一題簡單的泛函分析

Question

The Volterra operator V on L^2[0,1] is define by
(Vf)(t)=∫(0~t)f(s)ds
a.Show that V has no eigenvalue
b.Prove by induction that
(V^n)(f)(t)=∫(0~t)(t-s)^(n-1)f(s)ds/(n-1)!

Answer 1

b.
n=k+1時
(V^(k+1))(f)(t)=∫(0~t)∫(0~x)(x-s)^(k-1)f(s)ds/(k-1)! dx
(交換積分順序)
=∫(0~t)∫(s~t)(x-s)^(k-1)f(s)dx/(k-1)! ds
=∫(0~t)(t-s)^kf(s)/k! ds
故得證

2007-11-29 18:16:30 補充：
a.
設∫(0~t)f(s)ds=k*f(t), k≠0,
則f(0)=0且f(t)=C*exp(t/k) a.e. (不可能)
故沒有eigenvalue & eigenfn.

註:偷偷用了微分方程式(可能有點問題吧!?)