\(p\)-determinant regularization method for elliptic boundary problems
Let \(H\) be a Hilbert space and \(A : H \to H\) a compact operator. If \(A\) is of trace class, \(\text{det}_ 1(I - A)\) is defined as \(\prod^ \infty_{j = 1} (1 - \lambda_ j)\), where \(\lambda_ j\) are the eigenvalues of \(A\). If \(A\) is in the Schatten class \(J_ p\) (i.e. if \(\sum| \lambda_ j |^ p < \infty)\), then one defines the \(p\)- determinant by \[ \text{det}_ p(I - A) = \text{det}_ 1[(I - (I - A) \exp (A + A^ 2/2 + \cdots + A^{p-1}/(p-1))]. \] The author derives an expression for the \(p\)-determinant of the ``quotient'' of two elliptic differential operators with boundary conditions in terms of the boundary values of their solutions.
scientific article; zbMATH DE number 617392