Rank one perturbations of unitary operators and Clarks model in - PowerPoint PPT Presentation

Outline Rank one perturbations of unitary operators and Clark’s model in general situation Sergei Treil Department of Mathematics Brown University March 7, 2016 1

Outline Main objects: rank one perturbations and models 1 Rank one perturbations Functional models for c.n.u. contractions Characteristic function and defects for U γ A universal formula for the adjoint Clark operator 2 Spectral representation of unitary perturbations Adjoint Clark operator, freedom of choice A universal representation formula Representations in different transcriptions 3 Representation for the Sz.-Nagy–Foia¸ s transcription Representation in the de Branges–Rovnyak transcription Bounds on the normalized Cauchy transforms 2

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Rank one perturbations For a unitary U = U 1 let U γ := U + ( γ − 1) bb ∗ b 1 := U ∗ b, 1 , � b � = 1 , γ ∈ C . 3

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Rank one perturbations For a unitary U = U 1 let U γ := U + ( γ − 1) bb ∗ b 1 := U ∗ b, 1 , � b � = 1 , γ ∈ C . If | γ | = 1 then we have all rank one unitary perturbations (if the range of the perturbation is fixed). 3

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Rank one perturbations For a unitary U = U 1 let U γ := U + ( γ − 1) bb ∗ b 1 := U ∗ b, 1 , � b � = 1 , γ ∈ C . If | γ | = 1 then we have all rank one unitary perturbations (if the range of the perturbation is fixed). Indeed U + K = ( I + KU ∗ ) U , and it is easy to describe all unitary perturbations of I : KU ∗ = ( γ − 1) bb ∗ , � b � = 1 , | γ | = 1 . 3

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Rank one perturbations For a unitary U = U 1 let U γ := U + ( γ − 1) bb ∗ b 1 := U ∗ b, 1 , � b � = 1 , γ ∈ C . If | γ | = 1 then we have all rank one unitary perturbations (if the range of the perturbation is fixed). Indeed U + K = ( I + KU ∗ ) U , and it is easy to describe all unitary perturbations of I : KU ∗ = ( γ − 1) bb ∗ , � b � = 1 , | γ | = 1 . WLOG: b is cyclic, so U = M ξ in L 2 ( µ ) , µ ( T ) = 1 ; b ≡ 1 , therefore b 1 ( ξ ) = ξ . 3

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Rank one perturbations For a unitary U = U 1 let U γ := U + ( γ − 1) bb ∗ b 1 := U ∗ b, 1 , � b � = 1 , γ ∈ C . If | γ | = 1 then we have all rank one unitary perturbations (if the range of the perturbation is fixed). Indeed U + K = ( I + KU ∗ ) U , and it is easy to describe all unitary perturbations of I : KU ∗ = ( γ − 1) bb ∗ , � b � = 1 , | γ | = 1 . WLOG: b is cyclic, so U = M ξ in L 2 ( µ ) , µ ( T ) = 1 ; b ≡ 1 , therefore b 1 ( ξ ) = ξ . If | γ | < 1 , U γ is a completely non-unitary (c.n.u.) contraction with defect indices 1 - 1 , rank( I − U ∗ γ U γ ) = rank( I − U γ U ∗ γ ) = 1 . 3

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Models for U γ If | γ | = 1 then U γ is unitary, so U γ ∼ = M z , M z : L 2 ( µ γ ) → L 2 ( µ γ ) , M z f ( z ) = zf ( z ) . If | γ | < 1 then U γ is a c.n.u. contraction and admits the functional model, U γ ∼ = M θ , � � K θ ; M θ : K θ → K θ , M θ = P K θ M z here θ ∈ H ∞ , � θ � ∞ ≤ 1 is the characteristic function of U γ , and K θ is the model space Goal: Want to describe unitary operators intertwining U γ and its model. 4

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Characteristic function For detail see Sz.-Nagy–Foia¸ s [9]. For T , � T � ≤ 1 let D T = ( I − T ∗ T ) 1 / 2 , D T ∗ := ( I − TT ∗ ) 1 / 2 D = D T := clos Ran D T , D ∗ = D T ∗ := clos Ran D T ∗ . Characteristic function θ ∈ H ∞ ( D ; B ( D ; D ∗ )) is defined as � � � � − T + zD T ∗ ( I − zT ∗ ) − 1 D T θ T ( z ) = � D , z ∈ D . Note that � θ � ∞ ≤ 1 . Usually θ is defined up to constant unitary factors (choice of bases in D and D ∗ ); spaces E ∼ = D and E ∗ ∼ = D ∗ are used. 5

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Functional model(s) Following Nikolskii–Vasyunin [4] the functional model is constructed as follows: 1 For a contraction T : K → K consider its minimal unitary dilations U : H → H , K ⊂ H , T n = P K U n � � K , n ≥ 0 . 2 Pick a spectral representation of U 3 Work out formulas in this spectral representation 4 Model subspace K = K θ is usually a subspace of a weighted space L 2 ( E ⊕ E ∗ , W ) , E ∼ = D , E ∗ ∼ = D ∗ with some operator-valued weight. Specific representations give us a transcription of the model. Among common transcriptions are: the Sz.-Nagy–Foia¸ s transcription, the de Branges–Rovnyak transcription, Pavlov transcription. 6

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Sz.-Nagy–Foia¸ s and de Branges–Rovnyak transcriptions s: H = L 2 ( E ⊕ E ∗ ) (non-weighted, W ≡ I ). Sz.-Nagy–Foia¸ � � � θ � H 2 H 2 E ∗ K θ := ⊖ E , clos ∆ L 2 ∆ E where ∆( z ) := (1 − θ ( z ) ∗ θ ( z )) 1 / 2 , z ∈ T . de Branges–Rovnyak: H = L 2 ( E ⊕ E ∗ , W [ − 1] ) , where θ � � I θ ( z ) W θ ( z ) = θ ( z ) ∗ I and W [ − 1] is the Moore–Penrose inverse of W θ . K θ is given by θ �� g + � � : g + ∈ H 2 ( E ∗ ) , g − ∈ H 2 − ( E ) , g − − θ ∗ g + ∈ ∆ L 2 ( E ) . g − 7

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Characteristic function and defects for U γ Recall: U γ = U 1 + ( γ − 1) bb ∗ 1 , b 1 = U ∗ 1 b , | γ | < 1 . D U γ and D U ∗ γ are spanned by the vectors b 1 and b respectively. Characteristic function θ T of a contraction T is defined as � � � � − T + zD T ∗ ( I − zT ∗ ) − 1 D T θ T ( z ) = � D , z ∈ D . To compute it use Rank one inversion formula (Sherman–Morrison formula) ( I − bc ∗ ) − 1 = I + 1 dbc ∗ , d = ( b, c ) = c ∗ b. I − zU ∗ γ is a rank one perturbation of I − zU ∗ 1 = I − zM ξ ; The inverse of I − zM ξ is multiplication by (1 − zξ ) − 1 , so Cauchy integrals appear. 8

Main objects: rank one perturbations and models Rank one perturbations A universal formula for the adjoint Clark operator Functional models for c.n.u. contractions Representations in different transcriptions Characteristic function and defects for U γ Characteristic function and defects for U γ Define Cauchy integrals ξλdτ ( ξ ) 1 + ξλ ˆ ˆ R 1 τ ( λ ) := 1 − ξλ , R 2 τ ( λ ) := 1 − ξλdτ ( ξ ) . T T Characteristic function θ γ of U γ in the bases b 1 , b : θ γ ( λ ) = − γ + (1 − | γ | 2 ) R 1 µ ( λ ) 1 + (1 − γ ) R 1 µ ( λ ) = (1 − γ ) R 2 µ ( λ ) − (1 + γ ) (1 − γ ) R 2 µ ( λ ) + (1 + γ ) , Note that θ γ (0) = − γ , because R 1 µ (0) = 0 For γ = 0 1 + R 1 µ ( λ ) = R 2 µ ( λ ) − 1 R 1 µ ( λ ) θ 0 ( λ ) = R 2 µ ( λ ) + 1 , λ ∈ D . 9