ABSTRACT
Since the kernel I belongs to 3g, the corresponding integral operator L is compact in L'J' Moreover, since the sequence (zn)n converges weakly in L'J to 0, (Lzn)n converges in the norm of L'J to 0, and hence (Lzn, zn) - 0 as n - 00. But
hence
(5.39) lim 1616'(S,0')Zn(0')Zn(s)dO'ds = o. n-oo 0 0
Combining (5.38) and (5.39), we conclude that
lim ~(zn) = ess inf d(8), n_oo 0$.$6
which implies (5.37). •
We remark that Lemma 5.4 may also be proved by means of general perturbation theorems for spectra of linear operators (see e.g. DUNFORD-SCHWARTZ [1962] or KATO [1966)). As already observed, the application of Lemma 5.3 is difficult, since it is hard to compute the spectrum of the operator (5.34). It is in general much easier to study the integral operator L generated by the kernel I = l(s,O'). For instance, sometimes one can determine the greatest lower bound m_(L) of L which is, at least for compact operators L, a non-negative number. In particular, m_(L) = 0 if L in non-negative definite. The estimate
(5.40) essinf des) +m_(L) > 0 0~'$6
implies then obviously the estimate (5.33). So far we have analyzed condition (a) in the definition of a Ljapunov function ~ as given in (5.32); let us now discuss condition (b). If the operator T is given by (5.34), a straightforward calculation shows that
where we have put
(5,41) [T, A(t)] := TA(t) +A·(t)T. Using Lemma 3.5 we may write the functional .(x) = .'(x)(A(t)x) in the form
.(x) = 162C(t,s)d(s)x2(s)ds
This shows that i is also a quadratic functional like ., i.e.
