AbstractsMathematics

The spatial numerical range of an operator on a normed linear space and the algebra numerical range of an element of a unital Banach algebra, as developed by G. Lumer and F. F. Bonsall, are considered and the theory of such numerical ranges applied to Banach algebra. The first part of the thesis is largely expository as in it we introduce the basic results on numerical ranges. For an element of a unital Banach algebra, the question of approximating its spectrum by numerical ranges has been considered by F. F. Bonsall and J. Duncan. We give an alternative proof that the convex hull of the spectrum of an element may be approximated by its numerical range defined with respect to equivalent renormings of the algebra. In the particular case of operators on a Hilbert space, this leads to a sharper version of a result by J. P. Williams. An element is hermitian if it has real numerical range. Such an element is characterized in terms of the linear subspace spanned by the unit, the element and its square. This is used to characterize Banach*–algebras in which every self–adjoint element is hermitian. From this an elementary proof that such algebras are B*-algebras in an equivalent norm is given. As indicated by T. W. Palmer, a formula of L. Harris is then used to show that the equivalent renorming is unnecessary, thus giving a simple proof of Palmer's characterization of B*-algebras among Banach algebras. The closure properties of the spatial numerical range are studied. A construction of B. Berberian is extended to normed linear spaces, however because the numerical range need not be convex, the result obtained is weaker than that of Berberian for Hilbert spaces. A Hilbert space or an Lp-space, for p between one and infinity, is seen to be finite dimensional if and only if all the compact operators have closed spatial numerical range. The spatial numerical range of a compact operator, on a Hilbert space or an Lp-space, for p between one and infinity, is shown to contain all the non–zero extreme points of its closure. So, for a compact operator on a Hilbert space the spatial numerical range is closed if and only if it contains the origin, thus answering a question of P. R. Halmos. Operators that attain their numerical radius are also considered. A result of D. Hilbert is extended to a class of Banach spaces. In a Hilbert space the hermitian operators, which attain their numerical radius, are shown to be dense among all the hermitian operators. This leads to a stronger form of a result by J. Lindenstrauss in the spatial case of operators on a Hilbert space. PhD Doctorate