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Title: K-numerical ranges of operators on Hilbert spaces
Author: Nashvadian Bakhsh, B.
Awarding Body: University College of Swansea
Current Institution: Swansea University
Date of Award: 1978
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The concept of k-numerical 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 well-known Toeplitz-Hausdorff 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 k-numerical ranges, whenever k > 1. Our thesis is concerned with a detailed analysis of k-numerical 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 k-numerical 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 k-numerical ranges. We study the relationship of the spectral properties of such an operator with its k-numerical range, as well as continuity properties of the map which assigns to each operator the closure of its k-numerical range. We devote several sections of Chapter 2 to specific examples: the k-numerical ranges of shift-operators (and their iterates) on finite-dimensional and separable Hilbert spaces are determined. The final chapter is concerned with the k-numerical radius of a given operator T. We study the relationship, for fixed k, between the k-numerical radius of T and the operator-norm of T. From the main result in Chapter 2, we deduce that the sequence of the k-numerical 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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available