The numerical range of derivations

Abstract

Let p, q, n be integers satisfying 1 ≤ p ≤ q ≤ n. The (p, q)-numerical range of an n×n complex matrix A is defined by Wp,q(A) = {Ep((UAU*)[q]): U unitary}, where for an n×n complex matrix X, X[q] denotes its q×q leading principal submatrix and Ep(X[q]) denotes the pth elementary symmetric function of the eigenvalues of X[q]. When 1 = p = q, the set reduces to the classical numerical range of A, which is well known to be convex. Many authors have used the concept of classical numerical range to study different classes of matrices. In this note we extend the results to the generalized cases. Besides obtaining new results, we collect existing ones and give alternative proofs for some of them. We also study the (p,q)-numerical radius of A defined by rp,q(A) = max{|μ|:μ ∈ Wp,q(A)}. © 1989.