Compressing Operators to Obtain Single-Point Numerical Ranges

Abstract

We are all familiar with vector spaces and their basic properties. We understand linear operators, dot products, and maybe even norms. What the reader may not be as familiar with, however, is the concept of numerical range. . . . This thesis mainly concerns numerical ranges of linear operators acting on Hilbert spaces. . . . We will begin the paper with a review of concepts from linear algebra. These concepts lay the groundwork for our more complex results. We will also introduce the concept of operator norm, since it will be useful in a study of numerical range. . . . We will study this concept further in the body of the paper. From there, we will move into a general introduction of numerical range, with analysis of some of its important properties. Finally, we will address Professor Terilla's problem and conclude with an example in which we use our augmentation theorem. [From Introduction]