  Products of shifted characteristic polynomials, and ratios of such products,
averaged over the classical compact groups are of great interest to number
theorists as they model similar averages of L-functions in families with the
same symmetry type as the compact group. We use Toeplitz and Toeplitz plus
Hankel operators and the identities of Borodin - Okounkov, Case - Geronimo, and
Basor - Erhardt to prove that, in certain cases, these unitary averages factor
as polynomials in the matrix size into averages over the symplectic group and
the orthogonal group. Building on these identities we present new proofs of the
exact formulas for these averages where the ``swap'' terms that are
characteristic of the number theoretic averages occur from the Fredholm
expansions of the determinants of the appropriate Hankel operator. This is the
fourth different proof of the formula for the averages of ratios of products of
shifted characteristic polynomials; the other proofs are based on
supersymmetry; symmetric function theory, and orthogonal polynomial methods
from Random Matrix Theory.
