Trace class operator

Summary

A trace class operator is a bounded operator on a Hilbert space whose singular values have a finite sum. For such operators the trace $\sum_i \ip{T e_i}{e_i}$ converges absolutely and does not depend on the orthonormal basis, so the trace of matrices extends to them. Trace class operators are central in the theory of Fredholm determinants. In quantum mechanics, density matrices are positive trace class operators of trace $1$.