Unattained Boundary Points of the Numerical Range of Hilbert Space Operators

Abstract

The numerical range of an operator on a complex Hilbert space is considered and we introduce various known results associated with different points of the numerical range. By the Toeplitz-Hausdorff theorem, the numerical range is a convex set in the complex plane, though not necessarily closed. Thus these results, though interesting, are inapplicable to the unattained boundary points of the numerical range. Therefore, generalizations of these results which may hold for all points of the closure of the numerical range seem to be called for. .... Embry has shown that the subsets associated with points of the numerical range behave in a particular fashion if the operator has special characteristics and vice versa. We achieve some easy generalizations of these results for subsets consisting of sequences. Several results of C.S. Lin, 5.G. Stampfli and G. de Barra concerning seminormal and convexoid operators are then extended to the unattained boundary points of the numerical range. K.C. Das and G. Garske gave a theorem concerning weak convergence to zero at the unattained extreme points of the closure of the numerical range. Das and Craven also gave a bound for the norm of the weak limit of sequences corresponding to points on a line segment on the boundary of the numerical range. We achieve all these results as a simple corollary of a generalized Cauchy-Schwartz inequality. In the concluding part of our thesis we investigate whether convexity holds for other numerical ranges as well. A restricted numerical range is defined and certain conditions are imposed so that this newly defined numerical range is convex. As a corollary to this result, we deduce the convexity of Stampfli's numerical range, a result proved differently by J. Kyle.