The Spectral Theorem for Self-Adjoint and Unitary Operators Michael Taylor Contents 1. Introduction 2. Functions of a self-adjoint operator 3. Spectral theorem for bounded self-adjoint operators 4. Functions of unitary operators 5. Spectral theorem for unitary operators 6. Alternative approach 7. From Theorem 1.2 to Theorem 1.1 A. Spectral projections B. Unbounded self-adjoint operators C. Von Neumann's mean ergodic theorem 1 2 1. Introduction If H is a Hilbert space, a bounded linear operator A : H ! H (A 2 L(H)) has an adjoint A∗ : H ! H defined by (1.1) (Au; v) = (u; A∗v); u; v 2 H: We say A is self-adjoint if A = A∗. We say U 2 L(H) is unitary if U ∗ = U −1. More generally, if H is another Hilbert space, we say Φ 2 L(H; H) is unitary provided Φ is one-to-one and onto, and (Φu; Φv)H = (u; v)H , for all u; v 2 H. If dim H = n < 1, each self-adjoint A 2 L(H) has the property that H has an orthonormal basis of eigenvectors of A. The same holds for each unitary U 2 L(H). Proofs can be found in xx11{12, Chapter 2, of [T3]. Here, we aim to prove the following infinite dimensional variant of such a result, called the Spectral Theorem. Theorem 1.1. If A 2 L(H) is self-adjoint, there exists a measure space (X; F; µ), a unitary map Φ: H ! L2(X; µ), and a 2 L1(X; µ), such that (1.2) ΦAΦ−1f(x) = a(x)f(x); 8 f 2 L2(X; µ): Here, a is real valued, and kakL1 = kAk. Here is the appropriate variant for unitary operators. Theorem 1.2. If U 2 L(H) is unitary, there exists a measure space (X; F; µ), a unitary map Φ: H ! L2(X; µ), and u 2 L1(X; µ), such that (1.3) ΦUΦ−1f(x) = u(x)f(x); 8 f 2 L2(X; µ): Here, juj = 1 on X. These results are proven in x1 of [T2]. The proof there makes use of the Fourier transform on the space of tempered distributions. Here we prove these results, assuming as background only xx1{5 in [T1]. We use from x5 of that appendix the holomorphic functional calculus, which we recall here. If V is a Banach space and T 2 L(V ), we say ζ 2 C belongs to the resolvent set ρ(T ) if and only if ζ − T : V ! V is invertible. We define σ(T ) = C n ρ(T ) to be the spectrum of T . It is shown that ρ(T ) is open and σ(T ) is closed and bounded, hence compact. If Ω is a neighborhood of σ(T ) with smooth boundary and f is holomorphic on a neighborhood of Ω, we set Z 1 (1.4) f(T ) = f(ζ)(ζ − T )−1 dζ: 2πi @Ω 3 It is shown there that k + k (1.5) fk(ζ) = ζ (k 2 Z ) =) fk(T ) = T ; and, if f and g are holomorphic on a neighborhood of Ω, then (1.6) (fg)(T ) = f(T )g(T ): To apply these results to self-adjoint A and unitary U in L(H), it is useful to have the following information on their spectra. Proposition 1.3. If A 2 L(H) is self-adjoint, then σ(A) ⊂ R. Proof. If λ = x + iy; y =6 0, then x 1 λ − A = y(i + B);B = − A = B∗; y y so it suffices to show that if B = B∗, then i + B : H ! H is invertible. First note that, given u 2 H, kuk = 1 ) k(i + B)uk ≥ j((i + B)u; u)j (1.7) = j((I − iB)u; u)j ≥ Ref(u; u) − i(Bu; u)g: Now (1.8) B = B∗ ) (Bu; u) = (u; Bu) = (Bu; u) ) (Bu; u) 2 R; so the last line of (1.7) equals kuk2, and we have (1.9) k(i + B)uk ≥ kuk; when B = B∗: Hence i + B : H ! H is injective and has closed range. To see that it is onto, assume w 2 H is orthogonal to this range, so v 2 H =) ((i + B)v; w) = 0 (.110) =) (v; (i − B)w) = 0: Hence (i + B)w = 0. Then (1.9), with B replaced by −B, gives w = 0. This establishes the asserted surjectivity, and proves Proposition 1.3. From Proposition 1.3 it follows that, if A = A∗, (1.11) σ(A) ⊂ [−∥Ak; kAk]: Here is the unitary counterpart of Proposition 1.3. 4 Proposition 1.4. If U 2 L(H) is unitary, then σ(U) ⊂ S1 = fζ 2 C : jζj = 1g: Proof. If U is unitary, kUk = 1, so jζj > 1 ) ζ 2 ρ(U). Meanwhile, jζj < 1 =) ζ − U = −U(I − ζU −1); and kζU −1k = jζj kU ∗k = jζj < 1, so again this operator is invertible. We proceed in the rest of this note as follows. In x2 we take I = [−∥Ak−δ; kAk+δ] (with δ > 0 small) and show that f 7! f(A) extends from the space O of functions f holomorphic on a complex neighborhood of I to C(I). To do this, we show that (1.12) f 2 O; f ≥ 0 on I =) f(A) ≥ 0; where, given T 2 H, we say (1.13) T ≥ 0 () T = T ∗ and (T v; v) ≥ 0; 8 v 2 H: We deduce from (1.12) that (1.14) f 2 O; jfj ≤ M on I =) kf(A)k ≤ M: From this, the extension of f 7! f(A) to f 2 C(I) will follow. We will have (1.15) 8 f 2 C(I); f ≥ 0 ) f(A) ≥ 0; jfj ≤ M ) kf(A)k ≤ M: We use this fact to prove Theorem 1.1 in x3. In x4, we use arguments parallel to those in x2 to extend f 7! f(U) from the space O of functions holomorphic on a complex neighborhood of S1 to C(S1). We show that (1.16) f 2 O; f ≥ 0 on S1 =) f(U) ≥ 0; and (1.17) f 2 O; jfj ≤ M on S1 =) kf(U)k ≤ M; and from there obtain the asserted extension, and an analogue of (1.15), namely (1.18) 8 f 2 C(S1) f ≥ 0 ) f(U) ≥ 0; jfj ≤ M ) kf(U)k ≤ M: Then, in x5, we use this extension to prove Theorem 1.2. 5 One ingredient in the analysis in x4 is the identity 1 Z X 1 (1.19) f(U) = f^(k)U k; f^(k) = f(eiθ) dθ; −∞ 2π k= S1 for f 2 O, which follows from (1.4), with Ω an annular neighborhood of S1 ⊂ C. x 1 In 6,P we take (1.19) as the definition of f(U), whenever f is a function on S such that jf^(k)j < 1, which in particular holds for all f 2 C1(S1). We demonstrate directly, without using any holomorphic functional calculus, that (1.20) f 2 C1(S1); jfj ≤ M on S1 =) kf(U)k ≤ M; and, from this obtain the extension f 7! f(U) to f 2 C(S1), satisfying (1.18). This again leads to a proof of Theorem 1.2, which in this setting avoids use of the holomorphic functional calculus. In x7, we show how to go from the Spectral Theorem for unitary operators (Theorem 1.2) to the Spectral Theorem for self-adjoint operators (Theorem 1.1). This, in conjunction with x6, yields a proof of Theorem 1.1 that does not require the holomorphic functional calculus. In Appendix A, we relate Theorems 1.1 and 1.2 to other formulations of the Spectral Theorem, involving spectral projections. In Appendix B, we define un- bounded self-adjoint operators and sketch an approach to the Spectral Theorem for such operators, somewhat parallel to the derivation of Theorem 1.1 from Theo- rem 1.2 in x7, referring to material in [T1] and [T2] for details. In Appendix C we use Theorem 1.2 to derive von Neumann's mean ergodic theorem. 6 2. Functions of a self-adjoint operator Let A 2 L(H) be self-adjoint. As previewed in x1, we take δ > 0 and set (2.1) I = [−∥Ak − δ; kAk + δ];I0 = [−∥Ak; kAk]; so σ(A) ⊂ I0 ⊂ I. Let Ω be a smoothly bounded neighborhood of I0 in C, and \ R ⊂ assume Ω I. Let f be holomorphicZ on a neighborhoood of Ω. We have 1 (2.2) f(A) = f(ζ)(ζ − A)−1 dζ: 2πi @Ω Now, given S; T 2 L(H), we have (ST )∗ = T ∗S∗, so (2.3) ((ζ − A)−1)∗ = (ζ − A)−1: Hence Z 1 (2.4) f(A)∗ = − f(ζ)(ζ − A)−1 dζ: 2πi @Ω We can assume Ω is invariant under the reflection ζ 7! ζ, which reverses orientation on @Ω. Hence Z 1 (2.5) f(A)∗ = f(ζ)(ζ − A)−1 dζ = f ∗(A); 2πi @Ω where (2.6) f ∗(ζ) = f(ζ): In particular, if f is holomorphic on a neighborhood Ω of I0 in C, (2.7) f real on Ω \ R =) f(A) self-adjoint. Here is our key positivity result. Proposition 2.1. If f is holomorphic on a neighborhood Ω of I0 ⊂ C, then (2.8) f ≥ 0 on Ω \ R =) f(A) ≥ 0: Proof. For " > 0, set f"(ζ) = f(ζ) + ".