In functional analysis, the Hilbert–Carleman determinant is an operator determinant for certain integral operators on Banach spaces, whose kernels are not necessarily continuous. Unlike Fredholm determinant which is generally not defined for integral operators whose kernels are discontinuous on the diagonal, the Hilbert–Carleman determinant can be defined even when this condition fails. Similarly to the Fredholm determinant, the Hilbert–Carleman determinant is defined for sums of the form formula_1 where formula_2 is the identity operator and formula_3 is an integral operator.
The Hilbert–Carleman determinant is named after David Hilbert and Torsten Carleman.
Hilbert–Carleman Determinant.
Let formula_4 and let formula_5 be the L^p space over a measure space formula_6 with Lebesgue measure formula_7, where formula_8. Consider the integral operator
formula_9
acting on the Banach space formula_10 and let formula_2 denote the identity operator. Then the Hilbert–Carleman determinant of formula_12 is defined by
formula_13
where
formula_14
formula_18