Spectral Theory

These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for compact operators comes from [Zim90, Chapter 3]. 1. The Spectral Theorem for Compact Operators The idea of the proof of the spectral theorem for compact self-adjoint operators on a Hilbert space is very similar to the finite-dimensional case. Namely, we first show that an eigenvector exists, and then we show that there is an orthonormal basis of eigenvectors by an inductive argument (in the form of an application of Zorn’s lemma). We first recall a few facts about self-adjoint operators. Proposition 1.1. Let V be a Hilbert space, and T : V → V a bounded, self-adjoint operator. (1) If W ⊂ V is a T -invariant subspace, then W⊥ is T -invariant. (2) For all v ∈ V , 〈Tv, v〉 ∈ R. (3) Let Vλ = {v ∈ V | Tv = λv}. Then Vλ ⊥ Vμ whenever λ 6= μ. We will need one further technical fact that characterizes the norm of a self-adjoint operator. Lemma 1.2. Let V be a Hilbert space, and T : V → V a bounded, self-adjoint operator. Then ‖T‖ = sup{|〈Tv, v〉| | ‖v‖ = 1}. Proof. Let α = sup{|〈Tv, v〉| | ‖v‖ = 1}. Evidently, α ≤ ‖T‖. We need to show the other direction. Given v ∈ V with Tv 6= 0, setting w0 = Tv/‖Tv‖ gives |〈Tv,w0〉| = ‖Tv‖. Thus, ‖T‖ = sup v∈V, ‖v‖=1 |〈Tv,w0〉| ≤ sup v,w∈V, ‖v‖=‖w‖=1 |〈Tv,w〉|. Thus, it suffices to show that |〈Tv,w〉| ≤ α‖v‖‖w‖ for all v, w ∈ V . Without loss of generality, we may multiply w by a unit complex number to make 〈Tv,w〉 is real and positive. We now compute 〈T (v + w), v + w〉 = 〈Tv, v〉+ 〈Tv,w〉+ 〈Tw, v〉+ 〈Tw,w〉 = 〈Tv, v〉+ 2 〈Tv,w〉+ 〈Tw,w〉 ,