Some new classes of kernels whose Fredholm determinants have order less than one

Abstract:

Let $K(s,t)$ be a complex-valued ${L_2}$ kernel on the square $a \leqq s,t \leqq b$ and $\{ {\lambda _v}\}$, perhaps empty, denote the set of finite characteristic values (f.c.v.) of $K$, arranged according to increasing modulus. Such f.c.v. are complex numbers appearing in the integral equation ${\phi _v}(s) = {\lambda _v}\int _a^b {K(s,t){\phi _v}(t)dt}$, where the ${\phi _v}(s)$ are nontrivial ${L_2}$ functions on $[a,b]$. Further let ${k_1} = \int _a^b {K(s,s)}$ be well defined so that the Fredholm determinant of $K,D(\lambda )$, exists, and let $\mu$ be the order of this entire function. It is shown that (1) if $K(s,t)$ is a function of bounded variation in the sense of Hardy-Krause, then $\mu \leqq 1$; (2) if in addition to the assumption (1), $K(s,t)$ satisfies a uniform Lipschitz condition of order $\alpha > 0$ with respect to either variable, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$; (3) if $K(s,t)$ is absolutely continuous as a function of two variables and ${\partial ^2}K/\partial s\partial t$ (which exists almost everywhere) belongs to class ${L_p}$ for some $p > 1$, then $\mu < 1$ and ${k_1} = {\Sigma _v}1/{\lambda _v}$. In (2) and (3), the condition ${k_1} \ne 0$ implies $K(s,t)$ possesses at least one f.c.v.