Determinant on I + Trace-Class

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

Key properties (paper use):

• Near $I$: $\Delta(T)=\exp(\mathrm{tr}(\log T))$ (Lemma 2.1(a)).
• $\Delta$ is continuous on $GL(H)_1=I+$trace-class.
• No continuous extension exists on $GL(H)_2=I+$Hilbert–Schmidt (Lemma 2.1(b)).

Example: If $A$ is finite rank, $\Delta(I+A)$ matches the usual finite-dimensional determinant.