The Volterra Operator

0 |f(s)|ds ≤ |t| ≤ 1. Therefore the image of Bp is (uniformly) bounded. By Arzela-Ascoli, V : L p[0, 1]→ C[0, 1] is compact. The preceeding argument does not go through when V acts on L1[0, 1]. In this case equicontinuity fails, as is demonstrated by the following family {fn} ⊂ B1: fn(s) = n1[0,1/n](s). This suffices to preclude compactness of V ; in particular, V fn has no Cauchy subsequence. Suppose V f = λf for some λ 6= 0. By definition λf is (absolutely) continuous. We deduce that V f (and therefore f) is continuously differentiable and that f = λf ′. It follows that f(s) = ces/λ. But then 0 = λf − V f = c so V has no eigenvalues and by the spectral theorem for compact operators, σ(V ) = {0}. We now turn our attention to the operator norm of V . For now we restrict V to the square integrable domain L2. Note that C[0, 1] ⊂ L∞[0, 1] ⊂ L2[0, 1] and the identity mapping I : C[0, 1] → L2[0, 1] is a bounded linear operator. It follows that IV : L2[0, 1] → L2[0, 1] is compact and we will proceed to consider V : L2[0, 1]→ L2[0, 1]. Recall that L2[0, 1] is a Hilbert space with an inner product defined by