of an n × n matrix polynomial P(λ) = Amλm + Am–1λm–1 + … + A1λ + A0 is defined by W(P) = {λ ∈ ℂ : x*P(λ)x = 0, x ∈ ℂn, x*x = 1}, and plays an important role in the study of matrix polynomials. In this paper, we describe a methodology for the illustration of its boundary, ∂W(P), using recent theoretical results on numerical ranges and algebraic curves. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
In this paper we prove that every nonlinear ∗ -Lie derivation from a factor von Neumann algebra into itself is an additive ∗ -derivation.