The Fredholm determinant

Let H be a Hilbert space with an inner product that is conjugate linear in the first variable. We do not presume unless we say so that H is separable. We denote by B(H) the set of bounded linear operators H → H. For any A ∈ B(H), A∗A is positive and one proves that it has a unique positive square root |A| ∈ B(H). We call |A| the absolute value of A. We say that U ∈ B(H) is a partial isometry if there is a closed subspace X of H such that the restriction of U to X is an isometry X → U(X) and kerU = X⊥. One proves that for any A ∈ B(H), there is a unique partial isometry U satisfying both kerU = kerA and A = U |A|, and A = U |A| is called the polar decomposition of A. Some useful identities that the polar decomposition satisfies are U∗U |A| = |A|, U∗A = |A|, UU∗A = A. If A ∈ B(H) is compact and self-adjoint, the spectral theorem tells us that there is an orthonormal set {en : n ∈ N} in H and λn ∈ R, |λ1| ≥ |λ2| ≥ · · · , such that Ax = ∑