Weighted composition operators on the Dirichlet space: boundedness and spectral properties C´aceres,Marzo 2016 Weighted composition operators on the Dirichlet space: boundedness and spectral properties C´aceres,Marzo 2016 Joint work with Isabelle Chalendar (Lyon, France) and Jonathan R. Partington (Leeds, U.K.) Introduction Given h and ' functions defined on X such that '(X ) ⊂ X , we may consider the linear map Introduction Given h and ' functions defined on X such that '(X ) ⊂ X , we may consider the linear map Wh;' : f ! h (f ◦ ') Wh;'f (x) = h(x) f ('(x)); (x 2 X ) Introduction Given h and ' functions defined on X such that '(X ) ⊂ X , we may consider the linear map Wh;' : f ! h (f ◦ ') Wh;'f (x) = h(x) f ('(x)); (x 2 X ) Weighted composition operator Why studying weighted composition operators? Introduction 1930, Banach Ifr X is a compact metric space and T : C(X ) !C(X ) f ! Tf is a surjective linear isometry, then Tf (t) = h(t) f ('(t)) where jh(t)j = 1 and ' is a homeomorphism of X onto itself. Introduction 1930, Banach Ifr X is a compact metric space and T : C(X ) !C(X ) f ! Tf is a surjective linear isometry, then Tf (t) = h(t) f ('(t)) where jh(t)j = 1 and ' is a homeomorphism of X onto itself. That is, T is the weighted composition operator Wh;'. Introduction 1950, Bishop 2 2 Letr α 2 (0; 1) be an irrational number and Tα : L [0; 1) ! L [0; 1) defined by (Tαh)(x) = xh(fx + αg); x 2 [0; 1) where for any real number y the symbol fyg denotes the fractional part of y, Introduction 1950, Bishop 2 2 Letr α 2 (0; 1) be an irrational number and Tα : L [0; 1) ! L [0; 1) defined by (Tαh)(x) = xh(fx + αg); x 2 [0; 1) where for any real number y the symbol fyg denotes the fractional part of y,namely write y = n + s with n 2 Z, s 2 [0; 1) and set fyg := s. Introduction 1950, Bishop 2 2 Letr α 2 (0; 1) be an irrational number and Tα : L [0; 1) ! L [0; 1) defined by (Tαh)(x) = xh(fx + αg); x 2 [0; 1) where for any real number y the symbol fyg denotes the fractional part of y,namely write y = n + s with n 2 Z, s 2 [0; 1) and set fyg := s. Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem. Introduction 1950, Bishop 2 2 Letr α 2 (0; 1) be an irrational number and Tα : L [0; 1) ! L [0; 1) defined by (Tαh)(x) = xh(fx + αg); x 2 [0; 1) where for any real number y the symbol fyg denotes the fractional part of y,namely write y = n + s with n 2 Z, s 2 [0; 1) and set fyg := s. Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem. 1974, Davie. Tα has non-trivial invariant subspaces for almost every α 2 [0; 1). Introduction 1950, Bishop 2 2 Letr α 2 (0; 1) be an irrational number and Tα : L [0; 1) ! L [0; 1) defined by (Tαh)(x) = xh(fx + αg); x 2 [0; 1) where for any real number y the symbol fyg denotes the fractional part of y,namely write y = n + s with n 2 Z, s 2 [0; 1) and set fyg := s. Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem. 1974, Davie. Tα has non-trivial invariant subspaces for almost every α 2 [0; 1). Open question: Does Tα have non-trivial invariant subspaces for every α 2 [0; 1)? Introduction D = fz 2 C : jzj < 1g r Given h and ' analytic functions in D such that '(D) ⊂ D, we may consider the linear map Wh;' : f 2 H(D) ! h (f ◦ ') 2 H(D) Weighted composition operator Introduction 1960, de Leeuw, Rudin and Wermer r Characterization of the isometries of H1. Introduction 1960, de Leeuw, Rudin and Wermer r Characterization of the isometries of H1. 1964, Forelli r Characterization of the isometries of Hp, 1 < p < 1 and p 6= 2. Classical Hardy spaces Hp, with 1 ≤ p ≤ 1, r Classical Hardy spaces Hp, with 1 ≤ p ≤ 1, r p f 2 H ; 1 ≤ p < 1 () f 2 H(D) and Z 2π 1=p iθ p dθ kf kp = sup jf (re )j < 1: 0≤r<1 0 2π Classical Hardy spaces Hp, with 1 ≤ p ≤ 1, r p f 2 H ; 1 ≤ p < 1 () f 2 H(D) and Z 2π 1=p iθ p dθ kf kp = sup jf (re )j < 1: 0≤r<1 0 2π 1 f 2 H () f is a bounded analytic function on D Boundedness of weighted composition operators Given h and ' analytic functions in D such that '(D) ⊂ D, we may consider the linear map p Wh;' : f 2 H ! h (f ◦ ') Weighted composition operator p • Question: When does Wh;' take H boundedly into itself? • A necessary condition: Boundedness of weighted composition operators Given h and ' analytic functions in D such that '(D) ⊂ D, we may consider the linear map p Wh;' : f 2 H ! h (f ◦ ') Weighted composition operator p • Question: When does Wh;' take H boundedly into itself? • A necessary condition: Boundedness of weighted composition operators Given h and ' analytic functions in D such that '(D) ⊂ D, we may consider the linear map p Wh;' : f 2 H ! h (f ◦ ') Weighted composition operator p • Question: When does Wh;' take H boundedly into itself? • A necessary condition: h 2 Hp. Boundedness of weighted composition operators on Hardy spaces • Suppose that h 2 H1. Boundedness of weighted composition operators on Hardy spaces 1 • Suppose that h 2 H . Then, Wh;' = Mh C'. Boundedness of weighted composition operators on Hardy spaces 1 • Suppose that h 2 H . Then, Wh;' = Mh C'. 1925, Littlewood Subordination Principle Boundedness of weighted composition operators on Hardy spaces 1 • Suppose that h 2 H . Then, Wh;' = Mh C'. 1925, Littlewood Subordination Principle p If ' 2 H(D) such that '(D) ⊂ D, then C' is bounded on H . ϕ R D D ϕ(D) 1 Boundedness of weighted composition operators on Hardy spaces 2003, Contreras and Hern´andez-D´ıaz r Necessary and sufficient condition for boundedness of Wh;' in terms of Carleson measures. Boundedness of weighted composition operators on Hardy spaces 2003, Contreras and Hern´andez-D´ıaz r p Wh;' is bounded in H , µh;' is a Carleson measure on D, where Z p µh;'(E) = jhj dm; '−1(E)\D for measurable subsets E ⊆ D. Boundedness of weighted composition operators on Hardy spaces 2006, Harper r Boundedness of weighted composition operators on Hardy spaces 2006, Harper r 2 Wh;' is bounded in H , (1 − jwj2)1=2h sup < 1: jwj<1 1 − w' 2 Boundedness of weighted composition operators on Hardy spaces 2006, Harper r 2 Wh;' is bounded in H , sup kWh;'kw k2 < 1; jwj<1 2 where kw is the normalized reproducing kernel at w in H . Boundedness of weighted composition operators on Hardy spaces 2006, Harper r 2 Wh;' is bounded in H , sup kWh;'kw k2 < 1; jwj<1 2 where kw is the normalized reproducing kernel at w in H . 2007, Cu˘ckovi´cand Zhao Generalizationsr to mappings between Hp and Hq spaces. Boundedness of weighted composition operators on Hardy spaces 2007, Jury Letr H(') denote the de Branges{Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w 1 − '(w)'(z) k'(z) = : w 1 − wz Boundedness of weighted composition operators on Hardy spaces 2007, Jury Letr H(') denote the de Branges{Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w 1 − '(w)'(z) k'(z) = : w 1 − wz Theorem. Let ' be an analytic self-map of D and h 2 H('). 2 Thenr Wh;' is bounded in H and kWh;'k2 ≤ khkH('): Boundedness of weighted composition operators on Hardy spaces 2007, Jury Letr H(') denote the de Branges{Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w 1 − '(w)'(z) k'(z) = : w 1 − wz Theorem. Let ' be an analytic self-map of D and h 2 H('). 2 Thenr Wh;' is bounded in H and kWh;'k2 ≤ khkH('): Remark. This is not a necessary condition for boundedness. r Boundedness of weighted composition operators on Hardy spaces 2010, Kumar, GG and Partington r \Inner-Outer" factorization of Hardy functions. r • B is a Blaschke product. \Inner-Outer" factorization of Hardy functions. r p Let f 2 H , then f = BSσF where • B is a Blaschke product. \Inner-Outer" factorization of Hardy functions. r p Let f 2 H , then f = BSσF where • B is a Blaschke product. \Inner-Outer" factorization of Hardy functions. r p Let f 2 H , then f = BSσF where • B is a Blaschke product. \Inner-Outer" factorization of Hardy functions. r p Let f 2 H , then f = BSσF where • B is a Blaschke product. Given a sequence of (not necessarily distinct) points fzk g in D n f0g satisfying the Blaschke condition 1 X (1 − jzk j) < 1; k=1 the infinite product 1 Y jzk j zk − z B(z) = ; zk 1 − zk z k=1 converges uniformly on compact subsets of D to a holomorphic function B called the Blaschke product with zero sequence fzk g. \Inner-Outer" factorization of Hardy functions. r p Let f 2 H , then f = BSσF where • B is a Blaschke product.