Numerical ranges are shadows

Abstract

We present a new perspective on the numerical range of $n\times n$ matrices as varying “shadows” of an embedding of $C{P}^{n-1}$. This framework gives us geometric proofs of the elliptic range theorem and the Toeplitz-Hausdorff theorem. We apply this perspective to the Berezin Range or linear operators on finite-dimensional supbspaces of the Hardy-Hilbert space $H^2\left(D\right)$ of the open unit disk $D$. We characterize the convexity of the Berezin range for two-dimensional subspaces and show that uncountably many unitary conjugates of a given operator are needed to “cover” the numerical range using Berezin ranges if the boundary of the numerical range is not smooth.