On generalisedk-numerical radii of schatten class operators

Abstract

Let T be a bounded linear operator on a complex infinite-dimensional Hilbert space. For a natural number k, we introduce the concept of a generalised k-numerical range, and associated k-numerical radius. Of T with respect to an infinite sequence of complex numbers. These concepts reduce to those of the classical k-numerical range and radius(originally due to Hatmos) m the case where c is the special sequence with cn = 1 for all n. For a wide class of sequences c. which includes this special sequence, we give a characterisation of trace-class operators in terms of the order of magnitude of their generalised k-numerical radii. Moreover, in terms involving this order of magnitude, we state a condition which is necessary, as well as a condition which is sufficient, for a compact operator T to belong to a given Schatten class Cp when p > 1. Neither of these conditions, however, is both necessary and sufficient.