Trace-Class Operator

A bounded operator $X$ is trace-class if its trace norm $\|X\|_1<\infty$ (equivalently, the singular values are $\ell^1$).

Key properties (paper use):

• The trace $\mathrm{tr}(X)$ is well-defined and basis-independent.
• The “determinant” $\Delta(I+X)$ extends to $I+$trace-class (Fredholm determinant ).

Example: On $\ell^2$, $\mathrm{diag}(a_n)$ is trace-class iff $\sum_n |a_n|<\infty$.