Fredholm determinant and decay properties of eigenvalues of hilbert-schmidt integral operators

Abstract

In this thesis, some properties of the eigenvalues of Hilbert-Schmidt integral operators (briefly H-S operators) are investigated. This thesis consists of three chapters. In the first two chapters, H-S operators of the type are considered, and certain growth properties of their eigenvalues are studied. In the third chapter, symmetric H-S operators of the type (Krf)(x)=?k(x,y)f(y)dy(f?L2(R), for r>0)(K_rf)(x) = \int k(x,y)f(y)dy \quad (f \in L^2(\mathbb{R}), \text{ for } r > 0)(Kr?f)(x)=?k(x,y)f(y)dy(f?L2(R), for r>0) are considered, and the Kac-Akhiezer formulae for the Fredholm determinant of KrK_rKr? are extended to a fairly general class of kernels and a family of domains of integration. When KKK is symmetric, from classical theory [23], it is known that KKK has a sequence of eigenvalues {?n}n=1?\{\lambda_n\}_{n=1}^\infty{?n?}n=1??. This implies that ?n\lambda_n?n? decays faster than any polynomial rate as n??n \to \inftyn??. It was observed by H. Weyl [30] that if III is a bounded open interval (a,b)(a, b)(a,b) and kkk is a real symmetric continuously differentiable kernel on I×II \times II×I, then lim?n??ns?n=0\lim_{n \to \infty} n^s \lambda_n = 0limn???ns?n?=0 and that when III is a closed bounded interval and kkk is a real symmetric sss-times continuously differentiable kernel on I×II \times II×I, then lim?n??ns?n=0(s=0,1,2,…?).\lim_{n \to \infty} n^s \lambda_n = 0 \quad (s = 0, 1, 2, \dots).limn???ns?n?=0(s=0,1,2,…). From this, it follows that if III is a closed interval and kkk is a real symmetric infinitely differentiable function on I×II \times II×I, then lim?n??ns?n=0(?s?N),\lim_{n \to \infty} n^s \lambda_n = 0 \quad (\forall s \in \mathbb{N}),limn???ns?n?=0(?s?N), which is equivalent to {?n}??p\{\lambda_n\} \in \ell^p{?n?}??p for all 0<p<?0 < p < \infty0<p<? [19]. This space is denoted by S1S_1S1?. The goal of the first chapter is to characterize H-S operators on L2(I)L^2(I)L2(I) (with III an open interval) whose eigenvalue sequences belong to S1S_1S1?. The main results proved in this direction are:

A necessary and sufficient condition for the eigenvalue sequence of a symmetric H-S operator KKK on L2(I)L^2(I)L2(I) (with III an open interval) to be in the space S1S_1S1? is that KKK should be unitarily equivalent to a H-S operator GGG on L2(T)L^2(T)L2(T) induced by a real symmetric square-integrable kernel

g?S(R2)g \in \mathcal{S}(\mathbb{R}^2)g?S(R2) if I=(??,?)I = (-\infty, \infty)I=(??,?) (Theorem 1.4.1); g?Wm,2(J0×J0)g \in W^{m,2}(J_0 \times J_0)g?Wm,2(J0?×J0?) if III is a bounded interval where J0=J?{midpoint of I}J_0 = J \setminus \{\text{midpoint of } I\}J0?=J?{midpoint of I} (Theorem 1.3.1); g?S?g \in S^-g?S? if I=(a,?)I = (a, \infty)I=(a,?) with a?Ra \in \mathbb{R}a?R (Theorem 1.4.2); g?S+g \in S^+g?S+ if I=(??,a)I = (-\infty, a)I=(??,a) with a?Ra \in \mathbb{R}a?R.

Where: S(R2)\mathcal{S}(\mathbb{R}^2)S(R2) is the Schwartz space on R2\mathbb{R}^2R2; Wm,2(J0×J0)W^{m,2}(J_0 \times J_0)Wm,2(J0?×J0?) is the L²-Sobolev space of order mmm; S?S^-S? and S+S^+S+ are linear spaces defined by logarithmic and polynomial decay conditions.

There are integral operators whose eigenvalues decay even faster than those considered above — for example, the integral operator with the Poisson kernel k(x,y)=1?h22?(1?2hcos?(x?y)+h2)(??<x,y<?),k(x,y) = \frac{1 - h^2}{2\pi(1 - 2h\cos(x - y) + h^2)} \quad (-\pi < x, y < \pi),k(x,y)=2?(1?2hcos(x?y)+h2)1?h2?(??<x,y<?), for a given h?(?1,1)h \in (-1, 1)h?(?1,1). The aim of the second chapter is to analyze such operators. Main results include:

A necessary condition for the eigenvalue sequence {?n}\{\lambda_n\}{?n?} of a symmetric H-S operator KKK on L2(I)L^2(I)L2(I) to satisfy lim?n??ns?n=0for s=0,1,2\lim_{n \to \infty} n^s \lambda_n = 0 \quad \text{for } s = 0, 1, 2limn???ns?n?=0for s=0,1,2 is that KKK is unitarily equivalent to a symmetric H-S operator GGG on L2(I)L^2(I)L2(I) induced by a kernel

g?Hr(R2)g \in \mathcal{H}_r(\mathbb{R}^2)g?Hr?(R2) if I=(??,?)I = (-\infty, \infty)I=(??,?) (Theorem 2.2.1); g?Hr(J0×J0)?Wm,2(J0×J0)g \in \mathcal{H}_r(J_0 \times J_0) \cap W^{m,2}(J_0 \times J_0)g?Hr?(J0?×J0?)?Wm,2(J0?×J0?) if III is bounded (Corollary 2.2.1); g?Hr(I×I)?S?g \in \mathcal{H}_r(I \times I) \cap S^-g?Hr?(I×I)?S? if I=(a,?)I = (a, \infty)I=(a,?) (Corollary 2.2.2); g?Hr(I×I)?S+g \in \mathcal{H}_r(I \times I) \cap S^+g?Hr?(I×I)?S+ if I=(??,a)I = (-\infty, a)I=(??,a). Where Hr\mathcal{H}_rHr? is the space of real analytic functions.

A sufficient condition for the eigenvalue sequence {?n}\{\lambda_n\}{?n?} of a symmetric H-S operator KKK on L2(I)L^2(I)L2(I) (with III a closed bounded interval) to satisfy lim?n??ns?n=0?s?N,??>0\lim_{n \to \infty} n^s \lambda_n = 0 \quad \forall s \in \mathbb{N}, \forall \varepsilon > 0limn???ns?n?=0?s?N,??>0 is that KKK is unitarily equivalent to a symmetric H-S operator GGG on L2(I)L^2(I)L2(I) induced by a kernel ggg which is real analytic on I×II \times II×I (Theorem 2.2.2).

A counterexample shows that this result does not hold in general when ?=0\varepsilon = 0?=0. In the third chapter, for a class of continuous kernels of convolution type, the Fredholm determinant Dt(?)D_t(\lambda)Dt?(?) of the operator (Tf)(x)=?k(x?y)f(y)dy(Tf)(x) = \int k(x - y)f(y)dy(Tf)(x)=?k(x?y)f(y)dy is given by the Kac-Akhiezer formula: Dt(?)=exp?(??0ta(0,s;?)ds),D_t(\lambda) = \exp\left(-\int_0^t a(0,s;\lambda)ds\right),Dt?(?)=exp(??0t?a(0,s;?)ds), where a(x,t;?)a(x,t;\lambda)a(x,t;?) solves the integral equation: a(x,t;?)=?k(x)+??k(x?y)a(y,t;?)dy.a(x,t;\lambda) = \lambda k(x) + \lambda \int k(x - y)a(y,t;\lambda)dy.a(x,t;?)=?k(x)+??k(x?y)a(y,t;?)dy. This formula is extended to non-convolution kernels and higher-dimensional domains under conditions (0.2)–(0.9), including symmetry, square integrability, and continuity.