Properties of kernels of integral equations whose iterates satisfy linear relations

Abstract

<p>The principle result obtained in this thesis is the theorem that if the iterated kernels of an integral equation satisfy a linear relation a1K1x,y +a2K2x,y +&cdots;+aKnx,y ≡0,a1≠0, then the kernel K(x,y) must be of the special form i=1N ui(x)vi(y). Using this result it is shown that only such kernels can have a Fredholm determinant D(lambda) and Fredholm first minor D(x,y; lambda) being polynomials in lambda of the same degree. Also in the particular case of a continuous symmetric kernel it is shown that if either D(lambda) or Dx,y;lambda) are polynomials in lambda that then K(x,y) must be of the specia1 form given above;Further properties of idempotent kernels, i.e. ones for which K2x,y≡ K1x,y, are deduced. The connection between such kernels and an idempotent Markoff process is pointed out thus indicating a possible application of the theory of this thesis to certain problems in probability. An alternate proof, of a known result concerning such Markoff processes is given;The results of the thesis are generally derived under the assumption of measurability and boundedness of the kernel K(x,y). Also integration is in the sense of Lebesgue.</p>