Hilbert–Schmidt and Trace Class Operators

Abstract

This chapter introduces two subspaces of the space of compact operators and presents their basic theory in substantial detail. These spaces of operators are important in various areas of functional analysis and in applications of operator theory to quantum physics. Accordingly, after the characterization of Hilbert-Schmidt and trace class operators has been presented, the spectral representation for these operators is derived. Furthermore the dual spaces (spaces of continuous linear functionals) of these two spaces of operators are determined and their rôle in the description of locally convex topologies on the space $\mathcal{B}(\mathcal{H})$ of all bounded linear operators on a Hilbert space $\mathcal{H}$ is explained. Finally two results are included which are mainly used in quantum physics: Partial trace for trace class operators on tensor products of separable Hilbert spaces and Schmidt decomposition.