Spectral analysis of semigroups in Banach spaces and applications to PDEs

Transcription

1 Spectral analysis of semigroups in Banach spaces and applications to PDEs S. Mischler (Paris-Dauphine & IUF) in collaboration with M. J. Caceres, J. A. Cañizo, G. Egaña, M. Gualdani, C. Mouhot, J. Scher works by K. Carrapatoso, I. Tristani 6th International Workshop on Kinetic Theory & Applications Karlstad University, May 12-14, 2014 S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

2 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

3 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

4 Revisit the spectral theory in an abstract setting Spectral theory for general operator and its semigroup in general (large) Banach space, without regularity ( eventually norm continuous), without symmetry ( Hilbert space and self-adjoint op) and without (or with) positivity (Banach lattice) Spectral map Theorem Σ(e tλ ) e tσ(λ) and ω(λ) = s(λ) Weyl s Theorem (quantified) compact perturbation Σ ess (A + B) Σ ess (B) Small perturbation Σ(Λ ε ) Σ(Λ) if Λ ε Λ Krein-Rutmann Theorem s(λ) = sup ReΣ(Λ) Σ d (Λ) when S Λ 0 functional space extension (enlargement and shrinkage) Σ(L) Σ(L) when L = L E tide of spectrum phenomenon Structure: operator which splits as Λ = A + B, A B, B dissipative Examples: Boltzmann, Fokker-Planck, Growth-Fragmentation operators and W σ,p (m) weighted Sobolev spaces S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

5 Applications / Motivations : (1) Convergence rate in large Banach space for linear dissipative and hypodisipative PDEs (ex: Fokker-Planck, growth-fragmentation) (2) Long time asymptotic for nonlinear PDEs via the spectral analysis of linearized PDEs (ex: Boltzmann, Landau, Keller-Segel) in natural ϕ space (3) Existence, uniqueness and stability of equilibrium in small perturbation regime in large space (ex: inelastic Boltzmann, Wigner-Fokker-Planck, parabolic-parabolic Keller-Segel, neural network) Is it new? Simple and quantified versions, unified theory (sectorial, KR, general) which holds for the principal part of the specrtrum first enlargement result in an abstract framework by C. Mouhot (CMP06) Unusual splitting Λ = A }{{} 0 + B 0 = A }{{} ε + A }{{} c ε + B 0 }{{} compact dissipative smooth dissipative The applications to these linear(ized) kinetic equations and to these nonlinear problems are clearly new S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

6 Old problems Fredholm, Hilbert, Weyl, Stone 1932 (Funct Analysis & sg Hilbert framewrok) Hyle, Yosida, Phillips, Lumer, Dyson (sg Banach framework & dissipative operators) and also Dunford, Schwartz Kato, Pazy, Voigt (analytic op., positive op.) Engel, Nagel, Gearhart, Metz, Diekmann, Prüss, Arendt, Greiner, Blake, Mokhtar-Kharoubi, Yao, Spectral analysis of the linearized (in)homogeneous Boltzmann equation and convergence to the equilibrium Hilbert, Carleman, Grad, Ukai, Arkeryd, Esposito, Pulvirenti, Wennberg, Guo, Strain,... Spectral tide/spectral analysis in large space Bobylev (for linearized Boltzmann with Maxwell molecules, 1975), Gallay-Wayne (for harmonic Fokker-Planck, 2002) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

7 Still active research field Semigroup school ( 0, bio): Arendt, Blake, Diekmann, Engel, Gearhart, Greiner, Metz, Mokhtar-Kharoubi, Nagel, Prüss, Webb, Yao,... Schrodinger school / hypocoercivity and fluid mechanic: Batty, Burq, Duyckaerts, Gallay, Helffer, Hérau, Lebeau, Nier, Sjöstrand, Wayne,... Probability school (as in Toulouse): Bakry, Barthe, Bobkov, Cattiaux, Douc, Gozlan, Guillin, Fort, Ledoux, Roberto, Röckner, Wang,... Kinetic school ( Boltzmann): Carlen, Carvalho, Toscani, Otto, Villani,... (log-sobolev inequality) Desvillettes, Villani, Mouhot, Baranger, Neuman, Strain, Dolbeault, Schmeiser,... (Poincaré inequality & hypocoercivity) Guo school related to Ukai, Arkeryd, Esposito, Pulvirenti, Wennberg,... (existence in small spaces and large spaces ) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

8 A list of related papers M., Mouhot, Stability, convergence to self-similarity and elastic limit for the Boltzmann equation for inelastic hard spheres, CMP 2009 Gualdani, M., Mouhot, Factorization for non-symmetric operators and exponential H-Theorem, arxiv 2010 Arnold, Gamba, Gualdani, M., Mouhot, Sparber, The Wigner-Fokker-Planck equation: Stationary states and large time behavior, M3AS 2012 Cañizo, Caceres, M., Rate of convergence to the remarkable state for fragmentation and growth-fragmentation equations, JMPA 2011 & CAIM 2011 Egaña, M. Uniqueness and long time asymptotic for the Keller-Segel equation - Part I. The parabolic-elliptic case, arxiv 2013 M., Mouhot, Exponential stability of slowing decaying solutions to the kinetic Fokker-Planck equation, in progress M., Scher Spectral analysis of semigroups and growth-fragmentation eqs, arxiv 2013 Carrapatoso, Exponential convergence... homogeneous Landau equation, arxiv 2013 Tristani, Boltzmann equation for granular media with thermal force in a weakly inhomogeneous setting, arxiv 2013 S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

9 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

10 Examples of operators - I 1) - Linear Boltzmann, e.g. k(v, v ) = σ(v, v ) M(v ), σ(v, v) = σ(v, v ), Λf = k(v, v ) f (v ) dv k(v, v) dv f (v) } {{ }} {{ } =:Af =:Bf 2) - Fokker-Planck, with E(v) v v γ 2, γ 1, Λ = v + div v (E(v) ) M χ R + M χ R (v) }{{}}{{} =:B =:A 3) - Inhomogeneous/kinetic Fokker-Planck Λ = T + C M χ }{{ R + M χ } R (x, v) }{{} =:B =:A with T := v x + F x, Cf := v f + div v (E(v) f ) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

11 Examples of operators - II 4) - Growth fragmentation with Λ = F + F + D = F + δ + F +,c δ F + D }{{}}{{} =:A =:B Df = τ(x) x f νf, (τ, ν) = (1, 0) or (x, 2) F + (f ) := x k(y, x) f (y) dy, Mass conservation of F + F implies Self-similarity in y/x with K(x) = x k(x, y) = K(x) x 1 θ(y/x), 0 y k(x, y) dy x F f := K(x) f 1 0 zθ(z) dz = 1, θ D(0, 1) or θ(z) = 2 δ z=1/2 or θ(z) = δ z=0 + δ z=1 S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

12 Examples of operators - III 5) - Linearized Boltzmann Λh = Q(h, M) + Q(M, h) = Q + (h, M) + Q + (M, h) L(h) M L(M)h = Q +, δ [h] + Q +,,c δ [h] L(M)h }{{}}{{} =:Ah =:Bh 6) - Inhomogeneous linearized Boltzmann (in the torus) Λh = Q +, δ [h] + Q +,,c δ [h] L(M)h + T h, T := v x }{{}}{{} =:Ah =:Bh 7) - other operators: homogeneous/inhomogeneous linearized inelastic Boltzmann, homogeneous linearized Landau, Fokker-Planck with fractional diffusion, linearized Keller-Segel (parabolic-elliptic), homogeneous Boltzmann for hard potential without angular cut-off S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

13 The Growth-Fragmentation equation (as an application of the KR theorem) Th 1. (M., Scher) Assume that for γ 0, x 0 0, 0 < K 0 K 1 < : K 0 x γ 1 x x0 K(x) K 1 x γ. There exists a (unique) (λ, f ) with λ R and f is the unique solution to Ff + Df = λ f, f 0, f, 1 = 1. There exists a < λ, C > 0 such that f 0 L 1 α, α > 1 fe Λt f 0 e λt Π 0 f 0 L 1 α C e at f 0 e λt Π 0 f 0 L 1 α, where Π 0 is the projector on the eigenspace Vect(f ). Improve and unify : Metz-Diekmann (1983), Escobedo-M-Rodriguez (2005), Michel-M-Perthame (2005), Perthame-Ryzhik (2005), Laurençot-Perthame (2009), Caceres-Cañizo-M (2010) &t (2011) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

14 The Fokker-Planck equation (as a consequence of extension or KR theorems) Consider t f = Λf = v f + div v (F f ) with a (friction) force field F such that F x x γ, divf C F x γ 2, x B c R Th 2. Gualdani-M.-Mouhot; M.-Mouhot; Ndao There exists a unique positive and unit mass stationary solution f, and for any σ { 1, 0, 1}, p [1, ], any m = v k, k > k (p, σ, γ) or m = e κ v s, s [2 γ, γ], γ 1, κ < 1/γ if s = γ, any a (a σ(p, m), 0), there exists C = C(a, p, σ, γ, m) such that for any f 0 W σ,p (m) e tλ f 0 f 0 f W σ,p (m) C e at f 0 f 0 f W σ,p (m). Generalizes similar results known in L 2 (f 1/2 ) The same result holds for the kinetic Fokker-Planck in the torus and in R d x with confinement potential Provides decay in Wasserstein distance (see also Bolley-Gentil-Guillin (2012)) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

15 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

16 Conditionally (up to time uniform strong estimate) exponential H-Theorem (f t ) t 0 solution to the inhomogeneous Boltzmann equation for hard spheres interactions in the torus with strong estimate ( ) ft H k + f t L 1 (1+ v s ) Cs,k <. sup t 0 Desvillettes, Villani proved [Invent. Math. 2005]: for any s s 0, k k 0 f t t 0 f t log T R d G 1 (v) dvdx C s,k (1 + t) τ s,k with C s,k <, τ s,k when s, k, G 1 := Maxwell function Th 3. Gualdani-M.-Mouhot s 1, k 1 s.t. for any a > λ 2 exists C a f t t 0 f t log T R d G 1 (v) dvdx C a e a 2 t, with λ 2 < 0 (2 nd eigenvalue of the linearized Boltzmann eq. in L 2 (G 1 1 )). S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

17 Global existence and uniqueness for weakly inhomogeneous initial data for the elastic and inelastic inhomogeneous Boltzmann equation for hard spheres interactions in the torus Th 4. Gualdani-M.-Mouhot; Tristani For any F 0 L 1 3 (Rd ) there exists e 0 (0, 1) and ε 0 > 0 such that if f 0 (T d ; L 1 3 (Rd )) satisfies f 0 F 0 ε 0 and if e [e 0, 1] then W k,1 x there exists a unique global mild solution f (t, x, v) starting from f 0 ; f (t) G 1 when t (with rate) when e = 1; f (t) Ḡe when t (with rate) when e < 1 (diffuse forcing). The case e 1 is proved thanks to a small perturbation argument in a large space because Ḡ e(v) e v 3/2 / L 2 (G 1/2 1 ). The case e = 1 has been treated by non constructive arguments by Arkeryd-Esposito- Pulvirenti (CMP 1987), Wennberg (Nonlinear Anal. 1993) and for the space homogeneous analogous by Arkeryd (ARMA 1988), Wennberg (Adv. MAS 1992) Extend to a larger class of initial data similar results due to Ukai, Guo, Strain and collaborators S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

18 More results about constructive exponential rate of convergence For homogeneous Boltzmann eq for hard spheres (Mouhot 2006) homogeneous weakly inelastic Boltzmann eq for hard spheres (M-Mouhot 2009) homogeneous Landau eq for hard potential (Carrapatoso 2013) parabolic-elliptic Keller-Segel eq (Egaña-M 2013) homogeneous Boltzmann eq for hard potential (Tristani, soon on arxiv) In all these cases, we prove that under minimal assumptions on the initial datum f 0 (bounded mass, energy, entropy,...) the associated solution f (t) satisfies f (t) G when t (with exponential rate) where G is the unique associated equilibrium/self-similar profile We know (except for the inelastic Boltzmann eq) that the associated linearized operator L is self-adjoint and has a spectral gap in the very small space L 2 (G 1/2 1 ) in which a general solution does not belong (even for large time). we start by enlarge the space in which L has a spectral gap and then we (classically) prove a nonlinear stability result for the weakly inelastic Boltzmann eq we additionally use perturbation argument S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

19 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

20 Main issue For a given operator Λ in a Banach space X, we want to prove (1) Σ(Λ) a = {ξ 1 }, ξ 1 = 0 with Σ(Λ) = spectrum, α := {z C, Re z > α} (2) Π Λ,ξ1 = finite rank projection, i.e. ξ 1 Σ d (Λ) (3) S Λ (I Π Λ,ξ1 ) X X C a e at, a < Reξ 1 Definition: We say that L a is hypodissipative iff e tl X X C e at. S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

21 Spectral mapping - characterization Th 1. (M., Scher) (0) Λ = A + B, where A is B ζ -bounded with 0 ζ < 1, (1) S B (AS B ) ( l) X X C l e at, a > a, l 0, (2) S B (AS B ) ( n) X D(Λ ζ ) C n e at, a > a, with ζ > ζ, (3) Σ(Λ) ( a \ a ) =, a < a, is equivalent to (4) there exists a projector Π which commutes with Λ such that Λ 1 := Λ X1 B(X 1 ), X 1 := RΠ, Σ(Λ 1 ) a In particular and S Λ (t) (I Π) X X C a e at, Σ(e tλ ) e at = e tσ(λ) a a > a t 0, a > a max(s(λ), a ) = max(ω(λ), a ) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

22 Weyl s theorem - characterization Th 2. (M., Scher) (0) Λ = A + B, where A is B ζ -bounded with 0 ζ < 1, (1) S B (AS B ) ( l) X X C l e at, a > a, l 0, (2) S B (AS B ) ( n) X Xζ C n e at, a > a, with ζ > ζ, (3) 0 (AS B ) ( n+1) X Y e at dt <, a > a, with Y X, is equivalent to (4) there exist ξ 1,..., ξ J a, there exist Π 1,..., Π J some finite rank projectors, there exists T j B(RΠ j ) such that ΛΠ j = Π j Λ = T j Π j, Σ(T j ) = {ξ j }, in particular Σ(Λ) a = {ξ 1,..., ξ J } Σ d (Σ) and there exists a constant C a such that J S Λ (t) e tt j Π j X X C a e at, j=1 a > a S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

23 Small perturbation Th 3. (M. & Mouhot; Tristani) Assume (0) Λ ε = A ε + B ε in X i, X 1 X 0 = X X 1, A ε B ε, (1) S Bε (A ε S Bε ) ( l) Xi X i C l e at, a > a, l 0, i = 0, ±1, (2) S Bε (A ε S Bε ) ( n) Xi X i+1 C n e at, a > a, i = 0, 1, (3) X i+1 D(B ε Xi ), D(A ε Xi ) for i = 1, 0 and A ε A 0 Xi X i 1 + B ε B 0 Xi X i 1 η 1 (ε) 0, i = 0, 1, (4) the limit operator satisfies (in both spaces X 0 and X 1 ) Then Σ(Λ 0 ) a = {ξ 1,..., ξ k } Σ d (Λ 0 ). Σ(Λ ε ) a = {ξ ε 1,1,..., ξ ε 1,d ε 1,..., ξε k,1,..., ξε k,d ε k } Σ d(λ ε ), ξ j ξ ε j,j η(ε) 0 1 j k, 1 j d j ; dimr(π Λε,ξ ε j, Π Λ ε,ξ ε j,d j ) = dimr(π Λ0,ξ j ); S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

24 Krein-Rutmann for positive operator Th 4. (M. & Scher) Consider a semigroup generator Λ on a Banach lattice of functions X, (1) Λ such as in Weyl s Theorem for some a R; (2) b > a and ψ D(Λ ) X +\{0} such that Λ ψ b ψ; (3) S Λ is positive (and Λ satisfies Kato s inequalities); (4) Λ satisfies a strong maximum principle. Defining λ := s(λ), there holds a < λ = ω(λ) and λ Σ d (Λ), and there exists 0 < f D(Λ) and 0 < φ D(Λ ) such that and then Λf = λ f, Λ φ = λ φ, RΠ Λ,λ = Vect(f ), Π Λ,λ f = f, φ f f X. Moreover, there exist α (a, λ) and C > 0 such that for any f 0 X S Λ (t)f 0 e λt Π Λ,λ f 0 X C e αt f 0 Π Λ,λ f 0 X t 0. S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

25 Change (enlargement and shrinkage) of the functional space of the spectral analysis and semigroup decay Th 5. (Moutot 06, Gualdani, M. & Mouhot) Assume L = A + B, L = A + B, A = A E, B = B E, E E (i) (B a) is hypodissipative on E, (B a) is hypodissipative on E; (ii) A B(E), A B(E); (iii) there is n 1 and C a > 0 such that (AS B ) ( n) (t) E E C a e at. Then the following for (X, Λ) = (E, L), (E, L) are equivalent: ξ j a and finite rank projector Π j,λ B(X ), 1 j k, which commute with Λ and satisfy Σ(Λ Πj,Λ ) = {ξ j }, so that t 0, S Λ (t) k X S(t) Π j,λ C Λ,a e a t X j=1 S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

26 Discussion / perspective In Theorem 1, 2, 3, 4, one can take n = 1 in the simplest situations (most of space homogeneous equations), but one need to take n = 2 for the equal mitosis equation or for the space inhomogeneous Boltzmann equation In Theorem 5, one need to take n > d/4 for the space homogeneous Fokker-Planck equation in order to extend the spectral analysis from L 2 (well-known) to L 1 Beyond the dissipative case? example of the Fokker-Planck equation when γ (0, 1) and relation with weak Poincaré inequality by Röckner-Wang Links with semi-uniform stability by Lebeau & co-authors, Burq, Liu-R, Bátkal-E-P-S, Batty-D,... applications to Boltzmann and Landau equation associated to soft potential inhomogeneous linearized Landau, linearized Keller-Segel (parabolicparabolic), neural network, Fokker-Planck in the subcritical case γ (0, 1) S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

27 Outline of the talk 1 Introduction 2 Examples of linear evolution PDE Gallery of examples Hypodissipativity result under weak positivity Hypodissipativity result in large space 3 Nonlinear problems Increasing the rate of convergence Perturbation regime 4 Spectral theory in an abstract setting 5 Elements of proofs The enlargement theorem The spectral mapping theorem S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

28 Proof of the enlargement theorem We split the semigroup into invariant linear sub-manifolds (eigenspaces) S L = Π S L + (I Π) S L (I Π) and write the (iterated) Duhamel formula or stopped Dyson-Phillips series (the Dyson-Phillips series corresponds to the choice n = ) n 1 S L = S B (AS B ) ( l) + S L (AS B ) ( n) l=0 These two identities together or + (AS B ) ( n) S L. n 1 S L = Π S L + (I Π) { S B (AS B ) ( l) } (I Π) l=0 + {(I Π) S L } (AS B ) ( n) (I Π) or + (I Π)(AS B ) ( n) {S L (I Π)} S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

29 Sketch of the proof of the spectral mapping theorem We introduce the resolvent R Λ (z) = (Λ z) 1 = 0 S Λ (t) e zt dt. Using the inverse Laplace formula for b > ω(λ) s(λ) = sup ReΣ(Λ) and the fact that Π R Λ (z) is analytic in a, Π := I Π, we get S Λ (t)π = i 2π = lim M b+i b i a+im i 2π e zt Π R Λ (z) dz a im e zt Π R Λ (z) dz Similarly as for the (iterated) Duhamel formula, we have N 1 R Λ = ( 1) l R B (AR B ) l + ( 1) N R Λ (AR B ) N l=0 S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

30 These two identities together N 1 S L (t)π = Π ( 1) l i 2π = l=0 +( 1) N Π N 1 l=0 +( 1) N i 2π a+i a i a+i a i Π S B (AS B ) ( l) i 2π a+i a i e zt R B (z)(ar B (z)) l dz e zt R Λ (z)(ar B (z)) N dz e zt Π R Λ (z)(ar B (z)) N dz and we have to explain why the last term is of order O(e at ). We clearly have sup Π R Λ (z)(ar B (z)) N C M z=a+iy, y [ M,M] and it is then enough to get the bound R Λ (z)(ar B (z)) N C/ y 2, z = a + iy, y M, a > a S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

31 The key estimate We assume (in order to make the proof simpler) that ζ = 1, namely (AS B ) ( n) X X1 = O(e at ) t 0, with X 1 := D(Λ) = D(B), which implies (AR B (z)) n X X1 C a z = a + iy, a > a. We also assume (for the same reason) that ζ = 0, so that A L(X ) and R B (z) = 1 z (R B(z)B I ) L(X 1, X ) imply AR B (z) X1 X C a / z z = a + iy, a > a. The two estimates together imply ( ) (AR B (z)) n+1 X X C a / z z = a + iy, a > a. In order to deal with the general case 0 ζ < ζ 1 one has to use some additional interpolation arguments S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32

32 We write with U := R Λ (1 V) = U n ( 1) l R B (AR B ) l, V := ( 1) n+1 (AR B ) n+1 l=0 For M large enough ( ) V(z) 1/2 z = a + iy, y M, and we may write the Neuman series R Λ (z) = U(z) }{{} bounded V(z) j j=0 } {{ } bounded For N = 2(n + 1), we finally get from ( ) and ( ) R Λ (z)(ar B (z)) N C/ y 2, z = a + iy, y M S.Mischler (CEREMADE & IUF) Semigroups spectral analysis May 12-14, / 32