Clark model in the general situation

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

Classical perturbation theory

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.}

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.

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

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).

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

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

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

Cnu contractions and functional models alla Nikolski Vasyunin

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}.

Cnu contractions ( γ < 1) and characteristic functions

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}.

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

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.

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

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.

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.

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

( 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).

Universal representation formulas for Φ γ

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).

Φ γ in Sz.-Nagy Foiaş transcription

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)

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) ).

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

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}.

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 )

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

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 0 ) ( (1 + γθγ )/T f + + 1 (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.

Φ γ in de Branges Rovnyak transcription

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

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.

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.

The Clark operator Φ γ

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.

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.

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.

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.