Hilbert-Carleman Determinants

Chapter X presents examples of algebras on which the regularized determinants are computed. We show that in these algebras the regularized determinant coincides with the Hilbert-Carleman determinant. The first section, which is of a preliminary nature, contains basic formulas for the Hilbert-Carleman determinant for integral operators with degenerate kernel. These formulas are then used in Section 2 to introduce the Hilbert-Carleman determinant for integral operators in Banach spaces. The Hilbert-Carleman determinant on the algebra of Hilbert-Schmidt operators appears in Section 3. In Section 4 the Hilbert-Carleman determinant is defined on the Mikhlin-Itskovich algebra. This algebra is a natural generalization of the algebra of Hilbert-Schmidt operators. The discrete analogue of these results appears in Section 7. Section 5 contains an analysis of a specific algebra of integral operators. The so-called diagonally modified Hilbert-Carleman determinant is introduced and studied in Section 6.