Trace-Class Operator

Let $H$ be a Hilbert space. A compact operator $X:H\to H$ is trace-class if its singular values are summable:

\|X\|_1:=\sum_{n=1}^{\infty}s_n(X)<\infty.

The singular values include multiplicities and a zero tail for finite-rank operators. The displayed sum is the trace norm.

Trace

For a trace-class operator, the operator trace is defined and independent of the orthonormal basis. Equivalently, $\|X\|_1=\operatorname{Tr}|X|$ , with the positive operator trace interpreted as a possibly infinite sum before imposing finiteness.

Remarks

The principal properties used in Shale's paper are:

The Fredholm determinant is defined for operators of the form $I+X$ with $X$ trace-class.

Examples

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

References

Dan-Virgil Voiculescu, Math 209: Von Neumann Algebras, notes by Leonard Tomczak, UC Berkeley, Spring 2024. Lecture notes, §4, pp. 7–8, Proposition 4.1 and the singular-value expansion.