Chapter 4 The Spectral Theorem So far, we have only exploited properties of Borel functional calculi but not seen so many examples of them. With the so-called \spectral theorem" for normal operators on Hilbert spaces, this will change drastically. 4.1 Three Versions of the Spectral Theorem The Spectral Theorem actually comes in different versions, three of which shall be presented here. Probably the most striking (and most powerful) is the following. Theorem 4.1 (Spectral Theorem, Multiplicator Version). Let A be a normal operator on a Hilbert space H. Then A is unitarily equivalent to a multiplication operator on some L2-space over a semi-finite measure space. To wit: To a normal operator A on a Hilbert space H there exists a semi- finite measure space Ω, a measurable function a on Ω and a unitary operator U : H ! L2(Ω) such that −1 A = U Ma U: We shall call this a multiplication operator representation of the normal operator A. So Theorem 4.1 can be rephrased as: Each normal operator on a Hilbert space has a multiplication operator representation. In this form, the spectral theorem can be seen as a far-reaching gener- alization of a well-known theorem about unitary diagonalization of normal matrices or the well-known spectral theorem for compact normal operators [1, Thm. 13.11]. Indeed, the unitary equivalence to a multiplication operator is a kind of continuous diagonalization. Actually, one can say much more: The measure space can be chosen to be a Radon measure on a locally compact space and the multiplier function is continuous. And even more, one can simultaneously \diagonalize" any family 55 56 4 The Spectral Theorem of pairwise commuting (one would have to define what that is) normal op- erators on a Hilbert space. However, we shall not treat these generalizations here. From the multiplicator version it is only a small step to the version which is of most interest to us in this course. Theorem 4.2 (Spectral Theorem, Functional Calculus Version). Let A be a normal operator on a Hilbert space H. Then A has a unique Borel functional calculus on σ(A). Proof. Fix a unitary equivalence, say U, to a multiplication operator Ma on 2 an L -space. Now carry over the functional calculus for Ma to one for A by −1 Φ(f) := U Mf◦aU: This yields the existence part of Theorem 4.2, the uniqueness has been shown in Theorem 3.13. The main advantage of this second formulation of the spectral theorem is that the functional calculus for a normal operator is unique, whereas the multiplication representation is not. Finally, here is the third version of the spectral theorem. Theorem 4.3 (Spectral Theorem, Spectral Measure Version). Let A be a normal operator on a Hilbert space H. Then there is a unique projection- valued measure E defined on the Borel subsets of C such that Z A = z E(dz): C This theorem follows from Theorem 3.13 by the one-to-one correspondence of measurable functional calculi and projection-valued measures that has been mentioned in Chapter 3. We shall not use this version of the spectral theorem. Working with the Spectral Theorem We shall prove the spectral theorem below. Before, let us assume its validity and work with it. From now on, if A is a normal operator on a Hilbert space H then we denote by ΦA its (unique) Borel functional calculus on s(A). Note that we may as well consider ΦA a Borel calculus on C (which is automatically con- centrated on s(A), cf. Remark 3.10). For a Borel measurable function f on s(A) we also write f(A) := ΦA(f) frequently. The ΦA-null sets are just called A-null sets and abbreviated by 4.1 Three Versions of the Spectral Theorem 57 NA := NΦA = fB 2 Bo(C): 1B(A) = 0g: Note that the presence of A-null sets accounts for the fact that the Borel functional calculus for A may be concentrated on proper subsets of s(A). Example 4.4. It is easy to see that λ 2 C is an eigenvalue of the normal operator A if and only if fλg is not a ΦA-null set. In this case, 1fλg(A) is the orthogonal projection onto the corresponding eigenspace (Exercise 4.1). It follows that if A has no eigenvalues then its functional calculus is con- centrated on each set s(A) n fλg for each λ 2 s(A). One can see this phenomenon most clearly in a multiplication operator representation of A: In the situation of Example 3.5, where H = L2(Ω), A = Ma and K = essran(a) = s(A), the original functional calculus is L0(K; ν) !C(H) but the Borel calculus is just the composition 0 ΦA : M(K) ! L (K; ν) !C(H): It is easy to see that the ν-null sets are precisely the ΦA-null sets, i.e., ν(B) = 0 () 1B(A) = 0 0 for each Borel set B ⊆ C. Hence, L (K; ν) = M(C)=NA, and this does not depend on the multiplication operator representation. For a normal operator A and a Borel measurable function f on C we let essranA(f) := fλ 2 C j 8 " > 0 : [ jf − λj < " ] 2= NAg be the A-essential range of f, cf. Exercise 3.7. Theorem 4.5 (Composition Rule and Spectral Mapping Theorem). Let A be a normal operator on a Hilbert space, let f 2 M(C) and B := f(A). Then, for each g 2 M(C) one has g(B) = g(f(A)) = (g ◦ f)(A): Moreover, s(f(A)) = essranA(f) ⊆ f(s(A)): (4:1) If f is continuous, one has even s(f(A)) = f(s(A)). Proof. The definition Φ(g) := ΦA(g ◦ f) = (g ◦ f)(A) for g 2 M(C) yields a Borel functional calculus on C for the operator B (Exercise 3.6). By unique- ness, Φ(g) = ΦB(g) = g(f(A)) for all g 2 M(C). 58 4 The Spectral Theorem The identity s(f(A)) = essranA(f) was the subject of Exercise 3.7. If λ2 = f(s(A)) then λ has a positive distance " > 0 to f(s(A)). Hence, s(A) ⊆ [ jf − λj ≥ " ] and so [ jf − λj < " ] is an A-null set. Therefore, λ2 = essranA(f), which concludes the proof of (4.1). Finally, suppose that f is continuous, λ 2 C and f(λ) 2= essranA(f). Then there is " > 0 such that [ jf − f(λ)j < " ] 2 NA. Since f is continuous, there is δ > 0 such that [ jz − λj < δ ] = B(λ, δ) ⊆ [ jf − f(λ)j < " ] : Since NA is closed under taking measurable subsets, [ jz − λj < δ ] 2 NA. This means that λ2 = essranA(z) = s(A), by what we have shown above. We note that for the last part of Theorem 4.5 it suffices that the function f is continuous on s(A). Indeed, in this case one can find a continuous function f~ on C that coincides with f on s(A) and hence satisfies f~(A) = f(A). Note further that, in the case that s(A) is compact (e.g., if A is bounded) and f is continuous, Theorem 4.5 yields the spectral mapping identity s(f(A)) = f(s(A)): See also Exercise 4.2. Proof of the Spectral Theorem In the remainder of this chapter we shall present a proof of the spectral theorem. This will happen in four steps. After the first three, the spectral theorem for self-adjoint operators will be established and this is sufficient in many cases. Normal but not self-adjoint operators are treated in an optional supplement section. Some of the arguments can be greatly simplified when one is willing to use Gelfand theory, in particular the commutative Gelfand{Naimark theorem. This approach can be found, e.g., in our ealier book [2, App.D], see also [3, Chap. 18]. However, for this course we have decided to avoid that theory. 4.2 Proof: Bounded Self-Adjoint Operators In the first step of the proof we establish a continuous functional calculus for a bounded self-adjoint operator. By this we mean a calculus involving continuous (instead of measurable) functions. We closely follow [4, Section 5.1]. Let A 2 L(H) be a bounded, self-adjoint operator on H and let a; b 2 R such that s(A) ⊆ [a; b]. Fix p 2 C[z], denote by p∗ the polynomial p∗(z) := p(z), 4.3 Proof: From Continuous Functions To Multiplication Operators 59 and let q := pp∗. By the spectral inclusion theorem for polynomials1, s(q(A)) ⊆ q(s(A)): Now observe that p(A)∗p(A) = p∗(A)p(A) = q(A). Hence, q(A) is self-adjoint 2 and therefore its norm equals its spectral radius. Since q = jpj on R, kp(A)k2 = kp(A)∗p(A)k = kq(A)k = r(q(A)) = supfjλj : λ 2 s(q(A))g 2 ≤ supfjq(µ)j : µ 2 s(A)g ≤ kqk1;s(A) ≤ kpk1;[a;b]: It follows that the polynomial functional calculus for A is contractive for the supremum-norm on [a; b]. By the Weierstrass approximation theorem, the polynomials are dense in C[a; b]. A standard result from elementary functional analysis now yields a bounded (in fact: contractive) linear map Φ : C[a; b] !L(H) such that Φ(p) = p(A) for p 2 C[z]. It is easily seen that Φ is a unital ∗-homomorphism.