Determinant on I + Trace-Class

Let $H$ be a Hilbert space and let $A$ be a trace-class operator on $H$ . The Fredholm determinant of $I+A$ is

\Delta(I+A)=\det(I+A):=\prod_j(1+\lambda_j(A)),

where the nonzero eigenvalues are counted with algebraic multiplicity. The product converges and extends the finite-dimensional determinant.

Remarks

On a neighborhood of $I$ where a logarithm is defined, $\Delta(T)=\exp\bigl(\operatorname{tr}(\log T)\bigr).$
The restriction of $\Delta$ to the invertible group $GL(H)_1=\{I+A:A\text{ trace class and }I+A\text{ invertible}\}$ is continuous in the trace norm.

Examples

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