Higher-dimensional numerical ranges of quadratic operators

Abstract

We show that for every positive integer k, the k-numerical range of a square-zero operator on a (separable) Hilbert space is an (open or closed) circular disc centered at the origin. The radius and the closedness of the disc can be completely determined in terms of the "singular numbers" of the operator. The k-numerical range of idempotent operators is more difficult to describe since its boundary is in general not any familiar curve. What we do is to give enough information, again in terms of the singular numbers of the idempotent operator under consideration, so as to have a general idea of its shape and location.