Knowlpedia

Trace-Class Operator

An operator with absolutely summable singular values (Schatten class 1)

A bounded operator XX is trace-class if its trace norm ∥X∥1<∞\|X\|_1<\infty (equivalently, the singular values are ℓ1\ell^1).

Remarks

Key properties (paper use):

  • The trace tr(X)\mathrm{tr}(X) is well-defined and basis-independent.
  • The "determinant" Δ(I+X)\Delta(I+X) extends to I+I+trace-class ().
Examples
  • On ℓ2\ell^2, diag(an)\mathrm{diag}(a_n) is trace-class iff ∑n∣an∣<∞\sum_n |a_n|<\infty.