Numerical ranges of nilpotent operators

Abstract

For any operator A on a Hilbert space, let W(A), w(A) and w(0)(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if A(n) = 0, then w(A) <= (n - 1)w(0)(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A) = (n - 1)w(0)(A), (2)A is unitarily equivalent to an operator of the form aA(n) circle plus A', where a is a scalar satisfying vertical bar a vertical bar = 2w(0)(A), A(n) is the n-by-n matrix [GRAPHICS] and A' is some other operator, and (3) W(A) = bW(A(n)) for some scalar b. (C) 2008 Elsevier Inc. All rights reserved.