Volterra operators on Banach spaces of analytic functions om, ˚ Mikael Lindstr¨ Abo Akademi University mikael.lindstrom@abo.fi Poznan, July 2018 Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 1 / 35
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Outline Introduction 1 Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 8 / 35
Outline Introduction 1 Boundedness and compactness results for T g : H ∞ v α → H ∞ 2 Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 8 / 35
Outline Introduction 1 Boundedness and compactness results for T g : H ∞ v α → H ∞ 2 Boundedness, weak compactness and compactness of T ϕ 3 g Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 8 / 35
Introduction The main object of study in this talk is the generalized Volterra operator T ϕ g , defined for a fixed function g ∈ H ( D ) and selfmap ϕ : D → D as � ϕ ( z ) T ϕ f ( ξ ) g ′ ( ξ ) d ξ, g ( f )( z ) = z ∈ D , f ∈ H ( D ) . 0 In the special case when ϕ is the identity map ϕ ( z ) = z we get the Volterra operator � z f ( ξ ) g ′ ( ξ ) d ξ, T g ( f )( z ) = z ∈ D , f ∈ H ( D ) , 0 which has been extensively studied on various spaces of analytic functions during the past decades, starting from papers by Pommerenke, Aleman, Cima and Siskakis. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 9 / 35
Introduction The present talk is inspired by two recent works on boundedness and compactness properties of the Volterra operator T g mapping into the space H ∞ of bounded analytic functions on the unit disc. Namely, Smith, Stolyarov and Volberg obtained a very nice necessary and sufficient condition for T g to be bounded on H ∞ when g is univalent. The main purpose is to demonstrate that similar conditions characterize v α → H ∞ when 0 ≤ α < 1 and g boundedness and compactness of T g : H ∞ is univalent. In the other work, Contreras, Pel´ aez, Pommerenke and R¨ atty¨ a studied boundedness, compactness and weak compactness of T g : X → H ∞ acting on a Banach space X ⊂ H ( D ). We will discuss a similar study of the generalized Volterra operator T ϕ g mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 10 / 35
Introduction The weighted Banach spaces of analytic functions H ∞ and H 0 v are defined v by � � H ∞ = f ∈ H ( D ) : || f || H ∞ := sup z ∈ v ( z ) | f ( z ) | < + ∞ v v � � H 0 f ∈ H ∞ v = : | z |→ 1 − v ( z ) | f ( z ) | = 0 lim , v where the weight v : D → R is a continuous, strictly positive function, radial, non-increasing with respect to | z | , and lim | z |→ 1 v ( z ) = 0. Moreover, we will study the Bloch-type spaces B v and B 0 v defined by � � z ∈ v ( z ) | f ′ ( z ) | < + ∞ B v = f ∈ H ( D ) : || f || B v := | f (0) | + sup � � B 0 | z |→ 1 − v ( z ) | f ′ ( z ) | = 0 v = f ∈ B v : lim . Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 11 / 35
Introduction ≈ l ∞ and H 0 In 2006 Lusky has shown that H ∞ v ≈ c 0 for a large class of v weights including the normal weights, i.e. weights v for which v (1 − 2 − n − k ) v (1 − 2 − n ) inf k lim sup < 1 and sup v (1 − 2 − n − 1 ) < ∞ . v (1 − 2 − n ) n n Therefore H ∞ and H 0 v are nice spaces whenever v is a normal weight. v and ˜ By the differentiation operator D : f �→ f ′ , the spaces H ∞ B v as well v v and ˜ as their subspaces H 0 B 0 v are isometrically isomorphic. The map ( f , λ ) �→ f + λ gives that also ˜ B v ⊕ 1 C ≈ B v and ˜ B 0 v ⊕ 1 C ≈ B 0 v are isometrically isomorphic. Hence also B v and B 0 v behave nicely for normal weights. Weights of the type v α ( z ) := (1 − | z | 2 ) α with α > 0 are called standard weights, and they are clearly normal. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 12 / 35
Introduction Furthermore, the Hardy space H p for 1 ≤ p < ∞ consists of all functions f analytic in the unit disc such that � 2 π � f � p 1 | f ( re i θ ) | p d θ < ∞ , H p := sup 2 π 0 ≤ r < 1 0 and the weighted Bergman spaces for constants α > − 1 and 1 ≤ p < ∞ are given by � � � f ∈ H ( D ) : || f || p A p | f ( z ) | p (1 − | z | 2 ) α dA ( z ) < ∞ α = α := (1 + α ) , A p where dA ( z ) is the normalized area measure on D . Finally, the disc algebra, A ( D ) is the space of functions analytic on D that extend continuously to the boundary ∂ D . Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 13 / 35
Introduction The following result is useful when characterizing compactness and weak compactness of the generalized Volterra operator. Lemma (Contreras, Pel´ aez, Pommerenke, R¨ atty¨ a ) Let X ⊂ H ( D ) be a Banach space such that the closed unit ball B X is compact with respect to the compact open topology co and Y ⊂ H ( D ) be a Banach space such that point evaluation functionals on Y are bounded. Assume that T : X → Y is a co-co continuous linear operator. Then T : X → Y is compact (respectively weakly compact) if and only if { T ( f n ) } ∞ n =1 converges to zero in the norm (respectively in the weak topology) of Y for each bounded sequence { f n } ∞ n =1 in X such that f n → 0 uniformly on compact subsets of D . The above lemma can be applied to the generalized Volterra operator T ϕ g : X → Y , since it is obvious that T ϕ g : H ( D ) → H ( D ) always is co - co continuous. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 14 / 35
Boundedness and compactness results for T g v α → H ∞ induced by a Consider the classical Volterra operator T g : H ∞ univalent function g for standard weights v α with 0 ≤ α < 1. The study of the case when α = 0, that is when T g : H ∞ → H ∞ , was initiated by Anderson, Jovovic and Smith and they conjectured that T [ H ∞ ] = { g ∈ H ( D ) : T g : H ∞ → H ∞ is bounded } would coincide with the space of functions analytic in D with bounded radial variation � � � 1 | f ′ ( re i θ ) | dr < ∞ BRV = f ∈ H ( D ) : sup . 0 ≤ θ< 2 π 0 Recently Smith, Stolyarov and Volberg confirmed this conjecture when the inducing function g is univalent, that is T [ H ∞ ] ∩ { g ∈ H ( D ) : g is univalent } = BRV . Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 15 / 35
Boundedness and compactness results for T g On the other side, they also obtained a counterexample to the general conjecture posed by Anderson, Jovovic and Smith meaning that BRV � T [ H ∞ ]. a showed that T g : H ∞ v 1 → H ∞ Contreras, Pel´ aez, Pommerenke and R¨ atty¨ is bounded precisely when g is a constant function. Since the size of the spaces H ∞ v α increases as the power α grows, one concludes that the only v α → H ∞ that can be bounded when α ≥ 1 is the Volterra operator T g : H ∞ zero operator. Hence, we are left to consider the remaining cases 0 ≤ α < 1, and will also restrict to univalent symbols g . Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 16 / 35
Boundedness and compactness results for T g Anderson, Jovovic and Smith also discussed compactness of the Volterra operator T g : H ∞ → H ∞ , and suggested the space � � � 1 | f ′ ( re i θ ) | dr = 0 BRV 0 = f ∈ H ( D ) : lim sup t → 1 − 0 ≤ θ< 2 π t of functions analytic in the unit disc with derivative uniformly integrable on radii as a natural candidate for the set of such functions g . v α → H ∞ when Therefore we are interested in compactness of T g : H ∞ 0 ≤ α < 1 and g is univalent. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 17 / 35
Boundedness and compactness results for T g � � Ω r For β > 0 and r > 0, let B denote the class of all functions F , β analytic in the open sector � � � � � ≤ C F z ∈ C : 0 < | z | < r and − β 2 < arg( z ) < β � F ′ ( z ) Ω r β := , with 2 | z | for z ∈ Ω r β . Here C F is a constant only depending on β , r and the function � 1 � F . Below Ω β := Ω 1 β and � u is the harmonic conjugate of u with � u = 0. 2 Theorem (Smith, Stolyarov,Volberg) Let 0 < γ < β < π and ε > 0 . Then there is a number δ ( ε ) > 0 such that � � Ω 1 / 2 there exists a harmonic function u : Ω β → R with for each F ∈ B γ the properties (1) | Re ( F ( x )) − u ( x ) | ≤ ε , for x ∈ (0 , δ ( ε )] . (2) | � u ( z ) | ≤ C ( ε, γ, β, C F ) < ∞ , for z ∈ Ω β . The number δ ( ε ) is independent of F , whereas the constant C ( ε, γ, β, C F ) does depend on F but only through C F as defined above. Mikael Lindstr¨ om () Volterra operators Poznan, July 2018 18 / 35
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