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Spectral Theory for Bounded Self-Adjoint Operators

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Spectral Theory for Bounded Self-Adjoint Operators Spectral Theory for Bounded Self-adjoint Operators Roland Strömberg U.U.D.M. Project Report 2006:5 Examensarbete i matematik, 20 poäng Handledare och examinator: Andreas Strömbergsson Juni 2006 Department of Mathematics Uppsala University 1 Introduction Spectral theory for a self-adjoint operator is a quite complicated topic. If the operator at hand is compact the theory becomes, if not trivial, less complicated. Consider first the case of a self-adjoint operator A : V V ! with V finite dimensional. The complete spectral decomposition of A can be stated in a quite elementary fashion: We know that we can represent A with a matrix [A] and compute its eigenvalues and corresponding eigenvec- tors. The spectral theorem says that there exists a total orthonormal set of eigenvectors such that [A] can be diagonalized, or equivalently 1 · · · n that there exists a unitary matrix [U] and eigenvalues λ ; ; λ so that 1 · · · n for each H, ([U][A][U −1] ) = λ . This result carries over to the 2 i i i case where V is a infinite dimensional separable Hilbert space if we assume that V is a compact. In my thesis I have focused on bounded self-adjoint operators in full generality. For these operators I present the spectral theorem in three different forms; functional calculus form, multiplication operator form and projections- valued measure form. The proofs of these theorems that I have used are classical results in functional analysis, but there are many more in the math- ematical literature. For example, a general version of the spectral theorem for self-adjoint operators by Stone and von Neumann (1929-1932), and more recently a version concerning unbounded operators in Hilbert spaces was stated by E.B. Davies in 1992. This version is discussed in a thesis titled "Functional calculus" by C. Illand at Uppsala university in 2004. In my the- sis I have chosen to follow the the proofs in the book "Functional analysis, Book 1" by Reed and Simon, [2], and my main work has been concerned with filling in several if not all of the details which are left to the reader in [2]. Of these three forms the second one is the most transparent, in the sense that it is reminiscent of the spectral theorem for compact self-adjoint matrices stated above. We can leisurely say that the second form says that every bounded self-adjoint operator is a multiplication operator on a suitable measure space. Now, it is quite evident that we need a rather heavy mathematical machinery even to state the theorem. We clearly need some measure theory and for that we introduce the so called spectral measures. But the first step in reaching the second formulation of the spectral theorem is to define the functional calculus. We formulate this calculus first for the continuous case then for Borel sets. The first theorem (Theorem 1) concerning continuous functional calculus states that there exists a map from the set of continuous functions on the spectrum of a bounded self-adjoint operator A, σ(A) to the set of bounded linear operators on a Hilbert space H, L(H), i.e. f f(A). This map has a number of desirable properties, 7! for example it is a "conjugate-homomorphism" and continuous. The first version of the spectral theorem is basically Theorem 1 extended to Borel 1 functions on R. The fact that we are dealing with a separable Hilbert space H enables us to decompose H into a direct sum of invariant subspaces. Then on each subspace we can find a unitary operator so that we get a relation analogous to the matrix case above. This enables us to state the spectral theorem in its second form. To state the spectral theorem in its third form, projection-valued mea- sure form, we need to introduce some projections as the name indicates. If we have a bounded self-adjoint operator A and a Borel set Ω at hand we call P χ (A) a spectral projection of A. We further call a family of such Ω ≡ Ω projections P Ω is an arbitrary Borel set a projection valued measure f Ω j g if it obeys a number of certain conditions. The third form of the spectral theorem says that there is a one-one correspondence between a bounded self-adjoint operator A and a bounded projection valued measure PΩ. The contents of my thesis also involve a section about unbounded op- erators. The theory of such operators are of uttermost importance since one of the most recurring operator in mathematical analysis, the differential operator, is unbounded. This section is merely a brief introduction and does not exhaust the topic of unbounded operators. 2 The Functional calculus Let A be a given fixed operator and f a continuous function. How can one n k define f(A) to make sense? If f is given by a finite polynomial k=0 akx n k P we want f(A) to be k=0 akA . We also want f to be well defined in the P 1 k sense of convergence of the sum. If f is given by k=0 akx with radius of convergence R and A < R then 1 a Ak conP verges in L(H) (the jj jj k=0 k set of bounded linear operators on H). PSo if f is given by an infinite series then f(A) is given by 1 a Ak. The theorem below states the existence k=0 k · and uniqueness of an opPerator between the family of all continuous complex functions defined on the spectrum of A, and L(H). Theorem 1 (Continuous functional calculus.) Let A be a bounded self-adjoint operator on a Hilbert space H. Then there exists a unique map Φ : C(σ(A)) L(H) with the following properties: ! (a) Φ is an algebraic homomorphism, that is Φ(fg) = Φ(f)Φ(g); ∗ − Φ(λf) = λ Φ(f); Φ(1) = I; Φ(f + g) = Φ(f) + Φ(g); Φ(f) = Φ(f)∗. (b) Φ is continuous, that is Φ(f) C f . jj jjL(H) ≤ jj jj1 (c) If f(x) = x then Φ(f) = A. Moreover Φ has the additional properties. (d) IfA = λψ then Φ(f) = f(λ) . (e) σ[Φ(f)] = f(λ) λ σ(A) . f j 2 g (f) If f 0, then Φ(f) 0. ≥ ≥ (g) Φ(f) = f . jj jj jj jj1 2 We first prove that (a) and (c) uniquely determine Φ(P ) for any poly- nomial P (x). First let P be identity, then Φ(P ) = P (A) = A. If P (x) = x2 then Φ(P ) = A2 (by property (a)). By induction one obtains that if P (x) = xn then Φ(P ) = An. Now the additive homomorphism property n k n k asserts that if P (x) = k=0 akx then Φ(P ) = k=0 akA . Weierstrass theoremPstates that the set of polynomialsP is dense in C(σ(A)) so the core of the proof is to prove that P (A) = P (x) jj jjL(H) jj jjC(σ(A)) ≡ sup (P (λ)) . λ2σ(A) j j Existence and uniqueness of Φ then follow from the Bounded Linear Transform theorem. To prove the equality between the norms above a special case of (e) that holds for arbitrary bounded operators is proved. n k n k Lemma 1 Let P (x) = k=0 akx and P (A) = k=0 akA . Then σ[P (A)] = P (λ) λ σ(A) : P P f j 2 g Proof. Take arbitrary λ σ(A). Then A λ does not have an inverse 2 − in L(H). The fact that x = λ is a root of P (x) P (λ) enables the factor- − ization of P (x) P (λ) as (x λ)Q(x), where Q is a polynomial with lower − − degree than P . Now P (A) P (λ) = (A λ) Q(A), so P (A) P (λ) fails − − − to have an inverse in L(H). Hence P (λ) σ(P (A)). For µ σ(P (A)) let 2 2 P (x) µ = a(x λ ) (x λ ). − − 1 · · · − n If no λ σ(A) then (P (A) µ)−1 = a−1(A λ )−1 (A λ )−1 exists. i 2 − − 1 · · · · − n But this contradicts that µ σ(A). 2 Thus λ σ(A) for some i and hence µ = P (λ) for some λ σ(A). i 2 2 5 In order to prove Theorem 1 one more Lemma is needed. Lemma 2 Let A be a bounded self-adjoint operator. Then P (A) = jj jj sup P (λ) . λ2σ(A) j j Proof. Before writing out the proof a few preliminaries are needed. First, for a bounded operator T on a Hilbert space it holds that T ∗T = 2 n n jj jj∗ T (cf. [1, Theorem 3.9-4(e)]). Also, if P (A) = k=0 anA then P (A) = jj jjn n ∗ n n P ( k=0 anA ) = k=0 an A = P (A). Further (P P )(A) is self-adjoint: (P P )(A)∗ = P (A)P∗(P (A))∗ = P (A)P (A)∗∗ = P (A)P (A) = (P P )(A). So sup λ = r(P P (A)) = P P (A) , where r is the spectral radius of λ2σ(P P (A)) j j jj jj P P (A). And then by Lemma 1 : sup λ = sup P P (λ) . So λ2σ(P P (A)) j j λ2σ(A) j j P (A) 2 = P (A)∗P (A) = P P (A) = sup λ jj jj jj jj jj jj λ2 σ(P P (A)) j j = sup P P (λ) = sup P (λ) P (λ) λ2σ(A) j j λ2σ(A) j jj j = sup P (λ) 2 = ( sup P (λ) )2: λ2σ(A) j j λ2σ(A) j j 3 5 Proof of Theorem 1.
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