  We consider a family of linear operators, diagonalized by the Hankel
transform. The Fredholm determinants of these operators, restricted to $L_2[0,
R]$, are expressed in a convenient form for asymptotic analysis as
$R\to\infty$. The result is an identity, in which the determinant is equal to
the leading asymptotic multiplied by an asymptotically small factor, for which
an explicit formula is derived. We apply the result to the determinantal point
process with the Bessel kernel, calculating the speed of the convergence of
additive functionals with respect to the Kolmogorov-Smirnov metric.
