Clark model in the general situation

Transcription

1 Constanze Liaw (Baylor University) at UNAM April 2014 This talk is based on joint work with S. Treil.

2 Classical perturbation theory

3 Idea of perturbation theory Question: Given operator A, what can be said about the spectral properties of A + B for B Class X? Often Class X = {trace cl.}, {Hilbert Schmidt} or {comp.}

4 Idea of perturbation theory Question: Given operator A, what can be said about the spectral properties of A + B for B Class X? Often Class X = {trace cl.}, {Hilbert Schmidt} or {comp.} In this talk A and B are self-adjoint and Class X = {rank one} or {rank n}. Definition Operator A + B is a rank n perturbation of A : H H, if B = n α i (, f i )f i i=1 for some f i H and scalar α i R.

5 Trivial examples A α = A α = A α,β = ( ) α(, e )e 1 = ( ) 1 + α ( ) ( ) α α + α(, e e 2 )(e 1 + e 2 ) = α 3 + α ( ) ( ) α 0 + α(, e )e 1 + β(, e 2 )e 2 = β For an k k matrix, the k eigenvalues depend on α, β.

6 Classical perturbation theory applies Theorem (Weyl vonneuman) σ ess (S) = σ ess (T ) S T (Mod compact operators) Theorem (Kato Rosenblum, Carey Pincus 1976) S T (Mod trace class) S ac T ac, conditions

7 Classical perturbation theory applies Theorem (Weyl vonneuman) σ ess (S) = σ ess (T ) S T (Mod compact operators) Theorem (Kato Rosenblum, Carey Pincus 1976) S T (Mod trace class) S ac T ac, conditions Theorem (Aronszajn Donoghue Theory) Singular spectrum is not stable under rank one perturbations. Provides complete information about the eigenvalues, but only a set outside which A α has no singular continuous spectrum.

8 Classical perturbation theory applies Theorem (Weyl vonneuman) σ ess (S) = σ ess (T ) S T (Mod compact operators) Theorem (Kato Rosenblum, Carey Pincus 1976) S T (Mod trace class) S ac T ac, conditions Theorem (Aronszajn Donoghue Theory) Singular spectrum is not stable under rank one perturbations. Provides complete information about the eigenvalues, but only a set outside which A α has no singular continuous spectrum. Theorem (Poltoratski 2000) Conditions on purely singular operators S T (Mod rank 1).

9 Classical perturbation theory applies Theorem (Weyl vonneuman) σ ess (S) = σ ess (T ) S T (Mod compact operators) Theorem (Kato Rosenblum, Carey Pincus 1976) S T (Mod trace class) S ac T ac, conditions Theorem (Aronszajn Donoghue Theory) Singular spectrum is not stable under rank one perturbations. Provides complete information about the eigenvalues, but only a set outside which A α has no singular continuous spectrum. Theorem (Poltoratski 2000) Conditions on purely singular operators S T (Mod rank 1). Barry Simon: The cynic might feel that I have finally sunk to my proper level [...] to rank one perturbations maybe something so easy that I can say something useful! We ll see even this is hard and exceedingly rich.

10 What are rank one perturbations related to? In mathematical physics Half-line Schrödinger operator with changing boundary condition

11 What are rank one perturbations related to? In mathematical physics Half-line Schrödinger operator with changing boundary condition Random Hamiltonians: A + α i (, f i )f i for random α i Theorem (L., submitted) In some sense, rank one perturbations are as hard as random Hamiltonians. i=1

12 What are rank one perturbations related to? Within analysis Nehari interpolation problem Holomorphic composition operators Rigid functions Functional models Two weight problem for Hilbert/Cauchy transform

13 Cnu contractions and functional models alla Nikolski Vasyunin

14 Rank one unitary perturbations on H Given unitary U, all rank one K = (, ϕ) H ψ =: ψϕ for which the perturbed operator U + K is unitary can be parametrized by complex γ = 1: U + K = U + (γ 1)bb 1 where b 1 = U 1 b and b H = 1. WLOG: b cyclic, i.e. H = span{u k b : k Z}.

15 Rank one unitary perturbations on H Given unitary U, all rank one K = (, ϕ) H ψ =: ψϕ for which the perturbed operator U + K is unitary can be parametrized by complex γ = 1: U + K = U + (γ 1)bb 1 where b 1 = U 1 b and b H = 1. WLOG: b cyclic, i.e. H = span{u k b : k Z}. In the spectral representation of U wrt b we have U γ = M ξ + (γ 1)1 ξ on L 2 (µ) where µ = µ U,b.

16 Cnu contractions ( γ < 1) and characteristic functions

17 Cnu contractions ( γ < 1) and characteristic functions Recall U γ = M ξ + (γ 1)1 ξ on L 2 (µ). From now on consider γ < 1. Then U γ is cnu contraction with defect operators D Uγ = (I U γ U γ ) 1/2 = ( 1 γ 2) 1/2 b1 b 1, D Uγ = (I U γ U γ ) 1/2 = ( 1 γ 2) 1/2 bb and defect spaces D = span{b 1 } and D = span{b}.

18 Cnu contractions ( γ < 1) and characteristic functions Recall U γ = M ξ + (γ 1)1 ξ on L 2 (µ). From now on consider γ < 1. Then U γ is cnu contraction with defect operators D Uγ = (I U γ U γ ) 1/2 = ( 1 γ 2) 1/2 b1 b 1, D Uγ = (I U γ U γ ) 1/2 = ( 1 γ 2) 1/2 bb and defect spaces D = span{b 1 } and D = span{b}. Consider the characteristic function θ = θ γ HD D, θ 1 given by ( ) θ(z) := U γ + zd U γ (Id zuγ ) 1 D D Uγ, z D.

19 Free functional model K θ is subspace of L 2 (D D, W ) w/ matr. weight W = W θ, dz f L 2 (D D,W ) := (W (z)f(z), f(z)) D D 2π. T

20 Free functional model K θ is subspace of L 2 (D D, W ) w/ matr. weight W = W θ, dz f L 2 (D D,W ) := (W (z)f(z), f(z)) D D 2π. T Cnu contraction U γ is unitarily equivalent to its functional model, compression of the shift operator M θ := P θ M z Kθ.

21 Free functional model K θ is subspace of L 2 (D D, W ) w/ matr. weight W = W θ, dz f L 2 (D D,W ) := (W (z)f(z), f(z)) D D 2π. T Cnu contraction U γ is unitarily equivalent to its functional model, compression of the shift operator M θ := P θ M z Kθ. Clark operator Φ γ : K θ L 2 (µ) is a unitary operator with Φ γ M θ = U γ Φ γ, which maps defect spaces D Mθ D and D M θ D. Φ γ only unique up to multiplication by unimodular constant.

22 Sz.-Nagy Foiaş transcription for the weight W (z) I:

23 Sz.-Nagy Foiaş transcription for the weight W (z) I: ( K θ = H 2 D clos L 2 D ) ( θ ) H 2 D, := (1 θθ ) 1/2.

24 Sz.-Nagy Foiaş transcription for the weight W (z) I: ( K θ = H 2 D clos L 2 D ) ( θ ) H 2 D, := (1 θθ ) 1/2. Classical Clark theory considers the case where θ is inner (or equivalently, µ is purely singular). In this case K θ = H 2 θh 2

25 Sz.-Nagy Foiaş transcription for the weight W (z) I: ( K θ = H 2 D clos L 2 D ) ( θ ) H 2 D, := (1 θθ ) 1/2. Classical Clark theory considers the case where θ is inner (or equivalently, µ is purely singular). In this case K θ = H 2 θh 2, and the adjoint Φ γ : L 2 (µ) K θ can be represented using the normalized Cauchy transform (z D) Φ γf := C fµ, where C µ (z) := C µ Theorem (Poltoratski) T dµ(ξ) 1 ξz, C fµ(z) := For f L 1 (µ), lim r 1 C fµ (rz) C µ(rz) = f(z), for (µ) s-a.e. z T. T f(ξ)dµ(ξ) 1 ξz.

26 ( 0 θ de Branges Rovnyak transcription for W = θ 0 ) [ 1] :

27 ( 0 θ de Branges Rovnyak transcription for W = θ 0 ) [ 1] : The model space is given by {( ) } g+ K θ = : g + HD 2, g HD 2, g θ g + L 2 D g A function in K θ is determined by the boundary values of two functions g + and g analytic in D and ext(d) respectively.

28 ( 0 θ de Branges Rovnyak transcription for W = θ 0 ) [ 1] : The model space is given by {( ) } g+ K θ = : g + HD 2, g HD 2, g θ g + L 2 D g A function in K θ is determined by the boundary values of two functions g + and g analytic in D and ext(d) respectively. If θ is an extreme point of the unit ball in H (or equivalently if T ln 2 dz =,) (or equivalently T ln w dz = where dµ = wdm + dµ s), then this transcription reduces to studying g 1 H(θ).

29 Universal representation formulas for Φ γ

30 Representation formula for Φ γ in the free functional model Recall U γ = M ξ + (γ 1)1 ξ on L 2 (µ) uniquely determines θ, Φ γ M θ = U γ Φ γ, The 1-dimensional defect spaces are mapped Φ γ : D D Mθ, Φ γ : D D M θ,

31 Representation formula for Φ γ in the free functional model Recall U γ = M ξ + (γ 1)1 ξ on L 2 (µ) uniquely determines θ, Φ γ M θ = U γ Φ γ, The 1-dimensional defect spaces are mapped Φ γ : D D Mθ, Φ γ : D D M θ, WLOG c γ = Φ γb D Mθ and c γ 1 = Φ γb 1 D Mθ.

32 Representation formula for Φ γ in the free functional model Recall U γ = M ξ + (γ 1)1 ξ on L 2 (µ) uniquely determines θ, Φ γ M θ = U γ Φ γ, The 1-dimensional defect spaces are mapped Φ γ : D D Mθ, Φ γ : D D M θ, WLOG c γ = Φ γb D Mθ and c γ 1 = Φ γb 1 D Mθ. Theorem (Universal representation formula; L. Treil, subm.) For all f C 1 (T) we have f(ξ) f(z) Φ γf(z) = A γ (z)f(z) + B γ (z) dµ(ξ) 1 ξz where A γ (z) = c γ (z), B γ (z) = c γ (z) zc γ 1 (z).

33 Φ γ in Sz.-Nagy Foiaş transcription

34 For Sz.-Nagy Foiaş transcription the defect spaces D M θ and D Mθ, respectively, are spanned by c γ (z) = ( 1 γ 2) ( ) 1/2 1 + γθγ (z), γ γ (z) c γ 1 (z) = ( 1 γ 2) ( 1/2 z 1 ) (θ γ (z) + γ) z 1. γ (z)

35 For Sz.-Nagy Foiaş transcription the defect spaces D M θ and D Mθ, respectively, are spanned by c γ (z) = ( 1 γ 2) ( ) 1/2 1 + γθγ (z), γ γ (z) c γ 1 (z) = ( 1 γ 2) ( 1/2 z 1 ) (θ γ (z) + γ) z 1. γ (z) So in the universal representation formula (f C 1 (T)) f(ξ) f(z) Φ γf(z) = A γ (z)f(z) + B γ (z) dµ(ξ) 1 ξz we have A γ (z) = ( (1 γ 2 ) 1/2 1 γθ 0 (z) γ 0 (z) 1 γθ 0 (z) ) ( (1 γ 2 ) 1/2 (1 θ 0 (z)), B γ (z) = (γ 1) 0 (z) (1 γθ 0 (z)) 1 γθ 0 (z) ).

36 Vector valued singular integral operators (SIO) Consider Radon measures µ and ν on a Haussdorff space X, and the vector-valued spaces L 2 (µ, E 1 ) and L 2 (ν, E 2 ). Locally off the diagonal {(s, t) X X : s = t}, let K(s, t) : E 1 E 2 be bounded linear operators. T : L 2 (µ, E 1 ) L 2 (ν, E 2 ) is a SIO with kernel K(s, t), if for bounded functions f and g with disjoint compact supports: ( ) (T f, g) = K(s, t)f(t), g(s) dµ(t)dν(s). L 2 (ν) E 2 X X

37 Vector valued singular integral operators (SIO) Consider Radon measures µ and ν on a Haussdorff space X, and the vector-valued spaces L 2 (µ, E 1 ) and L 2 (ν, E 2 ). Locally off the diagonal {(s, t) X X : s = t}, let K(s, t) : E 1 E 2 be bounded linear operators. T : L 2 (µ, E 1 ) L 2 (ν, E 2 ) is a SIO with kernel K(s, t), if for bounded functions f and g with disjoint compact supports: ( ) (T f, g) = K(s, t)f(t), g(s) dµ(t)dν(s). L 2 (ν) E 2 X X Theorem (L. Treil, submitted) The operator Φ γ : L 2 (µ) L 2 (C 2 ) in the Sz.-Nagy Foiaş transcription is a singular integral operator with kernel K, 1 K(z, ξ) = B γ (z) 1 ξz.

38 A singular kernel K is restrictedly bounded if for f L 2 (µ) and g L 2 (ν) with disjoint compact supports (T f, g) L 2 (ν) C f L 2 (µ) g L 2 (ν). The restricted norm is K restr := inf{c}.

39 A singular kernel K is restrictedly bounded if for f L 2 (µ) and g L 2 (ν) with disjoint compact supports (T f, g) L 2 (ν) C f L 2 (µ) g L 2 (ν). The restricted norm is K restr := inf{c}. Corollary The kernel K(z, ξ) = 1 1 ξz is restrictedly bounded from L2 (µ) to L 2 (v), v(z) = B γ (z) 2, z T with the restricted norm at most 1.

40 A singular kernel K is restrictedly bounded if for f L 2 (µ) and g L 2 (ν) with disjoint compact supports (T f, g) L 2 (ν) C f L 2 (µ) g L 2 (ν). The restricted norm is K restr := inf{c}. Corollary The kernel K(z, ξ) = 1 1 ξz is restrictedly bounded from L2 (µ) to L 2 (v), v(z) = B γ (z) 2, z T with the restricted norm at most 1. ( ) T K(z, ξ)f(ξ), g(z) dµ(ξ)dm(z) f C g 2 L 2 (µ) L 2 (C 2 )

41 A singular kernel K is restrictedly bounded if for f L 2 (µ) and g L 2 (ν) with disjoint compact supports (T f, g) L 2 (ν) C f L 2 (µ) g L 2 (ν). The restricted norm is K restr := inf{c}. Corollary The kernel K(z, ξ) = 1 1 ξz is restrictedly bounded from L2 (µ) to L 2 (v), v(z) = B γ (z) 2, z T with the restricted norm at most 1. ( ) T K(z, ξ)f(ξ), g(z) dµ(ξ)dm(z) f C g 2 L 2 (µ) L 2 (C 2 ) WLOG consider g = (ϕ/ B γ ) B γ, ϕ L 2 scalar-valued, then with ψ = ϕ/ B γ we have ψ = ϕ = g L 2 (v) L 2 L 2 (C 2 )

42 A singular kernel K is restrictedly bounded if for f L 2 (µ) and g L 2 (ν) with disjoint compact supports (T f, g) L 2 (ν) C f L 2 (µ) g L 2 (ν). The restricted norm is K restr := inf{c}. Corollary The kernel K(z, ξ) = 1 1 ξz is restrictedly bounded from L2 (µ) to L 2 (v), v(z) = B γ (z) 2, z T with the restricted norm at most 1. ( ) T K(z, ξ)f(ξ), g(z) dµ(ξ)dm(z) f C g 2 L 2 (µ) L 2 (C 2 ) WLOG consider g = (ϕ/ B γ ) B γ, ϕ L 2 scalar-valued, then with ψ = ϕ/ B γ we have ψ = ϕ = g L 2 (v) L 2 L 2 (C 2 ) So T K(z, ξ)f(ξ)ψ(z)v(z)dµ(ξ)dm(z) f L 2 (µ) ψ L 2 (v)

43 Singular integral operators (Sz.-Nagy Foiaş transcription) Lemma Operators T r : L 2 (µ) L 2 (v), r [0, ) \ {1} with kernel 1 K r (z, ξ) =, r [0, )\{1} 1 rξz have the norm at most 4. The limit T ± = w.o.t.- lim r 1 T r exists as a bounded operator from L 2 (µ) L 2 (v).

44 Singular integral operators (Sz.-Nagy Foiaş transcription) Lemma Operators T r : L 2 (µ) L 2 (v), r [0, ) \ {1} with kernel 1 K r (z, ξ) =, r [0, )\{1} 1 rξz have the norm at most 4. The limit T ± = w.o.t.- lim r 1 T r exists as a bounded operator from L 2 (µ) L 2 (v). Idea of proof: For r 1 we compute 1 ξz 1 r ξz = 1 ± n=1 (r ±n r ±n 1 )(ξz) ±n. Further (ξz) n is a product of unimodular functions, and Restricted norm norm 1 + n=1 r±n r ±n 1 2

45 Formula for Φ γ on L 2 (µ) (Sz.-Nagy Foiaş transcription) Theorem For all f L 2 (µ) we have Φ γf(z) = A γ (z)f(z) + B γ (z) ( (T + f)(z) f(z)(t + 1)(z) ). Or more explicitly (1 γ 2 ) 1/2 Φ γf ( 0 = (γ (γ 1)T + 1) γ ( ) ( 0 = 1 γθ 0 1 γθ 0 T f ) ( (1 + γθγ )/T f (γ 1) γ 1 γ 2 1 γθ 0 1 T + 1 (γ 1) (1 γ 2 ) 1/2 1 γθ 0 0 ) T + f ) T + f.

46 Φ γ in de Branges Rovnyak transcription

47 Recall the model space in de Branges Rovnyak transcription: {( ) } g+ K θ = : g + H 2, g H, 2 g θ g + L 2. g

48 Recall the model space in de Branges Rovnyak transcription: {( ) } g+ K θ = : g + H 2, g H, 2 g θ g + L 2. g In the Sz.-Nagy Foiaş transcription a function ( ) ( g1 H g = 2 ) g 2 clos L 2 ( H is in K θ = 2 ) ( ) θ clos L 2 H 2 if and only if g := θg 1 + g 2 H 2 := L 2 H 2.

49 Recall the model space in de Branges Rovnyak transcription: {( ) } g+ K θ = : g + H 2, g H, 2 g θ g + L 2. g In the Sz.-Nagy Foiaş transcription a function ( ) ( g1 H g = 2 ) g 2 clos L 2 ( H is in K θ = 2 ) ( ) θ clos L 2 H 2 if and only if g := θg 1 + g 2 H 2 := L 2 H 2. Note, that g 2 = g θg 1.

50 Recall the model space in de Branges Rovnyak transcription: {( ) } g+ K θ = : g + H 2, g H, 2 g θ g + L 2. g In the Sz.-Nagy Foiaş transcription a function ( ) ( g1 H g = 2 ) g 2 clos L 2 ( H is in K θ = 2 ) ( ) θ clos L 2 H 2 if and only if g := θg 1 + g 2 H 2 := L 2 H 2. Note, that g 2 = g θg 1. The pair g + := g 1 and g belongs to the de Branges Rovnyak space and the norms coincide.

51 Φ γ in de Branges Rovnyak transcription Theorem Let µ be not the Lebesgue measure. Then the function g = g γ is given by g γ = (1 γ 2 ) 1/2 ( θ γ + γ ) T f T 1 = (1 γ 2 ) 1/2 θ 0 1 γθ 0 T f T 1.

52 The Clark operator Φ γ

53 Consider the Radon Nikodym decomposition dµ = wdm + dµ s, w = dµ/dm L 1. Any function f L 2 (µ) can be decomposed f = f s + f a where f s = d(fµ) s /dµ s, f a = d(fµ) a /dµ a.

54 Consider the Radon Nikodym decomposition dµ = wdm + dµ s, w = dµ/dm L 1. Any function f L 2 (µ) can be decomposed f = f s + f a where f s = d(fµ) s /dµ s, f a = d(fµ) a /dµ a. Theorem Let g = ( g1 g 2 let f = Φ γ g L 2 (µ). ) K θ (in the Sz.-Nagy Foiaş transcription) and

55 Consider the Radon Nikodym decomposition dµ = wdm + dµ s, w = dµ/dm L 1. Any function f L 2 (µ) can be decomposed f = f s + f a where f s = d(fµ) s /dµ s, f a = d(fµ) a /dµ a. Theorem Let g = ( g1 g 2 let f = Φ γ g L 2 (µ). Then ) K θ (in the Sz.-Nagy Foiaş transcription) and 1 the non-tangential boundary limits f s (ξ) = lim z ξ,z D 1 γ (1 γ 2 ) 1/2 g 1(z) µ s a.e. ξ T.

56 Consider the Radon Nikodym decomposition dµ = wdm + dµ s, w = dµ/dm L 1. Any function f L 2 (µ) can be decomposed f = f s + f a where f s = d(fµ) s /dµ s, f a = d(fµ) a /dµ a. Theorem Let g = ( g1 g 2 let f = Φ γ g L 2 (µ). Then ) K θ (in the Sz.-Nagy Foiaş transcription) and 1 the non-tangential boundary limits f s (ξ) = lim z ξ,z D 1 γ (1 γ 2 ) 1/2 g 1(z) µ s a.e. ξ T. 2 for the absolutely continuous part f a of f (1 γ 2 ) 1/2 wf a = 1 γθ 0 1 θ 0 g γθ 0 1 θ 0 g a.e. on T; here g := g 1 θ γ + γ g 2.

57 Some possible future questions Clark model for dissipative perturbations (include singular form bounded perturbations, where 1 / L 2 (µ)). More singular perturbations: U γ defined up to equivalence class. Higher rank perturbations.

58 Let H be a self-adjoint operator on a separable Hilbert space H. Let {ϕ n } H be a sequence of mutually linearly independent unit vectors in H, and let ω = (ω 1, ω 2,...) be a random variable corresponding to a probability measure Ω = i=1 Ω i on R. Assume that Ω satisfies Kolmogorov s 0-1 law (e.g satisfied, if Ω i are all purely absolutely continuous). Theorem Assume that (H ω ) ess is cyclic with respect to Ω almost surely. Let µ denote the spectral measure of the operator (H ω ) ess with respect to some cyclic vector. If ess-supp (µ ω ) ac = 0 almost surely, then (H ω ) ess (H η ) ess (Mod rank one) almost surely with respect to the product measure Ω Ω.