Let $\mathcal{H}$ be a Hilbert space, and let $T: \mathcal{H} \rightarrow \mathcal{H}$ be a trace-class operator.  Define
$$ f_T(z) = \sum_{i=0}^\infty \mbox{Tr}(\wedge^k T) \cdot z^k, $$

the ordinary generating function for traces of exterior powers of $T$.  Expressed another way,

$$ f_T(z) = \mbox{Det}(I + zT) $$

where $\mbox{Det}$ is the Fredholm determinant.  This function is entire, and can be considered a generalization of the characteristic polynomial.

I am wondering if 
$$f_T(z) = e^z $$
for some natural choice of $T$ on some nice incarnation of Hilbert space.

The motivation for this problem comes from representation theory, where finite minors of such a $T$ provide a formula for dimensions of irreps of $S_n$, the symmetric group on $n$ letters.