Higher rank numerical ranges of normal operators and unitary dilations

Abstract

Here we give a closure free description of the higher rank numerical range of a normal operator acting on a separable Hilbert space. This generalizes a result of Avendaño for self-adjoint operators. It has several interesting applications. We show using Durszt's example that there exists a normal contraction T for which the intersection of the higher rank numerical ranges of all unitary dilations of T contains the higher rank numerical range of T as a proper subset. We strengthen and generalize a result of Wu by providing a necessary and sufficient condition for the higher rank numerical range of a normal contraction being equal to the intersection of the higher rank numerical ranges of all possible unitary dilations of it. We also show that the above condition is necessary for a general contraction.