Knowlpedia

Determinant on I + Trace-Class

Extension of det via det(I+A)=exp(tr log(I+A)) for trace-class A

For trace-class AA, one defines a determinant-like map Δ(I+A)\Delta(I+A) ("Fredholm determinant") extending from finite rank.

Remarks

Key properties (paper use):

  • Near II: Δ(T)=exp⁡(tr(log⁡T))\Delta(T)=\exp(\mathrm{tr}(\log T)) (Lemma 2.1(a)).
  • Δ\Delta is continuous on GL(H)1=I+GL(H)_1=I+trace-class.
  • No continuous extension exists on GL(H)2=I+GL(H)_2=I+Hilbert–Schmidt (Lemma 2.1(b)).
Examples
  • If AA is finite rank, Δ(I+A)\Delta(I+A) matches the usual finite-dimensional determinant.