Spectra of weighted composition operators on spaces of analytic functions

Transcription

1 Spectra of weighted composition operators on spaces of analytic functions by Paweł Mleczko Adam Mickiewicz University in Poznań, Poland contribution to the conference New perspectives in the theory of function spaces and their applications Będlewo, September 17 23,

2 Coauthors and references This is a joint work with Mikael Lindtröm (Åbo Akademi University, Turku, Finland) Ted Eklund (Åbo Akademi University, Turku, Finland) Michał Rzeczkowski (Adam Mickiewicz University in Poznań, Poland) T. Eklund, M. Lindström, P. Mleczko Spectral properties of weighted composition operators on the Bloch and Dirichlet spaces Studia Math. 232 (2016), T. Eklund, M. Lindström, P. Mleczko, M. Rzeczkowski Spectra of weighted composition operators on abstract Hardy spaces submitted (2017), 13 pp. 2/21

3 Introduction Introduction 3/21

4 Composition operators Let φ H(D) be a self-map of D, i.e., φ(d) D. The map C φ : H(D) H(D) given by C φf(z) = f φ(z), is called a composition operator. f H(D), z D Introduction 4/21

5 Composition operators properties Boundedness (due to Littlewood, 20 of XX century) every self-map of D generates a bounded composition operator C φ : H p H p (irrespective of p, p [1, ]). Compactness (due to J. Shapiro, 80 of XX century) a composition operator C φ is compact on H p (irrespective of p, p [1, )) if and only if N φ (z) lim z 1 log 1 = 0, z where N φ is a Nevanlinna counting function Introduction 5/21

6 Weighted composition operators Let φ H(D) be a self-map of D and u H(D). Then the map uc φ : H(D) H(D) is called a weighted composition operator. A map uc φ is a bounded operator on H p is and only if u is a bounded function. Introduction 6/21

7 Research task Challenge Study the spectral properties of weighted composition operators on spaces of holomorphic functions (e.g., Hardy, Bloch spaces etc). Introduction 7/21

8 Results Results 8/21

9 Notation For a self-map of D the symbol φ n denotes the n-th iterate of φ, i.e., For any n φ n = φ φ φ }{{} n times (uc φ ) n f(z) = u(z)u(φ(z))... u(φ n 1 (z))f(φ n (z)) f H(D), z D and where (uc φ ) n = u (n) C φn, n 1 u (n) = u φ j H(D), n N. j=0 Results 9/21

10 Köthe function spaces X is a Köthe function spaces if X L 0 (Ω, Σ, µ) the space of real valued measurable functions on Ω. The order x y means x(ω) y(ω) for µ-almost all ω Ω. There exists u X with u > 0 µ-a.e. on Ω and x y with x L 0 (Ω) and y X implies x X with x X y X. If x X, then for any g equimeasurable with f, f X = g X We will consider complex Köthe function spaces. The role model for X are Lebesgue spaces, other important examples Orlicz spaces Lorentz spaces Marcinkiewicz spaces Köthe function space X is maximal (or has the Fatou property) if whenever {x n} is a norm bounded sequence in X such that 0 x n x L 0 (Ω), then x X and x = lim n x n. Results 10/21

11 Abstract Hardy spaces Let X be the Köthe space on T := [0, 2π). We define abstract Hardy spaces HX(D) in the following way. HX(D) = { f H(D) : f HX(D) := sup f r X < } r [0,1) where f r (t) := f(re it ), t T. r Results 11/21

12 Multipliers of abstract Hardy spaces For any function u H(D), an operator M u : H(D) H(D), M u f(z) = u(z)f(z), f H(D) is called a multiplication operator. Theorem Let X be a maximal r.i. space on T and suppose that u H(D). Then the following statements are equivalent: uhx HX u HX and the operator M u is bounded u H. Moreover, M u : HX HX = u. Results 12/21

13 Automorphisms of the discs φ nontrivial (i.e., not identity) automorhism of D elliptic φ has a unique fixed point in D, parabolic φ has a unique fixed (Denjoy Wolf) point a D, φ (a) = 1, hiperbolic φ has two distinct fixed points in D, an attraxtive fixed (Denjoy Wolf) point a and a repulsive fixed point b, φ (a) (0, 1) and φ (b) = 1/φ (a). When φ is a parabolic or hyperbolic automorphism, then ( lim ) 1 φn(0) 1 n = φ (a). n Results 13/21

14 Automorphisms of the disc pictures elliptic hyperbolic parabolic Results 14/21

15 Abstract Hardy spaces elliptic case Theorem Suppose that u A(D) and φ is an elliptic automorphism with the unique fixed point a D. If there is a positive integer j such that φ j (z) = z for all z D, then letting m be the smallest such integer, we have σ HX (uc φ ) = {λ C: u (m) (z) = λ m for some z D}. If φ n Id for every n N and if uc φ : HX HX is invertible, then σ HX (uc φ ) = {λ C: λ = u(a) }. Results 15/21

16 Abstract Hardy spaces parabolic case Theorem Let X be a maximal r.i. space on T and uc φ : HX HX be an invertible composition operator and assume that φ is a parabolic automorphism of the disc D with a unique fixed point a D and u A(D). Then r HX (uc φ ) = u(a) and σ HX (uc φ ) = { z C : z = u(a) }. Results 16/21

17 Invertible weighted composition operators Theorem Let X be an r.i. space on T and suppose that uc φ : HX HX is a weighted composition operator. Then uc φ is invertible if and only if u H, u is bounded away from zero on D, and φ is an automorphism of D. Moreover, if this is the case, then the inverse is also a weighted composition operator and ( ucφ ) 1 = 1 u φ 1 C φ 1. P. Bourdon Invertible weighted composition operators Proc. Amer. Math. Soc. 142 (2013), Results 17/21

18 Hardy Lorentz spaces stability of the spectrum Given a non-atomic measure space (Ω, Σ, µ) and p (1, ), q [1, ], the Lorentz space L p,q on (Ω, Σ, µ) consists of all L 0 (Ω) such that f L p,q = ( µ(ω) 0 sup t (0,µ(Ω)) ( t 1/p f (t) ) q dt t ) 1/q, q < t 1/p f (t), q =, where f (t) = 1 t t 0 f (s)ds, t > 0 and f denotes the non-increasing rearrangement of f (i.e., f (t) = inf { λ > 0 : µ f (λ) t }, t 0). Recall that for p (1, ), q [1, ], the Hardy Lorentz space H p,q is given by H p,q = HL p,q. Theorem Let φ be a holomorphic self-map of D, u H, and p, q (1, ). Then σ H p,q(uc φ ) = σ H p(uc φ ). Results 18/21

19 Hardy Lorentz spaces automorphisms Corollary Let uc φ be an invertible operator on Hardy Lorentz space H p,q, p, q (1, ) and u A(D). Then if φ is a parabolic automorphism with fixed point s D, then σ H p,q(uc φ ) = { λ C : λ = u(a) }, if φ is a hyperbolic automorphism with attractive point a D and repulsive point b D, then σ H p,q(uc φ ) = { { u(a) λ C : min φ (a), 1/p } { u(b) u(a) λ max φ (b) 1/p φ (a), 1/p }} u(b). φ (b) 1/p O. Hyvärinen, I. Nieminen Essential spectra of weighted composition operators with hyperbolic symbols Concr. Oper. 2 (2015), Results 19/21

20 Hardy Lorentz spaces non-automorphisms Corollary Assume that u, φ H(D), φ(d) D, and a is a Denjoy Wolff point with φ n a uniformly in D as n and φ (a) < 1. If u is bounded in D and continuous at a, then σ H p,q(uc φ) = { λ C : λ u(a) }. φ (a) 1/p O. Hyvärinen, I. Nieminen Essential spectra of weighted composition operators with hyperbolic symbols Concr. Oper. 2 (2015), C. Cowen, E. Ko, D. Thompson, F. Tian Spectra of some weighted composition operators on H 2 Acta Sci. Math. (Szeged) 82 (2016), no. 1 2, Results 20/21

21 Proofs ingredients interpolation and interpolation Complex functions theory. A sequence {z j } D is called interpolating for H (D) if for any bounded sequence of complex numbers {c j }, there is a function f H (D) such that f(z j ) = c j. It is known that if z 0 D, then the sequence of iterates {φ n (z 0 )}, n N, is an interpolating sequence for H (D). Function spaces methods. In particular fundamental function of Köthe function space X t ϕ(t) = χ [0,t], t {µ(a) : A Σ}. Interpolation of operators. For all p 0, p 1 (1, ), p 0 p 1, q 0, q 1, r [1, ], s (0, 1), where 1/p s = (1 s)/p 0 + s/p 1. (H p0,q0, H p1,q1 ) s,r = H ps,r, Results 21/21

22 Proofs ingredients interpolation and interpolation Complex functions theory. A sequence {z j } D is called interpolating for H (D) if for any bounded sequence of complex numbers {c j }, there is a function f H (D) such that f(z j ) = c j. It is known that if z 0 D, then the sequence of iterates {φ n (z 0 )}, n N, is an interpolating sequence Thank for Hyou (D). for your attention! Function spaces methods. Thank In you particular for yourfundamental attention! function of Köthe function space X t ϕ(t) = χ [0,t], t {µ(a) : A Σ}. Interpolation of operators. For all p 0, p 1 (1, ), p 0 p 1, q 0, q 1, r [1, ], s (0, 1), where 1/p s = (1 s)/p 0 + s/p 1. (H p0,q0, H p1,q1 ) s,r = H ps,r, Results 21/21