Unitarily Invariant Norms and Inequalities

In this chapter, we first introduce the notion of the trace of a matrix and its interesting properties. We then study the concept of the trace in a broader context. We explore the unitarily invariant norms on Hilbert space operators, which are symmetric gauge functions of singular values. Each of these norms is defined on a two-sided ideal of the algebra of bounded linear operators acting on a Hilbert space transforming the ideal into a Banach space. Examples of unitarily invariant norms include the operator norm, the Hilbert–Schmidt norm, the trace norm, and the Ky Fan norms. The study of unitarily invariant norms is traced back to von Neumann [1], and Fan and Hoffman [2]. We also present several elegant inequalities that are related to unitarily invariant norms, particularly the Hilbert–Schmidt and trace norms.