Compressions of Linear Operators Yielding a Single Point Numerical Range

Abstract

This thesis will focus on finding subspaces M such that the compression of T to M, denoted TM, has a single point numerical range. . . . The motivation for finding such subspaces lies in a theorem from quantum coding theory, which states that an error process is correctable on a subspace M if the compression to M of each member of a particular collection of linear operators associated with that error yields a single point numerical range. It is, in particular, desireable to find such subspaces M of highest possible dimension. . . . We will conclude with an investigation of Wr(T) for Ta normal operator, finding subsets of the numerical range of T in which Wr(T) must be contained, given r an integer such that Wr(T) is non-empty, and establishing a complete characterization of the subsets Wr(T) of W(T) when T : Cn ---> Cn has distinct eigenvalues forming a convex n-gon and n <= 6. We will finally conjecture that for T normal, Wr(T) is a convex set for every r. [From Introduction]