具正規核的積分方程

<正> §1.引言 凡合條件即是說凡合條件kkx,y]=kkx,y](1.1)的核k(x,y)叫做正規核(normal kernel).這種核顯然包括實對稱核、實畸對稱核、艾氏核及畸艾氏核等為特例。在本文中,我們將討論具此種核之積分方程之性質及解法尤其是關於此種核之特值及奇值(即希米特(E.Schmidt)的特值)之性質

INTEGRAL EQUATIONS WITH NORMAL KERNELS

By a normal kernel, we mean a kernel k(x, y) satisfying the condition kk x, y] =kk x, y], i.ie.,Evidently real symmetric kernels, real skew-symmetric kernels,Hermitian kernels, skew-Hermitian kernels etc. are normal kernels. In this paper, we discuss the properties and solutions of the integral equations with normal kernels, especially the properties of the singular values (E.Schmidt’s characteristic values), the characteristic values of such kernels and their relations.The main results are:1. If k(x, y) is a L~2 nonmal kernel of which the set of singular values is {λ_h}, then the set of singular values of the n-th iterated kernel k~nx, y] is {λ_h~n}.2.Let λ_h be a singular value of rank h_ρ of the kernel k(x, y), i.e., in the complete orthonormal system of adioint singular function {Φ_h(x), ψ_h(y)} there are h_ρ and only h_ρ pairs of functions Φ_(hi)(x), ψ_(hi)(y) (i=1,2,…h_ρ) with the same singular value λ_h such that (φ_(hi),φ_(hj)=δ_(ij), (ψ_(hi),ψ_(hj)=δ_(ij)(i, j = 1, 2,…, h_ρ) Then a necessary and sufficient condition for k(x,y) to be normal is that for each h, we should have where the a_(h,ij) ′s are constants such that the matrix △ = (a_(h,ij)) (i,j=1,2,…h_ρ) is unitary.3.If k(x, y) is normal,then where α_(h,ij)~((1))=α_(h,ij),(α_(h,ij)~((n)))=(α_(h,ij)~n=△~n=(α_(h,ij)~(-n)=△~(1-n). 4.For any L~2 normal kernel k(x,y) there exists a one to one correspondence between the set of characteristic values {μ_h} ahd the set of singular values {λ_h} each arranged in order of non-decreasing absolute value, such that |μ_h|=λ_h (h=1, 2, 3,…).5.Every L~2 normal kernel K(x, y) is expressible in the following form: where ～ denotes convergence in mean square, each k_h(x,y) is an algebraic kernel of the form with an unitary matrix (α_(h,ij)) (i,j = 1, 2,…h_ρ) so that and i) the rank of each characteristic value of k_h(x,y) is equal to its multiplicity; ii) corresponding to the characteristic values μ_(ih)(i = 1, 2,…, h_ρ) (equal or distinct) of k_h(x,y) there exists h_ρ linearly independent characteristic functions u_(hi)(x)(i=1,2,…, h_ρ) and h_ρ linearly independent transpose characteristic functions ν_(hi)(x)(i=1, 2,…, h_ρ) of k(x, y) so that u_(hi)(x) =μ_(hi)ku_(hi)x],ν_(hi)(x)=μ_(hi)kν_(hi)x],(i=1,2,…,h_ρ); iii) each of the u_(hi) (x)'s and of the ν_(hi)(x)'s is a linear combination of the φ_(hi)(x)'s and therefore is also a linear combination of the ψ_(hi)(x)′s;iv)if (u_(hi). u_(hj)) = δ_(ij) = (ν_(hi),ν_(hj)) (i, j=1, 2,…, h_ρ),then u_(hi)(x)=v_(hi)(x) (i=1,2,…h_ρ).(It is only known previously that7.If k(x,y)～k_h(x,y) is a L~2 normal kernel,let u_(hi)(x)(i=1,2,…,h_ρ)denote the complete orthonormal characteristic functions of K_h(x, y), then K(x, y) and all its iterated kernels and the solutions of the integral equation q(x) = φ(x) - λ k φ x] are expressible in terms of the functions u_(hi)(x) (i= 1, 2.…, h_ρ) such that when λ is not a characteristic value of k(x, y), but, if λ=μ_(ts) is a characteristic value of k(x, y), then a necessary and sufficient condition for the solvability of the above integral equation is that q(x) should be orthogonal to all transpose characteristic functions of k(x, y)belonging to μ_(ts), and when this condition is satisfied, the solution is given by where stands for those values of h and i which make μ_(hi)=μ_(ts); the coefficients of u_(hi) (x) may take arbitrary values.8. Suppose k(x, y) is an L~2 normal kernel, and p(x), q(x) ε L~2, then (It is a generalization of a formula of Hilbert for symmetric kernels).As applications we give new proofs to some classical theorems, for exampie, 1) the existence theorem of characteristic values of normal kernels, and 2) the singular points of the resolvent kernel of any L~2 normal kernel are all simple poles, etc.