Title:

Knumerical ranges of operators on Hilbert spaces

The concept of knumerical range for an operator on a Hilbert space wan originally suggested by P.R. Halmos as a suitable generalization of the spatial numerical range of such an operator. Hero k is a natural number, and the spatial numerical range corresponds to the case where k = 1. The wellknown ToeplitzHausdorff theorem assorts that the spatial numerical range is always a convex subset of the complex plane, and it was proved by C.A. Forger that this result is still true for knumerical ranges, whenever k > 1. Our thesis is concerned with a detailed analysis of knumerical ranges, and in particular of the properties of such ranges which generalise familiar facts concerning the spatial numerical range. In a brief appendix we comment on certain aspects of spatial numerical range which do not seem to be shared by knumerical ranges in general. Chapter 1 contains no new results, and is a concise summary of known facts required for subsequent investigation. The main result in Chapter 2 asserts that the essential numerical range of a Hilbert space operator coincides with the intersection (over all k) of the closures of its knumerical ranges. We study the relationship of the spectral properties of such an operator with its knumerical range, as well as continuity properties of the map which assigns to each operator the closure of its knumerical range. We devote several sections of Chapter 2 to specific examples: the knumerical ranges of shiftoperators (and their iterates) on finitedimensional and separable Hilbert spaces are determined. The final chapter is concerned with the knumerical radius of a given operator T. We study the relationship, for fixed k, between the knumerical radius of T and the operatornorm of T. From the main result in Chapter 2, we deduce that the sequence of the knumerical radii of T converges to the essential numerical radius of T, as k approaches infinity. In particular, when T is compact, this sequence converges to zero. The rate of convergence to zero when T lies in a specified von Neumann Schatten class is investigated.
