Factorization of fredholm operators on analytic functions

Abstract

AbstractLet Ω be a simply connected domain in the complex plane, and A(Ωn), the space of functions which are defined and analytic on Ωn, if K is the operator on elements u(t, a1, …, an) of A(Ωn + 1) defined in terms of the kernels ki(t, s, a1, …, an) in A(Ωn + 2) by Ku = ∑i = 1n ∝aitk i(t, s, a1, …, an) u(s, a1, …, an) ds ϵ A(Ωn + 1) and I is the identity operator on A(Ωn + 1), then the operator I − K may be factored in the form (I − K)(M − W) = (I − Π K)(M − ΠW). Here, W is an operator on A(Ωn + 1) defined in terms of a kernel w(t, s, a1, …, an) in A(Ωn + 2) by Wu = ∝ant w(t, s, a1, …, an) u(s, a1, …, an) ds. Π W is the operator; Π Wu = ∝an − 1 w(t, s, a1, …, an) u(s, a1, …, an) ds. Π K is the operator; Π Ku = ∑i = 1n − 1 ∝ait ki(t, s, a1, …, an) ds + ∝an − 1t kn(t, s, a1, …, an) u(s, a1, …, an) ds. The operator M is of the form m(t, a1, …, an)I, where m ϵ A(Ωn + 1) and maps elements of A(Ωn + 1) into itself by multiplication. The function m is uniquely derived from K in the following manner. The operator K defines an operator K∗ on functions u in A(Ωn + 2), by K∗u = ∑i = 1n − 1 ∝ait ki(t, s, a1, …, an) u(s, a, …, an + 1) ds + ∝an + 1t kn(t, s, a1, …, an) u((s, a1, …, an + 1) ds. A determinant δ(I − K∗) of the operator I − K∗ is defined as an element m∗(t, a1, …, an + 1) of A(Ωn + 2). This is mapped into A(Ωn + 1) by setting an + 1 = t to give m(t, a1, …, an). The operator I − Π K may be factored in similar fashion, giving rise to a chain factorization of I − K. In some cases all the matrix kernels ki defining K are separable in the sense that ki(t, s, a1, …, an) = Pi(t, a1, …, an) Qi(s, a1, …, an), where Pi is a 1 × pi matrix and Qi is a pi × 1 matrix, each with elements in A(Ωn + 1), explicit formulas are given for the kernels of the factors W. The various results are stated in a form allowing immediate extension to the vector-matrix case