Spectral Theory in Hilbert Space

EXERCISES FOR CHAPTER 11 81
Note that there are no eigenvalues in either of these cases; the
spectrum is purely continuous.
Exercises for Chapter 11
Exercise 11.1. Show that if K is a compact interval, then C 1 (K)
is a Banach space with the norm sup x∈K|u(x)| + sup x∈K|u ′ (x)|.
If you know some topology, also show that if I is an arbitrary interval,
then C(I) is a Fréchet space (a linear Hausdorff space with
the topology given by a countable family of seminorms, which is also
complete), under the topology of locally uniform convergence.
Exercise 11.2. With the assumptions of Corollary 11.4 the Fourier
series for u in the domain of T actually converges absolutely and locally
uniformly to u. If λ1, λ2, . . . are the eigenvalues and e1, e2, . . . the corresponding
orthonormal eigenfunctions, use Parseval’s formula to show
that, pointwise in x, g(x, ·, λ)2 = | ej(x)
λj−λ |2 , with natural notation.
Then show that as an L2 (I)-valued function x ↦→ g(x, ·, λ) is locally
bounded, i.e., x ↦→ g(x, ·, λ) is bounded on any compact subinterval
of I.
If v = Rλu and ûj is the j:th Fourier coefficient of u, then ˆvj =
〈Rλu, ej〉 = 〈u, Rλej〉 = ûj/(λj
− λ). Show that this implies that
j>n |ˆvjej(x)| tends locally uniformly to 0.