Composition operators: the essential norm and norm-attaining

Composition operators: the essential norm and norm-attaining Mikael Lindström Department of Mathematical Sciences University of Oulu Valencia, April, 2011

The purpose of this talk is to first discuss the essential norm of composition operators acting between standard weighted Bergman spaces. In particular, I give an elementary proof for characterizing compactness of such operators. Moreover, I will discuss norm-attaining weighted composition operators on weighted Banach spaces of holomorphic functions, these spaces are sometimes called weighted Bergman spaces of infinite order. 2

We denote by H(D) the space of holomorphic functions on the open unit disk D in the complex plane. For 1 p < and α > 1, the standard weighted Bergman space A p α := A p α(d) is the set of all analytic functions on D such that f p α,p = (α + 1) f(z) p (1 z 2 ) α da(z) <, D where da(z) = π 1 dxdy is the normalized Lebesgue area measure in D. For 1 p <, this defines a norm, and hence A p α is a Banach space. From now on, let da α (z) denote the normalized area measure (α + 1)(1 z 2 ) α da(z). 3

The weighted Banach spaces of holomorphic functions H v := {f H(D); f v = sup z D and on the smaller spaces v(z) f(z) < } H 0 v := {f H v ; lim v(z) f(z) = 0} z 1 endowed with norm. v, where the weight v : D R is bounded, continuous and strictly positive function. These weighted Banach spaces of analytic functions have been studied intensively by several people during the last four decades. Especially I want to mention Bierstedt, Bonet, Lusky, Shields and Williams. 4

Let ϕ : D D be an analytic mapping and ψ H(D). Each such pair (ϕ, ψ) induces via composition and multiplication a linear weighted composition operator C ϕ,ψ (f) = ψ(f ϕ) which preserves H(D). If ψ = 1, then C ϕ,1 = C ϕ is the usual composition operator. Operators of this type have been studied on various spaces of analytic functions. For a discussion of composition operators on classical spaces of analytic functions we refer to the excellent monographs of Shapiro and Cowen-MacCluer. 5

The reason why I am interested in studying the essential norm of composition operators between standard weighted Bergman spaces is that different estimates of the essential norm of a composition operator have been obtained as you soon will see. Recall that the essential norm T e of a bounded operator T L(X, Y ) is the distance in the operator norm from T to the compact operators, that is, T e = inf{ T K : K is compact}. Thus the essential norm of T equals zero if and only if T is compact. 6

It is known that the following quantities are comparable for C ϕ : A p α A p β, p > 1, α, β > 1: ( )1 Nϕ,2+β (z) p C ϕ e lim sup (1 z 2 ) 2+α (Shapiro, 1987) lim sup z 1 z 1 ( D (Čučković and Zhao, 2007), (1 z 2 ) α+2 )1 p 1 zϕ(w) 2(α+2) da β(w) where N ϕ,2+α is the generalized Nevanlinna counting function defined by N ϕ,2+α (x) = z ϕ 1 (x) (log(1/ z ))2+α, x D \ {ϕ(0)}. 7

The result Čučković and Zhao is based on the Carleson measure theorem in Bergman spaces and Shapiro uses a technique involving the generalized Nevanlinna counting function to obtain his result. This means that both proofs are using advanced tools. These two estimates of the essential norm of a composition operator are in my opinion neat results, but in practice it is still pretty hard to evaluate the above mention integral as well to evaluate the expression involving the generalized Nevanlinna counting function. So both methods used are somewhat complex! 8

Boundedness and the essential norm of composition operators We now state our first main result. Theorem 1. Let p > 1 and α, β > 1. Assume that ϕ : D D is an analytic function such that sup z D (1 z 2 ) β+1 <. (1 ϕ(z) 2 ) α+1 Then the operator C ϕ : A p α A p β is bounded and lim sup z 1 (1 z 2 ) β+2 p (1 ϕ(z) 2 ) α+2 p C ϕ e lim sup z 1 (1 z 2 ) β+1 p (1 ϕ(z) 2 ) α+1 p. 9

If one uses the method on pages 140-141 in the book of Cowen-MacCluer, then the result of Shapiro involving the generalized Nevanlinna counting function gives our above mentioned result. Furthermore, when ϕ is univalent, it reduces to the formula C ϕ p e lim sup z 1 (1 z 2 ) β+2 (1 ϕ(z) 2 ) α+2. On the contrary, our argument is straightforward and only based on Littlewood s subordination principle. Indeed, as a direct consequence of Littlewood s subordination principle we get: Lemma 1. Let p > 0, 0 < r < 1 and f H(D). Assume that ϕ is an analytic selfmap of D such that ϕ(0) = 0. Suppose that for some ρ (0, 1) holds ϕ(re iθ ) ρ for every 0 θ < 2π. Then 1 2π 2π 0 (f ϕ)(re iθ ) p dθ 1 2π 10 2π 0 f(ρe iθ ) p dθ.

Theorem 2. Let p > 1 and η > 0, λ > 1. Assume that ϕ : D D is analytic such that ϕ(0) = 0 and sup z D 1 z 2 (1 ϕ(z) 2 ) η <. Then the operator C ϕ : A p (λ+1)η 1 Ap λ ( C ϕ p e lim sup z 1 is bounded and ) λ+1 1 z 2. (1 ϕ(z) 2 ) η Proof. We only consider the statement concerning the essential norm. 11

Fix 0 < δ < 1 and set M δ := sup z >δ 1 z 2 (1 ϕ(z) 2 ) η > 0. Define the function ρ : [δ, 1] [0, 1] by ρ(r) = 1 1 (1 r 2 ) η 1 M 1 η δ By the definition of M δ, ϕ(re iθ ) ρ(r) for every 0 θ < 2π and δ r < 1. Let D δ = {z D : z < δ} and γ = (λ + 1)η 1. It is easy to see that we have the following estimate C ϕ p e lim sup f ϕ(z) p da λ. δ 1 f γ,p 1 D\D δ 1 2. 12

Let f A p γ, f γ,p 1 be arbitrary. Applying our above lemma and Fubini s theorem, we get f ϕ(z) p da λ = λ + 1 1 2π (f ϕ)(re iθ ) p dθr(1 r 2 ) λ dr D\D δ π δ 0 λ + 1 1 2π f(ρ(r)e iθ ) p dθr(1 r 2 ) λ dr π δ 0 = λ + 1 2π 1 f(ρ(r)e iθ ) p r(1 r 2 ) λ drdθ. π 0 δ Making the substitution r := ρ(r), we get r dr = ηm δ (1 r 2 ) η 1 r dr, so the previous inequality yields 13

D\D δ f ϕ(z) p da λ λ + 1 π = M λ+1 δ 2π 1 0 ρ(δ) f(r e iθ ) p ηm δ (1 r 2 ) η 1 r M λ δ (1 r 2 ) λη dr dθ D\D ρ(δ) f(z) p da γ. Now we conclude from the above inequalities that C ϕ p e lim M λ+1 δ sup δ 1 f γ,p 1 ( = lim sup z 1 (1 z 2 ) (1 ϕ(z) 2 ) η f(z) p da γ D\D ρ(δ) ) λ+1. 14

Put α > 1, λ = β > 1 and η = α+1 theorem, if then C ϕ : A p α A p β sup z D 1 z 2 β+1 (1 ϕ(z) 2 ) α+1 β+1 is bounded and C ϕ p e lim sup z 1. According to the previous <, (1 z 2 ) β+1 (1 ϕ(z) 2 ) α+1, which means that the upper estimate in our first main result is proved. 15

The following example shows that the finiteness of the supremum in the assumption of our main result is not equivalent with the boundedness of the composition operator. Example 1. Let α > β > 1, p > 1 and ϕ(z) = 1 (1 z) b, where b = (β + 2)/(α + 2). It is easy to show using the Carleson measure theorem that the operator C ϕ maps A p α into Ap β non-compactly. However, sup z D (1 z 2 ) β+1 =. (1 ϕ(z) 2 ) α+1 Note that in the case α < β, the limit supremums in our main result are vanish, since for any analytic selfmap ϕ of D, sup z D 1 z 2 1 ϕ(z) 2 <. 16

In a recent paper Ueki has presented an argument that is claimed to show that the essential norm of a bounded composition operator C ϕ : A 2 α A 2 β, β α, is equivalent to the lower bound obtained in our main result for p = 2. But this argument seems to be in error. More precisely, if C ϕ : A 2 α A 2 β is bounded, then A2 β A2 2(β+1) so C ϕ can also be considered as a bounded operator from A 2 α into A 2 2(β+1). Now in Ueki s paper, it is stated that the adjoint operators of these two composition operators coincide when restricted to functions from the unit ball of A 2 β. These adjoint operators depend on the duality, and they are explicitly given by and (C ϕ ) f(z) = (1 + β) (C ϕ ) t f(z) = (3 + 2β) respectively. D D f(w)(1 z 2 ) β (1 zϕ(w)) α+2da(w), f A2 β, f(w)(1 z 2 ) 2(β+1) (1 zϕ(w)) α+2 da(w), f A 2 2(β+1), 17

Now, for example, if we pick the normalized kernel functions then it is easily verified that kz β (w) := (1 z 2 ) 2+β 2 (1 zw) 2+β A2 β, lim z 1 C ϕ(kz β ) 2 (1 z 2 ) β+2 α,2 = lim z 1 (1 ϕ(z) 2 ) α+2, which give the lower bound of the essential norm of C ϕ : A 2 α A 2 β, but on the other hand, lim z 1 C t ϕ(k β z ) 2 α,2 = 0. 18

Norm-attaining weighted composition operators The question of norm-attaining composition operators was first explicitely studied by Hammond (2003) in the setting of the Hardy space H 2. In 2005 Harmond also studied norm-attaining composition operators on the classical Dirichlet space for a certain special class of self-maps. Motivated by this, Martín (2009) characterized norm-attaining composition operators acting on the classical Bloch space B as well as on the little Bloch space B 0. I will now discuss how her results can be generalized to the setting of weighted composition operators acting on weighted Banach spaces of analytic functions. 19

Recall that a bounded linear operator T on a Banach space X attains its norm on X if there exists a function f X with norm 1 such that T = T f. We say that a function f with these properties is an extremal function for the norm of T. James proved that a Banach space X is reflexive if and only if every compact operator on X is norm-attaining. The so called associated weight is an important tool to handle problems in the setting of weighted Banach spaces of analytic functions. For a weight v the associated weight ṽ is defined by ṽ(z) := (sup{ f(z) ; f H v, f v 1}) 1 = ( δ z v ) 1, z D, where δ z denotes the point evaluation of z. 20

The classical Bloch space B = {f H (D) : f(0) = 0, f B = sup(1 z 2 ) f (z) < }, z D and the little Bloch space B 0 = {f B : lim z 1 (1 z 2 ) f (z) = 0} are Banach spaces endowed with the norm B. For a bounded weighted composition operator C φ,ψ : H v H w, the norm of C φ,ψ is given by C φ,ψ H v H w = sup z D w(z) ψ(z), ṽ(φ(z)) and, if v is a typical weight, then it is known the essential norm of such an operator is given by C φ,ψ e,h v H w = lim r 1 21 sup φ(z) >r w(z) ψ(z). ṽ(φ(z))

The following result can be obtained: Theorem 3. If the weighted composition operator C φ,ψ : B B is bounded, then C φ,ψ B B max { C φ,ψ H (log 1 1 z 2 ) 1 H 1 z 2, C φ,φ ψ H 1 z 2 H 1 z 2 }. Martín s result that every (bounded) composition operator C φ on the Bloch space is norm-attaining is based on the following result due to Gorkin-Mortini (2004): Lemma 2. Let (z n ) D be a sequence such that z n 1, when n. Then there exists an infinite Blaschke product B whose zeros are elements of the sequence (z n ) and such that B (z n ) (1 z n 2 ) 1, when n. 22

Our second main result is the following: Theorem 4. Every bounded weighted composition operator C φ,ψ : Hv Hw is norm-attaining. For the proof we need the following result. Lemma 3. Let (z m ) D be a sequence such that z m 1, when m. Then there is a subsequence (z n ) of (z m ) and a function g H v, g v 1, such that g(z n ) ṽ(z n ) 1, when n. In particular, there is a h B, h B 1, with h (z n ) (1 z n 2 ) 1, when n. For the second statement of the above lemma we apply the first result with the weight v 1 (z) = 1 z 2. Consider now the bounded operators S : B Hv 1, S(h) = h and S 1 : Hv 1 B, (S 1 h)(z) = z 0 h(ξ)dξ. Clearly S, S 1 are isometric onto maps. 23

Then, let h := S 1 (g) B, so h B = g v1 1 and h = g. Consider a composition operator C φ : B B. Then we can use Theorem 4 to find an extremal function g Hv 1 for the norm of C φ,φ : Hv 1 Hv 1, and therefore C φ B B = C φ,φ H v1 H v 1 = SC φ S 1 H v1 H v 1 = (SC φ S 1 )g v1. Now using that S is an isometry, it follows that h := S 1 (g) B is an extremal function for the norm of C φ : B B. Thus we obtain Martin s Theorem: Corollary 1. Every composition operator C φ : B B is norm-attaining. 24

The following result is a consequence of the above theorems 3 and 4. Corollary 2. For every bounded weighted composition operator C φ,ψ : B B, there are norm-one functions f H (log 1 1 z 2) 1, g H 1 z 2 such that C φ,ψ B B max { C φ,ψ f 1 z 2, C φ,φ ψg 1 z 2}. 25

Finally let me mention the corresponding results for the small spaces H 0 v : Theorem 5. The bounded operator C φ,ψ : H 0 v H 0 w is norm-attaining if and only if we can find a point b D and a sequence (z n ) n D such that lim n φ(z n ) = b and sup z D ψ(z) w(z) ṽ(φ(z)) ψ(z n ) w(z n ) = lim. n ṽ(φ(z n )) Moreover, if the bounded operator C φ,ψ is norm-attaining on Hv 0, then the function f Hv 0 is norm-attaining if and only if there is a point b as in the first part and such that additionally the condition f(b)ṽ(b) = 1 is satisfied. Corollary 3. Let v be a typical weight and C φ,ψ be a bounded operator on Hv 0. If C φ,ψ e,h 0 v Hv 0 < C φ,ψ H 0 v Hv 0 is norm-attaining., then the operator 26

The next is the proof of Lemma 1: Proof. From the assumption and the maximum modulus principle of analytic functions we deduce, that ϕ maps the disk D r into the disk D ρ. By a classical theorem of complex analysis, there exist a function H : D ρ R harmonic in D ρ such that H(ρe iθ ) = f(ρe iθ ) p for every 0 θ < 2π. By the assumption, f p is subharmonic in D, so f p H in D ρ. We also assumed that ϕ is analytic and ϕ(d r ) D ρ. Hence the function H ϕ is harmonic in D r and f ϕ p H ϕ in D r. Therefore 1 2π 2π 0 2π (f ϕ)(re iθ ) p dθ 1 (H ϕ)(re iθ ) dθ = (H ϕ)(0) 2π 0 = H(0) = 1 2π H(ρe iθ ) dθ = 1 2π f(ρe iθ ) p dθ 2π 2π by the meanvalue property of harmonic functions. 0 27 0