Spectral Theory in Hilbert Space

44 7. THE SPECTRAL THEOREM
as a closed, densely defined operator. We need only show that this
inverse is bounded, to see that its domain is all of H so that λ ∈ ρ(T ).
But (T − λ)u2 = ∞
−∞ (t − λ)2 d〈Etu, u〉 ≥ ε2 ∞
−∞d〈Etu, u〉 = ε2u2 so the inverse of T − λ is bounded by 1/ε. Conversely, assume that
Et is not constant near λ. Then there are arbitrarily short intervals ∆
containing λ such that E∆ = 0, i.e., there are non-zero vectors u such
that E∆u = u. But then (T − λ)u ≤ |∆|u, where |∆| is the length
of ∆. Hence we can find a sequence of unit vectors uj, j = 1, 2, . . .
for which (T − λ)uj → 0 (a ‘singular sequence’). Consequently either
T −λ is not injective, or else the inverse is unbounded so λ /∈ ρ(T ).
Exercises for Chapter 7
Exercise 7.1. Suppose σ is increasing and ∞
−λ ∞
−∞
dσ(t)
t−λ → ∞
−∞
the origin. In particular, ∞
−∞
−∞
dσ < ∞. Show that
dσ as λ → ∞ along any non-real ray originating in
dσ(t)
t−λ
→ 0.
Exercise 7.2. Suppose B(·, ·) is a sesqui-linear form on a complex
linear space. Show the polarization identity
B(u, v) = 1
3
i
4
k B(u + i k v, u + i k v) .
k=0
Exercise 7.3. Show that if T is a closed operator on H and S is
bounded and everywhere defined, then T S, but not necessarily ST , is
closed.
Exercise 7.4. Show that if T is selfadjoint and f is a continuous
function defined on σ(T ), then f(T ) = ∞
−∞f(t) dEt defines a densely
defined operator, which is bounded if f is and selfadjoint if f is realvalued.
Also show that (f(T )) ∗ = f(T ), that (f(T )) ∗ has the same domain
as f(T ) and commutes with it in a reasonable sense, and that fg(T ) =
f(T )g(T ). This is the functional calculus for a selfadjoint operator, and
also makes sense for arbitrary Borel functions. The integral is made
sense of in the same way as in the statement of the spectral theorem.
Exercise 7.5. Let T be selfadjoint and put H(t) = e −itT , t ∈ R,
the exponential being defined as in the previous exercise. Show that
H(t + s) = H(t)H(s) for real t and s (a group of operators), that
H(t) is unitary and that if u0 ∈ D(T ), then u(t) = H(t)u0 solves the
Schrödinger equation T u = iu ′ t with initial data u(0) = u0.
Similarly, if T ≥ 0 and t ≥ 0, show that K(t) = e −tT is selfadjoint
and bounded, that K(t + s) = K(t)K(s) for s ≥ 0 and t ≥ 0 (a semigroup
of operators) and that if u0 ∈ H then u(t) = K(t)u0 solves the
heat equation T u = u ′ t for t > 0 with initial data u(0) = u0.