Type: Article

Infinite numerical range is convex

Roldán-Pensado, Edgardo;

Abstract:

In a Hilbert space H, the n-rank numerical range of a bounded linear operator T:H?H is defined as the set of ??C such that there exists a projection P of rank n that satisfies PTP=?P. This is a generalization of the classical numerical range. It has been proved in [H. J. Woerdeman, Linear Multilinear Algebra 56 (2008), no. 1-2, 65–67; MR2378302] that every finite rank numerical range is convex. We prove that the infinite rank numerical range is convex. To do this, we show that the infinite rank numerical range is the intersection of all the finite rank numerical ranges.