Some results on finite section linear integral operators

Abstract

This thesis deals mainly with some spectral properties of finite section linear integral operators from L2[0,t]L^2[0,t]L2[0,t] into itself, defined as Ttf(x)=?0tK(x,y)f(y)?dyT_t f(x) = \int_0^t K(x,y) f(y) \, dyTt?f(x)=?0t?K(x,y)f(y)dy under suitable conditions on the kernel K(x,y)K(x,y)K(x,y). If TtT_tTt? is symmetric and completely continuous, then the spectrum of TtT_tTt? consists only of at most a countable number of real eigenvalues, with zero as the only possible limit point. Let {?j+(t)}\{\lambda_j^+(t)\}{?j+?(t)} and {?j?(t)}\{\lambda_j^-(t)\}{?j??(t)} be the sets of all positive and negative eigenvalues, respectively, arranged as ?1+(t)>?2+(t)>…\lambda_1^+(t) > \lambda_2^+(t) > \dots?1+?(t)>?2+?(t)>… In Chapter 1, we consider TtT_tTt? to be symmetric and completely continuous for 0<t<M<?0 < t < M < \infty0<t<M<?, for a sufficiently large MMM, and prove a regularity property of the multiplicity of the eigenvalues of TtT_tTt?, namely: for fixed mmm, the set {t?(0,M):?m+(t)?B}\{ t \in (0,M) : \lambda_m^+(t) \in B \}{t?(0,M):?m+?(t)?B} has multiplicity mmm, where m=1,2,…m = 1, 2, \dotsm=1,2,…, and is open. In the above definition, BBB denotes a suitably defined closed countable set. Then we prove a continuity property of the eigenfunction ?m(x,t)\phi_m(x,t)?m?(x,t) as a function of ttt when m=1,2m = 1, 2m=1,2. We also extend these results for general symmetric and a class of non-symmetric kernels. In Chapter 2, we consider the operator to be normal for all ttt and analyze the properties of the eigenvalues and eigenfunctions as functions of ttt. In this connection, we define a new function ?i(t)\Lambda_i(t)?i?(t) for each i?N i \in \mathbb{N}i?N, which takes its values from the spectrum St={?n(t)}n=1?S_t = \{\lambda_n(t)\}_{n=1}^\inftySt?={?n?(t)}n=1?? of TtT_tTt? for each ttt, and prove some continuity and differentiability properties for that function. Assuming TtT_tTt? to be symmetric, Vittal Rao [J. Math. Anal. Appl., 51 (1976), 79–88 and Math. Anal. Appl., 59 (1977), 60–68] proved the Kac–Akhiezer formula for the eigenvalues and related them to Chandrasekhar's X-function. In Chapter 3, we extend some of these results for normal operators using the results of Chapter 2. Chapter 4 deals with the integral equations of the third kind defined by g(t)?(t)=f(t)+?abK(t,t?)?(t?)?dt?,g(t)\phi(t) = f(t) + \int_a^b K(t,t')\phi(t') \, dt',g(t)?(t)=f(t)+?ab?K(t,t?)?(t?)dt?, where ?(t)\phi(t)?(t) is the unknown and g(t)g(t)g(t) vanishes at a finite number of points in (a,b)(a,b)(a,b). In general, the Fredholm Alternative theory does not hold for this type of integral equation. However, imposing certain conditions on g(t)g(t)g(t) and K(t,t?)K(t,t')K(t,t?), the above integral equation was shown by Bant [J. Math. Anal. Appl., 79 (1981), 49–57] to obey a Fredholm-type theory except for a certain class of kernels. We prove in this chapter a Fredholm-type theory for kernels belonging to this class also.