Matricial ranges of quadratic operators

Abstract

We show that if T is a quadratic operator on a Hilbert space, then (1) the numerical range of T is an (open or closed) elliptical disc (or its degenerate form) and (2) for every n greater than or equal to 1, the nth matricial range of T consists of n x n matrices whose numerical ranges are contained in the closure of the numerical range of T.