INTRODUCTION TO SPECTRAL THEORY

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1 INTRODUCTION TO SPECTRAL THEORY WILLIAM CASPER Abstract. This is a abstract. 1. Itroductio Throughout this paper, we let H deote a Hilbert space, X a measureable space ad Ω a σ-algebra of subsets of X. By a operator T o H we will always mea a liear trasformatio such that the operator orm T := sup{ Tψ : ψ H, ψ = 1} is fiite. A operator will be called ivertible if it has a algebraic iverse T 1 which is also a operator o H Basic Defiitios ad Facts. Defiitio 1.1. The spectrum of a liear operator T o H is (1) spec(t) = {λ C : T λ is ot ivertible. }. The approximate poit spectrum of T is (2) aspec(t) = {λ C : (T λ) = 0}, where (T) = if{ Tψ : ψ H, ψ = 1}. Theorem 1.1. aspec(t) spec(t). Theorem 1.2. If T is a ormal operator, the aspec(t) = spec(t). Theorem 1.3 (Trasforms of Spectra). (i) If p C[x], the (ii) If T is ivertible, the spec(p(t)) = p(spec(t)) := {p(λ) : λ spec(t)}. spec(t 1 ) = (spec(t)) 1 := {λ 1 : λ spec(t)}. (iii) The spectrum of the adjoit of T satisfies Theorem 1.4. Defie (3) spec(t ) = (spec(t)) := {λ : λ spec(t)}. N T (f) = sup{ f(λ) : λ spec(t)}. I geeral, spec(t) is a compact subset of the complex plae ad N T (x) T. If T is Hermitia, the spec(t) is a subset of R ad N T (p(x)) = p(t) for ay polyomial p R[x]. Date: February 24,

2 2 WILLIAM CASPER 2. Spectral Measures Defiitio 2.1. A spectral measure E o a measureable space (X,Ω) is a projectio-valued (idempotet, hermitia) set fuctio satisfyig E : Ω { projectios o H} (i) E(X) = 1; (ii) for ay collectio of pairwise disjoit sets {A k } k=1, ( ) E A k = E(A k ). k=1 Theorem 2.1 (Properties of Spectral measures). If E is a spectral measure, the for all A,B Ω k=1 (i) if A B, the E(A) E(B); (ii) if A B, the E(B \ A) = E(B) E(A); (iii) E(A B) + E(A B) = E(A) + E(B); (iv) E(A B) = E(A)E(B). Theorem 2.2. A fuctio is a spectral measure if ad oly if E : Ω { projectios o H} (i) E(X) = 1; (ii) for ay two fixed elemets ψ, φ H,the fuctio µ : Ω C defied by µ(a) = E(A)ψ,φ for all A Ω is a complex measure. We use the otatio d E(λ)ψ,φ = dµ(λ) so that i geeral 1 A d E(λ)ψ,φ = E(A)ψ,φ for all A Ω. Theorem 2.3. If E is a spectral measure ad f is a E-measureable fuctio, the there exists a uique operator deoted by either fde or f(λ)de(λ) ad defied as (4) fdeψ,φ = f(λ)d E(λ)ψ,φ. Theorem 2.4 (Properties of fde). Give ay E-measureable fuctios f,g ad α C, (i) (αf)de = α fde; (ii) (f + g)de = fde + gde; (iii) ( fde ) = f de; (iv) fgde = ( fde ) ( fde ). Theorem 2.5. If E is a spectral measure ad E(A) commutes with T for every A Ω, the fde commutes with T.

3 INTRODUCTION TO SPECTRAL THEORY 3 3. Complex Spectral Measures For the remaider of the paper, we assume that X is a locally compact Hausdorff space ad that Ω is the Borel σ-algebra o X. Defiitio 3.1. A spectral measure is regular if for all A Ω, E(A) = sup{e(c) : C A, C is compact }. Defiitio 3.2. The spectrum of a spectral measure is spec(e) := X \{λ X : λ A, A is ope, E(A) = 0}. A spectral measure is compact if its spectrum is compact. Theorem 3.1. If E is a regular spectral measure, the spec(e) is closed ad E(X \ spec(e)) = 0 (ad therefore E(spec(E)) = 1). Theorem 3.2. For ay complex-valued, E-measureable fuctio bouded o spec(e), defie N E (f) = sup{ f(λ) : λ spec(e)}. The if E is a compact, regular spectral measure ad f is a cotiuous fuctio o X, the fde = NE (f). Defiitio 3.3. A spectral measure is called complex whe X = C. Theorem 3.3. Every complex spectral measure is regular. Theorem 3.4. If E is a compact, complex spectral measure ad if T = λde(λ), the spec(t) = spec(e). Theorem 3.5. A complex spectral measure is defied completely by the operator λde(λ). That is, give two complex spectral measures E1 ad E 2, E 1 = E 2 if ad oly if λde 1 (λ) = λde 2 (λ). Theorem 3.6. Let E be a complex spectral measure ad T a operator. The T commutes with E(A) for all A Ω if ad oly if T commutes with both λde(λ) ad λ de(λ). 4. The Spectral Theorem Theorem 4.1 (Spectral Theorem for Hermitia Operators). Let T be a Hermitia operator. The there exists a uique compact, complex spectral measure E such that T = λde(λ). Theorem 4.2 (Spectral Theorem for Normal Operators). Let T be a ormal operator. The there exists a uique compact, complex spectral measure E such that T = λde(λ). Defiitio 4.1. For ay ormal operator T, we call the spectral measure E satisfyig T = λde(λ) the spectral measure of T.

4 4 WILLIAM CASPER Appedix A. Applicatios of the Spectral Theorem A.1. Weak Mixig. As a first example applicatio of the spectral theorem, we will use it to show that a measure-preservig trasformatio T is weak mixig whe the oly eigefuctios of the uitary operator U T defied by U T f(x) = f(tx) are the costats. Defiitio A.1. Let (X,Ω,µ) be a probability space ad T a measure preservig trasformatio (mpt) o X (µ(a) = µ(t 1 (A)) for all A Ω). The T is called weakly mixig if (5) lim µ(t k (A B) µ(a)µ(b) 0 for all A,B Ω. Theorem A.1. A mpt T is weakly mixig if the oly measureable eigefuctios of U T are the costats. Proof. We first ote that T is weakly mixig if ad oly if (6) lim U k Tf,g f,1 1,g 0 for all f,g L 2 (µ). Let V be the closed liear subspace of the eigefuctios of T i L 2 (µ) ad let E be the spectral measure of T. The for ay λ 0 spec(u T ), we have that U T E({λ 0 }) = λde(λ) 1 λ0 (λ)de(λ) = λ1 {λ0}(λ)de(λ) = λ 0 E({λ 0 }), so that i particular U T E({λ 0 })f = λ 0 f for every f L 2 (µ). Thus E({λ 0 })f V for all f L 2 (µ). If f V, this implies that 0 = E({λ 0 })f,f = E({λ 0 }) 2 f,f = E({λ 0 })f,e({λ 0 })f, ad therefore E({λ 0 })f = 0. Now fix a f V ad g L 2 (µ) ad defie µ to be the complex Borel measure o the spectrum of T satisfyig dµ = d E(λ)f,g. The for all λ 0 spec(t), we have that µ({λ 0 }) = E({λ})f,g = 0. Settig = {(λ,ω) spec(u T ) spec(u T ) : λ = ω}, we fid U k Tf,g 2 = 1 λ k de(λ)f,g 2 = 1 λ k de(λ)f,g 2 = 1 λ k dµ(λ) 2 = 1 λ k dµ(λ) (ω) k dµ (ω) = 1 (λω) k dµ(λ)dµ (ω) = (λω) k d(µ µ )(λ,ω) 1 1 (λω) k = 1 λω d(µ µ )(λ,ω) + 1d(µ µ )(λ,ω). c

5 INTRODUCTION TO SPECTRAL THEORY 5 Additioally, 1d(µ µ )(λ,ω) = 1 (λ,ω)dµ(λ)dµ (ω) = µ({ω})dµ (ω) = 0. For (λ,ω) /, (1 (λω) k )/(1 λω) is a cyclotomic polyomial, ad is therefore bouded o the compact set spec(u T ). Thus by the bouded covergece theorem 1 1 (λω) k lim 1 λω d(µ µ )(λ,ω) = 0d(µ µ )(λ,ω) = 0. c c We coclude that lim U k Tf,g 2 = 0. By the Cauchy-Schwartz iequality, ( 2 UTf,g ) k UTf,g k 2. Dividig both sides by 2, we fid ( ) 2 U k Tf,g 1 U k Tf,g 2, ad therefore lim U k Tf,g = 0. Now for ay f,g L 2 (G), if the eigefuctios of U T are the costats, f f,1 V. Therefore lim U k T(f f,1 ),g = lim U k Tf f,1,g Therefore T is weakly mixig. A.2. Almost Periodic Fuctios. B.1. Proof for Sectio 1. = lim U k Tf,g f,1 1,g = 0. Appedix B. Proofs of Theorems Proof of Theorem (1.1). If λ / spec(t), the T λ is ivertible. Thus for ay ψ H with ψ = 1, we have that 1 = ψ = (T λ) 1 (T λ)ψ (T λ) 1 (T λ)ψ. Thus (as defied by Eq. (1)) satisfies (T λ) 1/ (T λ) 1 ad so λ / aspec(t). This proves our theorem. Lemma B.1. A operator T is ivertible if ad oly if its rage is dese i H ad there exists a positive real umber c > 0 such that Tψ c ψ for all ψ H.

6 6 WILLIAM CASPER Proof. If T is ivertible, the it is a bijectio ad therefore the rage must be H. Moreover, ψ = T 1 Tψ T 1 Tψ, so Tψ c ψ with c = 1/ T 1. Coversely, suppose the rage of T is dese i H ad there exists a positive real umber c > 0 such that Tψ c ψ for all ψ H. We first show that the rage of T is H. Let {φ i } i=1 be a coverget sequece i the rage of H covergig to φ. For every i > 0, there exists a ψ i H such that Tψ i = φ i. Moreover, ψ i ψ j φ i φ j /c. It follows that the sequece {ψ i } i=1 is Cauchy ad therefore coverges to a fuctio ψ H. Sice T is cotiuous, φ = T ψ, ad therefore φ is i the rage of T. We coclude that the rage of T is closed. Sice the rage of T is dese i H, the rage of T must be H. The kerel of T is trivial, sice if ψ ker(t), the ψ Tψ /c = 0, implyig that ψ = 0. Thus T is a bijectio, ad all that is left to show is that the algebraic iverse, which we call T 1, is bouded. We have that T 1 ψ TT 1 ψ /c ψ /c. It follows that T 1 1/c. This proves our theorem. Icidetally, this also shows us that c = 1/ T 1 is the sharpest value for c. Proof of Theorem (1.2). By Theorem (1.1), we eed oly prove spec(t) aspec(t). Suppose that λ / aspec(t). The there exists a costat c > 0 such that (T λ)ψ c ψ for all ψ H. By Lemma (B.1), we eed oly show that the rage of T λ is dese i H. Sice T commutes with T, T λ commutes with (T λ) = T λ, ad it follows that (T λ)ψ = (T λ )ψ for all ψ H. If φ ragle(t λ), the 0 = (T λ)ψ,φ = ψ,(t λ )φ for all ψ H, ad therefore (T λ )φ = 0. It follows that φ = 0, sice φ (T λ)φ /c = (T λ )φ /c = 0. Thus ragle(t λ) = {0} ad it follows that ragle(t λ) is dese i H. This proves our theorem. Proof of Theorem (1.3). (i) Let p C[x] ad λ spec(t). The λ is a root of r(x) = p(x) p(λ) ad therefore there exists a polyomial q C[x] such that q(x)(x λ) = p(x) p(λ). If r(t) is ivertible, the q(t) commutes with r 1 (T) ad (T λ)q(t)r(t) 1 = r(t)r(t) 1 = 1 = r(t) 1 r(t) = r(t) 1 (T λ)q(t) = r(t) 1 q(t)(t λ) = q(t)r(t) 1 (T λ). It follows that (T λ) is ivertible with (T λ) 1 = q(t)r(t) 1, which is a cotradictio. Thus r(t) is ot ivertible ad p(λ) spec(p(t)). Coversely, suppose λ spec(p(t)) ad let {r i } i=1 be the roots of the polyomial p(x) λ. We have that p(x) λ = (x r 1 )...(x r ) ad therefore p(t) λ = (T r 1 )...(T r ). Sice p(t) λ is ot ivertible, (T r j ) is ot ivertible for some j. Whece r j spec(t) ad p(r j ) λ = 0. We coclude that λ p(spec(t)). This proves (i). (ii) Note that for ay λ C, we have that T 1 λ 1 = T 1 λ 1 (T λ), ad it follows that T 1 λ 1 is ivertible if ad oly if T λ is ivertible. This proves (ii). (iii) If λ / spec(t), the T λ is ivertible. It follows that (T λ) = T λ is ivertible, ad therefore spec(t ) spec(t). By the same argumet with T replaced by T, spec(t) = spec((t ) ) spec(t ), ad therefore spec(t) (spec(t ) ) = spec(t ). This proves our theorem.

7 INTRODUCTION TO SPECTRAL THEORY 7 Lemma B.2. If T is a operator such that 1 T < 1, the T is ivertible. Proof. Defie c > 0 by c = 1 1 T. The Tψ = ψ (ψ Tψ) ψ (1 T)ψ ψ (1 T) ψ = c ψ. Thus by Lemma (B.1), we eed oly show that the rage of T is dese i H. Let φ H ad let δ = if{ Tψ φ : ψ H}. Suppose that δ > 0. The for all ǫ = δ c 1 c, there exists ψ H such that δ Tψ φ < δ + ǫ. Moreover δ T(Tψ φ) (Tψ φ) = (1 T)(Tψ φ) < (1 c) Tψ φ = (1 c)(δ+ǫ) δ. That is, δ < δ, which is a cotradictio. We coclude that δ = 0. Sice φ H was take arbitrarily, this meas that the rage of T is dese i H. This proves our lemma. Proof of Theorem (1.4). If λ 0 / spec(t), the T λ 0 is ivertible. If λ C with λ λ 0 < r := 1/ (T λ 0 ) 1, the 1 (T λ 0 ) 1 (T λ) = (T λ 0 ) 1 [(T λ 0 ) (T λ)] (T λ 0 ) 1 λ λ 0 < 1. Therefore by Lemma (B.2) (T λ 0 ) 1 (T λ) is ivertible ad it follows that (T λ) must be ivertible. We coclude that the ball B(λ 0 ;r) about λ 0 of radius r is cotaied i C \ spec(t). It follows that C \ spec(t) is ope ad therefore spec(t) is closed. Moreover, if λ C satisfies T < λ, the 1 (1 T/λ) = T/λ < 1 ad therefore 1 T/λ is ivertible by Lemma(B.2). It follows that T λ is ivertible, ad therefore λ / spec(t). Thus if λ spec(t), the λ T ecessarily. I particular, this shows that N T (x) T ad that spec(t) is a closed ad bouded subset of C (ad therefore compact). Suppose T is Hermitia ad λ spec(t). The T is ormal ad spec(t) = aspec(t) by Theorem (1.2). Thus there exists a sequece {ψ i } i=1 H such that ψ i = 1 for all i ad (T λ)ψ i 0. Thus λ λ = λ λ ψ i 2 = (T λ)ψ i,ψ i (T λ )ψ i,ψ i = (T λ)ψ i,ψ i ψ i,(t λ)ψ i 2 (T λ)ψ i ψ i = 2 (T λ)ψ i 0. It follows that λ is real. Moreover for ay λ R, sice T is Hermitia, we have the relatio T 2 ψ λ 2 ψ 2 = T 2 ψ λ 2 ψ,t 2 ψ λ 2 ψ = T 2 ψ 2 + λ 4 ψ 2 (λ 2 ) T 2 ψ,ψ λ 2 ψ,t 2 ψ = T 2 ψ 2 + λ 4 ψ 2 2λ 2 Tψ 2. Now let {ψ i } i=1 H be a sequece such that ψ i = 1 for all i ad Tψ i T. The takig λ = T i the above relatio, we fid that (T 2 T 2 )ψ i 2 = T 2 ψ i λ 2 ψ i 2 = T 2 ψ i 2 + λ 4 ψ i 2 2λ 2 Tψ i 2 = T 2 ψ i 2 + T 4 2 T 2 Tψ i 2 0. Thus T 2 spec(t 2 ), ad it follows from Theorem (1.3) that either T spec(t) or T spec(t). I particular, this proves N T (x) = T. If p(x) R[x], the

8 8 WILLIAM CASPER p(t) is Hermitia ad therefore N T (p(x)) = N p(t) (x) = p(t). This proves our theorem. B.2. Proofs for Sectio 2. B.3. Proofs for Sectio 3. B.4. Proofs for Sectio 4. Lemma B.3 (Weierstrass Approximatio Theorem). Let X be a compact subset of R ad let f be a cotiuous fuctio o X. The there exists a sequece of real polyomials {p i } i=1 such that p i f uiformly o X. Lemma B.4. Let L be a bouded liear fuctioal o R[x] ad let X be a compact subset of R. The there exists a uique Borel measure µ o X satisfyig L(p) = p(λ)dµ(λ) for all p R[x]. Sketch of proof. Let Ω be the collectio of all Borel subsets of X ad let A Ω. Let {p i } i=1 R[x] be a sequece of polyomials with p i 1 A uiformly o X. Defie µ(a) by µ(a) = lim L(p i ). i The µ is a well-defied complex Borel measure o X. Proof of Theorem (4.1). Let ψ,φ H ad defie a fuctio L : R[x] C by L(p) = p(t)ψ,φ. The L is liear ad L(p) p(t)ψ φ p(t) ψ φ N T (p(x)) ψ φ ad therefore L is a liear fuctioal o R[x]. The set R[x] is a dese subset of the collectio of all cotiuous, real-valued fuctios o spec(t), ad it follows that there exists a uique complex measure µ o X = spec(t) with σ-algebra Ω cosistig of all Borel subsets of spec(t) such that L(p) = p(λ)dµ(λ) for all p C[x]. For give ψ,φ H, we deote this measure by µ (ψ,φ). Let ψ 1,ψ 2,φ 1,φ 2 H ad let α C. p(λ)dµ (ψ1+ψ 2,φ)(λ) = p(t)(ψ 1 + ψ 2 ),φ = p(t)ψ 1,φ + p(t)ψ 2,φ = p(λ)dµ (ψ1,φ)(λ) + p(λ)dµ (ψ2,φ)(λ), from which it follows that Similarly, Lastly, for A Ω we have that µ (ψ1+ψ 2,φ) = µ (ψ1,φ) + µ (ψ2,φ). µ (ψ,φ1+φ 2) = µ (ψ,φ1) + µ (ψ,φ2); µ (αψ,φ) = αµ (ψ,φ) ad µ (ψ,αφ) = α µ (ψ,φ). µ (ψ,φ) (A) µ (ψ,φ) (X) = sup { 1 N T (p) } pdµ (ψ,φ) : p C[x] = sup { p(t)ψ,φ /N T (p) : p C[x]} = ψ φ.

9 INTRODUCTION TO SPECTRAL THEORY 9 For ay A Ω, we defie µ A (ψ,φ) = µ (ψ,φ) (A). The above properties show us that µ A is a symmetric, biliear fuctioal, ad therefore for every A Ω, there exists a uique Hermitia operator E(A) such that µ A (ψ,φ) = E(A)ψ,φ for all ψ,φ H. We first show that E(A) is idempotet for all A Ω by provig the more geeral result E(A B) = E(A)E(B) for all A,B Ω. Fix B Ω ad let {q i } i=1 R[x] be a fixed sequece of polyomials with q i (λ) 1 A (λ) uiformly o X. Also fix ψ,φ H. For each i, defie a measure ν i by dν i (λ) = q i (λ)dµ (ψ,φ) (λ). The for ay p C[x], q(t) is Hermitia commutes with p(t) ad we have that p(λ)dν i (λ) = p(λ)q i (λ)dµ (ψ,φ) (λ) = p(t)q i (T)ψ,φ = p(t)ψ,q i (T)φ = p(λ)dµ (ψ,qi(t)φ). Let A Ω ad {p i } i=1 R[x] be a sequece of polyomials with p i(λ) 1 A (λ) uiformly o X. The the domiated covergece theorem tells us that ν i (A) = lim p i (λ)q i (λ)dµ (ψ,φ) (λ) = lim p i (λ)dµ i i (ψ,qi(t)φ)(λ) = 1 A (λ)dµ (ψ,qi(t)φ)(λ) = µ A (ψ,q i (T)φ) = E(A)ψ,q i (T)φ = q i (T)E(A)ψ,φ = q i (λ)dµ (E(A)ψ,φ) (λ) The domiated covergece theorem also tells us that E(A B)ψ,φ = 1 A B (λ)dµ (ψ,φ) (λ) = 1 A (λ)1 B (λ)dµ (ψ,φ) (λ) = lim ν i (A) = lim q i (λ)dµ (E(A)ψ,φ) (λ) i = 1 B (λ)dµ (E(A)ψ,φ) (λ) = E(B)E(A)ψ,φ. i Sice A,B Ω ad ψ,φ H were arbitrary, this proves that E(A B) = E(A)E(B) for all A,B Ω. Thus E is idempotet. Lastly, we have that E(X)ψ,φ = µ X (ψ,φ) = µ (ψ,φ) (X) = 1dµ (ψ,φ) (λ) = ψ,φ, ad by Theorem (2.2) this meas that E(X) is a compact, complex spectral measure. A quick calculatio shows that λdµ (ψ,φ) (λ) = Tψ,φ. The uiqueess of the measure follows from Theorem (3.5). This proves our theorem. Sketch of proof of Theorem (4.2). Let T be ormal ad defie T 1 ad T 2 by T 1 = 1 2 (T + T ) ad T 2 = 1 2i (T T ). The T 1 ad T 2 are Hermitia with T = T 1 + it 2 ad there exist uique spectral measures E 1,E 2 such that T i = λde i (λ) for i = 1,2. Defie A = {A + ib : A,B real Borel sets } ad defie a projectio-valued set fuctio E by E(A + ib) = E 1 (A)E 2 (B). The A is a algebra of sets ad the

10 10 WILLIAM CASPER σ-algebra geerated by A is Ω, the collectio of all Borel subsets of C. Moreover, E(C) = E 1 (R)E 2 (R) = 1 ad E exteds uiquely to a projectio-valued measure E o C.