Weighted Composition Operators on the Dirichlet Space: Boundedness and Spectral Properties

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 ϕ functions deﬁned on X such that ϕ(X ) ⊂ X , we may consider the linear map Introduction

Given h and ϕ functions deﬁned on X such that ϕ(X ) ⊂ X , we may consider the linear map

Wh,ϕ : f → h (f ◦ ϕ)

Wh,ϕf (x) = h(x) f (ϕ(x)), (x ∈ X ) Introduction

Given h and ϕ functions deﬁned on X such that ϕ(X ) ⊂ X , we may consider the linear map

Wh,ϕ : f → h (f ◦ ϕ)

Wh,ϕf (x) = h(x) f (ϕ(x)), (x ∈ 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 (ϕ(t)) where |h(t)| = 1 and ϕ 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 (ϕ(t)) where |h(t)| = 1 and ϕ is a homeomorphism of X onto itself. That is, T is the weighted composition operator Wh,ϕ. Introduction

1950, Bishop 2 2 Letr α ∈ (0, 1) be an irrational number and Tα : L [0, 1) → L [0, 1) deﬁned by

(Tαh)(x) = xh({x + α}); x ∈ [0; 1) where for any real number y the symbol {y} denotes the fractional part of y, Introduction

1950, Bishop 2 2 Letr α ∈ (0, 1) be an irrational number and Tα : L [0, 1) → L [0, 1) deﬁned by

(Tαh)(x) = xh({x + α}); x ∈ [0; 1) where for any real number y the symbol {y} denotes the fractional part of y,namely write y = n + s with n ∈ Z, s ∈ [0; 1) and set {y} := s. Introduction

1950, Bishop 2 2 Letr α ∈ (0, 1) be an irrational number and Tα : L [0, 1) → L [0, 1) deﬁned by

(Tαh)(x) = xh({x + α}); x ∈ [0; 1) where for any real number y the symbol {y} denotes the fractional part of y,namely write y = n + s with n ∈ Z, s ∈ [0; 1) and set {y} := s.

Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem. Introduction

1950, Bishop 2 2 Letr α ∈ (0, 1) be an irrational number and Tα : L [0, 1) → L [0, 1) deﬁned by

(Tαh)(x) = xh({x + α}); x ∈ [0; 1) where for any real number y the symbol {y} denotes the fractional part of y,namely write y = n + s with n ∈ Z, s ∈ [0; 1) and set {y} := s.

Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem.

1974, Davie. Tα has non-trivial invariant subspaces for almost every α ∈ [0, 1). Introduction

1950, Bishop 2 2 Letr α ∈ (0, 1) be an irrational number and Tα : L [0, 1) → L [0, 1) deﬁned by

(Tαh)(x) = xh({x + α}); x ∈ [0; 1) where for any real number y the symbol {y} denotes the fractional part of y,namely write y = n + s with n ∈ Z, s ∈ [0; 1) and set {y} := s.

Conjecture: Tα are candidates as counterexamples to the Invariant Subspace Problem.

1974, Davie. Tα has non-trivial invariant subspaces for almost every α ∈ [0, 1).

Open question: Does Tα have non-trivial invariant subspaces for every α ∈ [0, 1)? Introduction

D = {z ∈ C : |z| < 1} r Given h and ϕ analytic functions in D such that ϕ(D) ⊂ D, we may consider the linear map

Wh,ϕ : f ∈ H(D) → h (f ◦ ϕ) ∈ 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 < ∞ and p 6= 2. Classical Hardy spaces Hp, with 1 ≤ p ≤ ∞, r Classical Hardy spaces Hp, with 1 ≤ p ≤ ∞, r p f ∈ H , 1 ≤ p < ∞ ⇐⇒ f ∈ H(D) and

Z 2π 1/p iθ p dθ kf kp = sup |f (re )| < ∞. 0≤r<1 0 2π Classical Hardy spaces Hp, with 1 ≤ p ≤ ∞, r p f ∈ H , 1 ≤ p < ∞ ⇐⇒ f ∈ H(D) and

Z 2π 1/p iθ p dθ kf kp = sup |f (re )| < ∞. 0≤r<1 0 2π

∞ f ∈ H ⇐⇒ f is a bounded analytic function on D Boundedness of weighted composition operators

Given h and ϕ analytic functions in D such that ϕ(D) ⊂ D, we may consider the linear map

p Wh,ϕ : f ∈ H → h (f ◦ ϕ)

Weighted composition operator

p • Question: When does Wh,ϕ take H boundedly into itself?

• A necessary condition: Boundedness of weighted composition operators

Given h and ϕ analytic functions in D such that ϕ(D) ⊂ D, we may consider the linear map

p Wh,ϕ : f ∈ H → h (f ◦ ϕ)

Weighted composition operator

p • Question: When does Wh,ϕ take H boundedly into itself?

• A necessary condition: Boundedness of weighted composition operators

Given h and ϕ analytic functions in D such that ϕ(D) ⊂ D, we may consider the linear map

p Wh,ϕ : f ∈ H → h (f ◦ ϕ)

Weighted composition operator

p • Question: When does Wh,ϕ take H boundedly into itself?

• A necessary condition: h ∈ Hp. Boundedness of weighted composition operators on Hardy spaces

• Suppose that h ∈ H∞. Boundedness of weighted composition operators on Hardy spaces

∞ • Suppose that h ∈ H . Then, Wh,ϕ = Mh Cϕ. Boundedness of weighted composition operators on Hardy spaces

∞ • Suppose that h ∈ H . Then, Wh,ϕ = Mh Cϕ.

1925, Littlewood Subordination Principle Boundedness of weighted composition operators on Hardy spaces

∞ • Suppose that h ∈ H . Then, Wh,ϕ = Mh Cϕ.

1925, Littlewood Subordination Principle

p If ϕ ∈ H(D) such that ϕ(D) ⊂ D, then Cϕ 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,ϕ in terms of Carleson measures. Boundedness of weighted composition operators on Hardy spaces

2003, Contreras and Hern´andez-D´ıaz

r p Wh,ϕ is bounded in H ⇔ µh,ϕ is a Carleson measure on D, where Z p µh,ϕ(E) = |h| dm, ϕ−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,ϕ is bounded in H ⇔

(1 − |w|2)1/2h

sup < ∞. |w|<1 1 − wϕ 2 Boundedness of weighted composition operators on Hardy spaces

2006, Harper r 2 Wh,ϕ is bounded in H ⇔

sup kWh,ϕkw k2 < ∞, |w|<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,ϕ is bounded in H ⇔

sup kWh,ϕkw k2 < ∞, |w|<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(ϕ) denote the de Branges–Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w

1 − ϕ(w)ϕ(z) kϕ(z) = . w 1 − wz Boundedness of weighted composition operators on Hardy spaces

2007, Jury Letr H(ϕ) denote the de Branges–Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w

1 − ϕ(w)ϕ(z) kϕ(z) = . w 1 − wz

Theorem. Let ϕ be an analytic self-map of D and h ∈ H(ϕ). 2 Thenr Wh,ϕ is bounded in H and

kWh,ϕk2 ≤ khkH(ϕ). Boundedness of weighted composition operators on Hardy spaces

2007, Jury Letr H(ϕ) denote the de Branges–Rovnyak space, that is, the reproducing kernel Hilbert space on D with reproducing kernel at w

1 − ϕ(w)ϕ(z) kϕ(z) = . w 1 − wz

Theorem. Let ϕ be an analytic self-map of D and h ∈ H(ϕ). 2 Thenr Wh,ϕ is bounded in H and

kWh,ϕk2 ≤ khkH(ϕ).

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 ∈ H , then f = BSσF where • B is a Blaschke product. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product.

Given a sequence of (not necessarily distinct) points {zk } in D \{0} satisfying the Blaschke condition

∞ X (1 − |zk |) < ∞, k=1 the inﬁnite product

∞ Y |zk | 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 {zk }. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product. A general expression for a Blaschke product is given by

∞ Y |zk | zk − z eiθ zN zk 1 − zk z k=1 where N ≥ 0 is an integer. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product.

• Sσ is a singular inner function. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product.

• Sσ is a singular inner function.

Z 2π eiθ + z Sσ(z) = exp − iθ dσ(θ) 0 e − z where σ is a positive singular measure in ∂D. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product.

• Sσ is a singular inner function. • F is an outer function. “Inner-Outer” factorization of Hardy functions. r p Let f ∈ H , then f = BSσF where • B is a Blaschke product.

• Sσ is a singular inner function. • F is an outer function.

Z 2π iθ 1 e + z iθ F (z) = λ exp iθ log |f (e )| dθ . 2π 0 e − z Boundedness of weighted composition operators on Hardy spaces

2010, Kumar, GG and Partington r Boundedness of weighted composition operators on Hardy spaces

2010, Kumar, GG and Partington Ifr ϕ is inner, then

2 2 H(ϕ) = Kϕ := H ϕH ,

and

PKϕ h = ϕP−(ϕh), 2 2 where P− is the orthogonal projection onto L H . Boundedness of weighted composition operators on Hardy spaces

2010, Kumar, GG and Partington Ifr ϕ is inner, then 2 2 H(ϕ) = Kϕ := H ϕH , and

PKϕ h = ϕP−(ϕh), 2 2 where P− is the orthogonal projection onto L H . In this case, if h ∈ Kϕ, 2 ∞ ∞ X n 2 X 2 anϕ h = khk2 |an| , n=0 2 n=0 since ( 0 for n 6= m, hϕnh, ϕmhi = 2 khk2 for n = m, Boundedness of weighted composition operators on Hardy spaces

2010, Kumar, GG and Partington r So, if ϕ is inner and h ∈ Kϕ

kWh,ϕf k2 = khk2kf k2. Boundedness of weighted composition operators on Hardy spaces

2010, Kumar, GG and Partington r So, if ϕ is inner and h ∈ Kϕ

kWh,ϕf k2 = khk2kf k2.

Thus the condition in Jury’s Theorem holds for h ∈ Kϕ, although not in general. Indeed, if ϕ(z) = z, then kWh,ϕk = kThk = khk∞. Boundedness of weighted composition operators on Hardy spaces

Deﬁnition. For ϕ : D → D analytic, the multiplier space of ϕ is deﬁnedr by

2 M(ϕ) = {h ∈ H : Wh,ϕ := ThCϕ is bounded}. Boundedness of weighted composition operators on Hardy spaces

Deﬁnition. For ϕ : D → D analytic, the multiplier space of ϕ is deﬁnedr by

2 M(ϕ) = {h ∈ H : Wh,ϕ := ThCϕ is bounded}.

Remark. H∞ ⊆ M(ϕ) ⊆ H2 for all analytic self-maps ϕ of the unitr disc. Boundedness of weighted composition operators on Hardy spaces

Deﬁnition. For ϕ : D → D analytic, the multiplier space of ϕ is deﬁnedr by

2 M(ϕ) = {h ∈ H : Wh,ϕ := ThCϕ is bounded}.

Remark. H∞ ⊆ M(ϕ) ⊆ H2 for all analytic self-maps ϕ of the unitr disc. It is easily veriﬁed that M(ϕ) is a Banach space with the norm khkM(ϕ) = kWh,ϕk. Boundedness of weighted composition operators on Hardy spaces

Deﬁnition. For ϕ : D → D analytic, the multiplier space of ϕ is deﬁnedr by

2 M(ϕ) = {h ∈ H : Wh,ϕ := ThCϕ is bounded}.

Remark. H∞ ⊆ M(ϕ) ⊆ H2 for all analytic self-maps ϕ of the unitr disc. It is easily veriﬁed that M(ϕ) is a Banach space with the norm khkM(ϕ) = kWh,ϕk.

Question. Determine M(ϕ). r Boundedness of weighted composition operators on Hardy spaces

Theorem (2010, Kumar, GG, Partington) M(ϕ) = H2 if and

onlyr if kϕk∞ < 1. Boundedness of weighted composition operators on Hardy spaces

Theorem (2010, Kumar, GG, Partington) M(ϕ) = H2 if and

onlyr if kϕk∞ < 1.

Theorem (2003, Contreras and Hernández-D´ıaz,2008 Matache) ∞ Mr (ϕ) = H if and only if ϕ is a finite Blaschke product. Angular derivative r Angular derivative r Let ϕ be an analytic self-map with ϕ(D) ⊂ D and α ∈ ∂D. Angular derivative r Let ϕ be an analytic self-map with ϕ(D) ⊂ D and α ∈ ∂D. ϕ has (finite) angular derivative at α, denoted by ϕ0(α), whenever the non-tangential limit

ϕ(z) − η ∠ l´ım (z ∈ D), z→α z − α exists and is ﬁnite for some η ∈ ∂D. Julia-Carath´eodory Theorem: r Let ϕ be an analytic self-map of D with ϕ(D) ⊂ D and α ∈ ∂D. The following conditions are equivalent

1 ϕ has finite angular derivative at α. 2 Both radial limits ϕ(α) and ϕ0(α) exist and are finite. 1−|ϕ(z)| 3 l´ıminfz→α 1−|z| < ∞, where the l´ıminf is calculated as z approaches α within D. Julia-Carathéodory Theorem: r Let ϕ be an analytic self-map of D with ϕ(D) ⊂ D and α ∈ ∂D. The following conditions are equivalent

1 ϕ has ﬁnite angular derivative at α. 2 Both radial limits ϕ(α) and ϕ0(α) exist and are ﬁnite. 1−|ϕ(z)| 3 l´ıminfz→α 1−|z| < ∞, where the l´ıminf is calculated as z approaches α within D. Moreover, under the above conditions it holds that ϕ(α) = η,

1 − |ϕ(z)| |ϕ0(α)| = l´ıminf z→α 1 − |z| Theorem (2010, Kumar, GG, Partington) Let ϕ be an analytic self-mapr of D. Let

Eϕ = {ζ ∈ T : ϕ has ﬁnite angular derivative at ζ}.

p If Wh,ϕ is bounded on H for some 1 ≤ p < ∞, then h is pointwise bounded on every Stolz domain whose vertex is a point of Eϕ. Theorem (2010, Kumar, GG, Partington) Let ϕ be an analytic self-mapr of D. Let

Eϕ = {ζ ∈ T : ϕ has ﬁnite angular derivative at ζ}.

p If Wh,ϕ is bounded on H for some 1 ≤ p < ∞, then h is pointwise bounded on every Stolz domain whose vertex is a point of Eϕ.

Remark. The converse does not hold. r Theorem (2010, Kumar, GG, Partington) Let ϕ be an analytic self-mapr of D. Let

Eϕ = {ζ ∈ T : ϕ has ﬁnite angular derivative at ζ}.

p If Wh,ϕ is bounded on H for some 1 ≤ p < ∞, then h is pointwise bounded on every Stolz domain whose vertex is a point of Eϕ.

Remark. The converse does not hold. r Corollary Let ϕ be an inner function . Then any function hr ∈ M(ϕ) is essentially bounded on all relatively compact subsets of T \ σ(ϕ), where σ(ϕ) denotes the spectrum of ϕ, namely, σ(ϕ) = {an}n ∪ supp µ. Boundedness of weighted composition operators

The Dirichlet space D Boundedness of weighted composition operators

The Dirichlet space D f ∈ D ⇐⇒ f ∈ H(D) and Z 2 2 0 2 kf kD = |f (0)| + |f (z)| dA(z) D Boundedness of weighted composition operators

The Dirichlet space D

Wh,ϕ : f ∈ D → h (f ◦ ϕ)

Weighted composition operator

• Question: When does Wh,ϕ take D boundedly into itself? Boundedness of weighted composition operators

The Dirichlet space D

Wh,ϕ : f ∈ D → h (f ◦ ϕ)

Weighted composition operator

• Question: When does Wh,ϕ take D boundedly into itself? Boundedness of weighted composition operators on the Dirichlet space

Remark. Not every composition operator Cϕ takes D into itself! r Boundedness of weighted composition operators on the Dirichlet space

Remark. Not every composition operator Cϕ takes D into itself! r 1980, C. Voas r Cϕ : D → D f → f ◦ ϕ Boundedness of weighted composition operators on the Dirichlet space

Remark. Not every composition operator Cϕ takes D into itself! r 1980, C. Voas r Cϕ : D → D f → f ◦ ϕ

nϕ(w) ≡ multiplicity of ϕ at w

S(ξ, δ) = {z ∈ D : |z − ξ| < δ} the Carleson disk centered at ξ ∈ ∂D of radius 0 < δ < 1.

Z 2 Cϕ is bounded on D ⇐⇒ nϕ(w)dA(w) ∼ O (δ ). S(ξ,δ) Boundedness of weighted composition operators on the Dirichlet space

It is clear that if Cϕ is a bounded operator on D and u is a multiplier of D, that is, the Toeplitz operator Tu : f 7→ uf is deﬁned everywhere on D and hence bounded, the weighted composition operator Wu,ϕ on D is obviously bounded. Multipliers of D. r . Multipliers of D. Not easy to describe. r . Multipliers of D. Not easy to describe. r 1980, Stegenga Characterization in terms of a condition involvingr the logarithmic capacity of their boundary values. Multipliers of D. Not easy to describe. r 1980, Stegenga Characterization in terms of a condition involvingr the logarithmic capacity of their boundary values. In particular, the strict inclusion

M(D) ⊂ D ∩ H∞ holds. Multipliers of D. Not easy to describe. r 1980, Stegenga Characterization in terms of a condition involvingr the logarithmic capacity of their boundary values. In particular, the strict inclusion

M(D) ⊂ D ∩ H∞ holds. 1999, Wu An equivalent condition in terms of Carleson measuresr for D (that is, there is a continuous injection from D 2 into L (D, µ)),

u ∈ M(D) ⇐⇒ u ∈ H∞ and dµ(z) = |u0(z)|2 dA(z) is a Carleson measure for D. Boundedness of weighted composition operators on the Dirichlet space

An example Boundedness of weighted composition operators on the Dirichlet space

An example Let u(z) = (1 − z)2 and let ϕ be the inﬁnite Blaschke product with zeroes

2 (1 − 1/n )n≥1. Boundedness of weighted composition operators on the Dirichlet space

An example Let u(z) = (1 − z)2 and let ϕ be the inﬁnite Blaschke product with zeroes

2 (1 − 1/n )n≥1.

Now ϕ 6∈ D, so Cϕ is clearly unbounded. Boundedness of weighted composition operators on the Dirichlet space

An example Let u(z) = (1 − z)2 and let ϕ be the inﬁnite Blaschke product with zeroes

2 (1 − 1/n )n≥1.

Now ϕ 6∈ D, so Cϕ is clearly unbounded. However, Wu,ϕ is bounded on D. Boundedness of weighted composition operators on the Dirichlet space

An example Let u(z) = (1 − z)2 and let ϕ be the inﬁnite Blaschke product with zeroes

2 (1 − 1/n )n≥1.

Now ϕ 6∈ D, so Cϕ is clearly unbounded. However, Wu,ϕ is bounded on D.

Conclusion One may construct self-maps of the unit disc ϕ such that ϕ 6∈ D and multipliers u ∈ M(D) such that Wu,ϕ is bounded in the Dirichlet space. Boundedness of weighted composition operators on the Dirichlet space

An example Let u(z) = (1 − z)2 and let ϕ be the inﬁnite Blaschke product with zeroes

2 (1 − 1/n )n≥1.

Now ϕ 6∈ D, so Cϕ is clearly unbounded. However, Wu,ϕ is bounded on D.

Conclusion One may construct self-maps of the unit disc ϕ such that ϕ 6∈ D and multipliers u ∈ M(D) such that Wu,ϕ is bounded in the Dirichlet space.Therefore, facing the problem of describing the weighted composition operators taking D boundedly into itself deals not only with the multipliers of D but also with those self-maps of the unit disc that may induce unbounded composition operators in D. Boundedness of weighted composition operators on the Dirichlet space