GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 4, 1997, 373-383

WEIGHTED COMPOSITION OPERATORS ON BERGMAN AND DIRICHLET SPACES

G. MIRZAKARIMI AND K. SEDDIGHI

Abstract. Let H(Ω) denote a functional Hilbert space of analytic functions on a domain Ω. Let w : Ω ⟶ C and ϕ : Ω ⟶ Ω be such that wf ∘ ϕ is in H(Ω) for every f in H(Ω). The operator wC_{ϕ} given by f ⟶ wf ∘ϕ is called a weighted composition operator on H(Ω). In this paper we characterize such operators and those for which (wC_{ϕ})^{∗} is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.

1. Introduction

A Hilbert space H(Ω) of analytic functions on a domain Ω is called a functional Hilbert space provided the point evaluation f ⟶ f(x) is continuous for every x in Ω. The Hardy space H^{2} and the Bergman space L_{a}^{2}(D) are the well-known examples of functional Hilbert spaces. An application of the Riesz representation theorem shows that for every x ∈ Ω there is a vector k_{x} in H(Ω) such that f(x) =< f, k_{x} > for all f in H(Ω). Let K = {k_{x} : x ∈ Ω}. An operator T on H(Ω) is a composition operator if and only if K is invariant under T^{∗} [1]. In fact, T^{∗}k_{x} = k_{ϕ(x)}, where T = C_{ϕ}. It is a multiplication operator if and only if the elements of K are eigenvectors of T^{∗} [2]. In this case T^{∗}k_{x} = \overline{ψ(x)}k_{x}, where T = M_{ψ} is the operator of multiplication by ψ. An operator T on H(Ω) is a weighted composition operator if and only if T^{∗}K ⊂ K̃, where K̃ = {λk_{x}|λ ∈ C, x ∈ Ω}. In this case T^{∗}k_{x} = \overline{w(x)}k_{ϕ(x)}, where T = wC_{ϕ}.

We note that the Hardy space H^{2} can be identified as the space of func-373 tions f analytic in the open unit disc D such that

1991 Mathematics Subject Classification. 47B37, 47B38.

Key words and phrases. Weighted composition operator, compact, functional Hilbert space, Carleson measure, angular derivative, Dirichlet space.

1072-947X/97/0700-0373$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

∥f∥^{2} = sup_{0<r<1} ∫_{−π}^{π} |f(re^{iθ})|^{2}dθ < ∞.

Actually, if f ∈ H^{2} and f(z) = ∑ a_{n}z^{n} then ∥f∥^{2} = ∑ |a_{n}|^{2}. Moreover, if f ∈ H^{2} then

< f, g >= ∑ a_{n}\overline{b_{n}},

where g(z) = ∑ b_{n}z^{n}. For λ ∈ D the function k_{λ}(z) = (1 − \overline{λ}z)^{−1} is the reproducing kernel for λ.

Let G be a bounded open subset of the complex plane C. For 1 ≤ p ≤ ∞, the Bergman space of G, L_{a}^{p}(G) is the set of all analytic functions f : G ⟶ C such that ∫_{G} |f|^{p}dA < ∞, where dA(z) = 1/πrdrdθ is the usual area measure on G. Note that L_{a}^{p}(G) is closed in L^{p}(G) and it is therefore a Banach space. When G = D the inner product in L_{a}^{2}(D) is given by

< f, g >= ∑ \frac{a_{n}\overline{b}_{n}}{n + 1},

where f = ∑ a_{n}z^{n} and g = ∑ b_{n}z^{n}. Therefore k_{λ}(z) = (1 − \overline{λ}z)^{−2} is the reproducing kernel for the point λ ∈ D.

Let λ_{α} (α > −1) be the finite measure defined on D by dλ_{α}(z) = (1 − |z|^{2})^{α}dA(z). For α > −1 and 0 < p < ∞ the weighted Bergman space A_{α}^{p} is the collection of all functions f analytic in D for which ∥f∥_{p,α}^{p} = ∫_{D} |f|^{p}dλ_{α} < ∞. The weighted Dirichlet space D_{α} (α > −1) is the collection of all analytic functions f in D for which the derivative f′ belongs to A_{α}^{2}. Note that A_{α}^{p} is a Banach space for p ≥ 1, and a Hilbert space for p = 2 [3]. The Dirichlet space D_{α} is a Hilbert space in the norm

∥f∥_{D_{α}}^{2} = |f(0)|^{2} + ∫_{D} |f′|^{2}dλ_{α}.

For these spaces the unit ball is a normal family and the point evaluation is bounded. Also, f(z) = ∑ a_{n}z^{n} analytic in D belongs to A_{α}^{2} if and only if ∑(n + 1)^{−1−α}|a_{n}|^{2} < ∞, and to D_{α} if and only if ∑(n + 1)^{1−α}|a_{n}|^{2} < ∞. We also note that if α > −1, then D_{α} ⊂ A_{α}^{2} and the inclusion map is continuous.

A function ϕ on D is said to have an angular derivative at ζ ∈ ∂D if there exist a complex number c and a point ω ∈ ∂D such that (ϕ(z) − ω)/(z − ζ) tends to c as z tends to ζ over any triangle in D with one vertex at ζ. Define d(ζ) = lim inf_{z→ζ} \frac{1−|ϕ(z)|}{1−|z|}, where z tends unrestrictedly to ζ through D. By [4, §5.3] the existence of an angular derivative at ζ ∈ ∂D is equivalent to d(ζ) < ∞.

For the proof of the next proposition see [5, Proposition 3.4].

Proposition 1.1. If ϕ is analytic in D with ϕ(D) ⊂ D, then C_{ϕ} is bounded on A_{α}^{p} for all 0 < p < ∞ and α > −1. Also, if w ∈ H^{∞} then wC_{ϕ} is bounded on A_{α}^{p} for all 0 < p < ∞ and α > −1.

In this paper we characterize such operators and those for which (wC_{ϕ})^{∗} is a composition operator. We also study the boundedness and compactness of the weighted composition operators on A_{α}^{p} or D_{α}. The relationship between the compactness of such operators and a special class of measures on the unit disc, Carleson measures, is shown. The main result is to determine, in terms of geometric properties of ϕ and w, when wC_{ϕ} is a compact operator on weighted Dirichlet spaces. For Bergman spaces we attack the problem in terms of an angular derivative of ϕ and an angular limit of w. We obtain some sufficient conditions for weighted Dirichlet spaces. Finally, we would like to acknowledge the fact that we are borrowing heavily the techniques of the proofs of [5].

2. Adjoint of Weighted Composition Operators

In this section we investigate when the adjoint of a weighted composition operator on some functional Hilbert space is a composition operator.

Theorem 2.1. Let T = wC_{ϕ} be a weighted composition operator on A_{α}^{2}, α > −1. Then T^{∗} = C_{ψ} if and only if w = k_{λ} and ϕ(z) = az(1−λ̄z)^{−1}, where λ = ψ(0) and a is a suitable constant. In particular, ψ has the form ψ(z) = \overline{a}z + λ.

Proof. Assume (wC_{ϕ})^{∗} = C_{ψ}. Then (wC_{ϕ})^{∗}k_{x} = C_{ψ}k_{x} or \overline{w(x)}k_{ϕ(x)}(y) = k_{x} ∘ ψ(y). It follows that

\frac{\overline{w(x)}}{(1 − \overline{ϕ(x)}y)^{α+2}} = \frac{1}{(1 − \overline{x}ψ(y))^{α+2}} , x, y ∈ D.

In short, (1−\overline{ϕ(x)}y)^{α+2} = \overline{w(x)}(1−\overline{x}ψ(y))^{α+2}. If we put y = 0 and ψ(0) = λ we have 1 = \overline{w(x)}(1−λ\overline{x})^{α+2}. Therefore w = k_{λ}. We also have (1−λ\overline{x})(1− \overline{ϕ(x)}y) = 1 − \overline{x}ψ(y) for all x, y ∈ D. Hence \overline{ϕ(x)}y + λ\overline{x} − λ\overline{ϕ(x)}\overline{x}y = \overline{x}ψ(y). Now if xy ≠ 0, then \overline{ϕ(x)}(1 − λ\overline{x})(\overline{x})^{−1} = (ψ(y) − ψ(0))y^{−1}. Since the right-hand side is independent of x, it should be a constant, say, \overline{a}, a ∈ C. Therefore ψ(z) = \overline{a}z + λ and ϕ(z) = az(1 − \overline{λ}z)^{−1}.

Conversely, suppose T = wC_{ϕ}, where w = k_{λ} and ϕ(x) = ax(1 − \overline{λ}x)^{−1} , a ∈ C. Then

T^{∗}k_{y}(x) = \overline{w(y)}k_{ϕ(y)}(x) = \frac{1}{(1 − λ\overline{y})^{α+2}} · \frac{1}{(1 − \overline{ϕ(y)}x)^{α+2}} =

= \frac{1}{(1 − λ\overline{y})^{α+2}} · \frac{1}{(1 − \overline{a}\overline{y}(1 − λ\overline{y})^{−1}x)^{α+2}} =

= \frac{1}{(1 − λ\overline{y} − \overline{a}\overline{y}x)^{α+2}} = C_{ψ}k_{y}(x) ,

where ψ(x) = \overline{a}x + λ.

Remark. By an analogous proof we can show that Theorem 2.1 is also true when T is a weighted composition operator on H^{2}.

We use the next theorem to give a sufficient condition for the subnormality of wC_{ϕ} on H^{2}.

Theorem 2.2 ([6]). If ϕ is a nonconstant analytic function defined on the unit disc D with ϕ(D) ⊂ D such that C_{ϕ}^{∗} is subnormal on H^{2} (and not normal), then there is a number c with |c| = 1 for which lim_{ρ→1} ϕ(ρc) = c and lim_{ρ⟶1^{−}} ϕ′(ρc) = s < 1. Moreover, if ϕ is analytic in a neighborhood of c, then C_{ϕ}^{∗} is subnormal on H^{2} if and only if

ϕ(z) = \frac{(r + s)z + (1 − s)c}{r(1 − s)\overline{c}z + (1 + sr)}

for some r, s with 0 ≤ r ≤ 1 and 0 < s < 1. Here, as above, s = ϕ′(c).

Applying Theorem 2.2 and the above remark we obtain

Corollary 2.3. If w = k_{λ} and ϕ(z) = szk_{λ}(z) with 0 < s < 1 and λ = (1 − s)c, where c is the number indicated in Theorem 2.2, then wC_{ϕ} is subnormal on H^{2}.

3. A Weighted Shift Analogy

As we shall see, for suitable w and ϕ the operator (wC_{ϕ})^{∗} (as well as the operator wC_{ϕ}) has an invariant subspace on which it is similar to a weighted shift.

We begin by defining the notions of forward and backward iteration sequences, see also [7].

Definition 3.1. A nonconstant sequence {z_{k}}_{k=0}^{∞} is a B-sequence for ϕ if ϕ(z_{k}) = z_{k−1} , k = 1, 2, . . .. A nonconstant sequence {z_{k}}_{k=0}^{∞} or {z_{k}}_{k=−∞}^{∞} is an F-sequence for ϕ if ϕ(z_{k}) = z_{k+1} for all k.

Theorem 3.2. If {z_{j}}_{j=0}^{∞} is a B-sequence for ϕ and

\frac{1 − |z_{j}|}{1 − |z_{j−1}|} ≤ r < 1

for all j, then {z_{j}}_{j=0}^{∞} gives rise to an invariant subspace of (wC_{ϕ})^{∗} on which it is similar to a backward weighted shift.

Proof. Let {z_{j}} be a B-sequence as in the statement of the theorem. By [7, p. 203], {z_{j}} is an interpolating sequence. Let u_{j} = (1 − |z_{j}|^{2})^{1/2}k_{j}, where k_{j} denotes the reproducing kernel at z_{j}. We keep this notation throughout the rest of this section. Let M be the closed linear span of {u_{j}}. By [6], {u_{j}} is a basic sequence in M equivalent to an orthonormal basis. Since

(wC_{ϕ})^{∗}u_{j} = (1 − |z_{j}|^{2})^{1/2}\overline{w(z_{j})}k_{j−1} = \overline{w(z_{j})}(\frac{1 − |z_{j}|^{2}}{1 − |z_{j−1}|^{2}} )^{1/2}u_{j−1},

(wC_{ϕ})^{∗} |_{M} is similar to a backward weighted shift with weights

\{(\frac{1 − |z_{j+1}|^{2}}{1 − |z_{j}|^{2}} )^{1/2}\overline{w(z_{j+1})}\}.

Recall that if ϕ is analytic in D with ϕ(D) ⊂ D and ϕ is not an analytic elliptic automorphism of D, then there is a unique fixed point a of ϕ (with |a| ≤ 1) such that |ϕ′(a)| ≤ 1. We will call the distinguished fixed point a the Denjoy–Wolff point [8] of ϕ. We note that if |a| = 1 then 0 < ϕ′(a) ≤ 1, and if |a| < 1 then 0 ≤ |ϕ′(a)| < 1.

Corollary 3.3. If ϕ has a Denjoy–Wolff point a in ∂D with ϕ′(a) < 1 then every F-sequence for ϕ gives rise to an invariant subspace of (wC_{ϕ})^{∗} on which it is similar to a forward weighted shift with weights

\{(\frac{1 − |z_{j−1}|^{2}}{1 − |z_{j}|^{2}} )^{1/2}\overline{w(z_{j−1})}\}.

Corollary 3.4. For 0 < s < 1 let w = k_{1−s} and ϕ(z) = szk_{1−s}(z). Then wC_{ϕ} has an invariant subspace M such that wC_{ϕ} |_{M} is similar to a weighted shift.

Proof. Let ψ(z) = sz + (1 − s). Then 1 is a Denjoy–Wolff point for ψ. Also, ψ′(1) = s < 1. So by Corollary 3.3 every F-sequence for ψ gives rise to an invariant subspace of C_{ψ}^{∗} on which it is similar to a weighted shift. Now by Theorem 2.1, C_{ψ}^{∗} = wC_{ϕ} where w = k_{1−s} and ϕ(z) = szk_{1−s}(z). The proof is now complete.

We note that if ϕ has a Denjoy–Wolff point a in ∂D with ϕ′(a) < 1, then for real θ, C_{ϕ} is similar to e^{iθ}C_{ϕ} [7]. In fact, much more is true. For the proof of the next corollary see [7].

Corollary 3.5. If ϕ is an analytic map of the disc to itself, ϕ(1) = 1 and ϕ′(1) < 1, then for any function w for which wC_{ϕ} is bounded we have wC_{ϕ} similar to λwC_{ϕ} for |λ| = 1.

4. Compactness on Weighted Bergman Spaces

In this section we will focus our attention on the relationship between compact weighted composition operators and a special class of measures on the unit disc. First, we will recall a few definitions.

For 0 < δ ≤ 2 and ζ ∈ ∂D let

S(ζ, δ) = {z ∈ D : |z − ζ| < δ}.

One can show that the λ_{α}-measure of the semidisc S(ζ, δ) is comparable with δ^{α+2} (α > −1). We can now give

Definition 4.1. Let α > −1 and suppose μ is a finite positive Borel measure on D. We call μ an α-Carleson measure if

∥μ∥_{α} = sup μ(S(ζ, δ))/δ^{α+2} < ∞,

where the supremum is taken over all ζ ∈ ∂D and 0 < δ ≤ 2. If, in addition,

lim_{δ→0} sup_{ζ∈∂D} μ(S(ζ, δ))/δ^{α+2} = 0,

then we call μ a compact α-Carleson measure.

The next theorem is stated and proved in [5]. Since we refer to it several times, we state it without proof.

Theorem 4.2. Fix 0 < p < ∞ and α > −1 and let μ be a finite positive Borel measure on D. Then μ is an α-Carleson measure if and only if A_{α}^{p} ⊂ L^{p}(μ). In this case the inclusion map I_{α} : A_{α}^{p} ⟶ L^{p}(μ) is a bounded operator with a norm comparable with ∥μ∥_{α}. If μ is an α-Carleson measure, then I_{α} is compact if and only if μ is compact.

In the next theorem we extend the result of [5, Corollary 4.4] by characterizing the compact weighted composition operators on the spaces A_{α}^{p} in terms of Carleson measures.

Theorem 4.3. Fix 0 < p < ∞ and α > −1. Then wC_{ϕ} is a bounded (compact) operator on A_{α}^{p} if and only if the measure μ_{α,p} ∘ ϕ^{−1} is an α-Carleson (compact α-Carleson) measure. Here dμ_{α,p} = |w|^{p}dλ_{α}.

Proof. We know that

∥(wC_{ϕ})f∥_{p,α}^{p} = ∫_{D} |f ∘ ϕ|^{p}|w|^{p}dλ_{α} = ∫_{D} |f|^{p}dμ_{α,p} ∘ ϕ^{−1},

for every f ∈ A_{α}^{p}. By Theorem 4.2 wC_{ϕ} is bounded on A_{α}^{p} if and only if μ_{α,p} ∘ ϕ^{−1} is an α-Carleson measure.

Now equip the space A_{α}^{p} with the metric of L^{p}(μ_{α,p} ∘ ϕ^{−1}) and call this (usually incomplete) space X. The above equation shows that wC_{ϕ} induces an isometry S of X into A_{α}^{p}. Thus wC_{ϕ} = SI_{α} is compact if and only if I_{α} is. An application of Theorem 4.2 completes the proof.

A modification of the proof of Theorem 5.3 of [5] will give

Theorem 4.4. Suppose α > −1, p > 0.

(a) If wC_{ϕ} is a compact operator on A_{α}^{p}, then ϕ does not have an angular derivative at those points of ∂D at which w has a nonzero angular limit.

(b) Suppose w has a zero angular limit at any point of ∂D at which ϕ has an angular derivative; then wC_{ϕ} is compact.

5. Boundedness on Weighted Dirichlet Spaces

In this section we study the relationship between the boundedness of weighted composition operators on weighted Dirichlet spaces and a special class of measures on the unit disc.

We recall that D_{1} = H^{2} and if α > 1 then D_{α} = A_{α−2}^{2} and the characterization of bounded (compact) weighted composition operators on D_{α} for α > 1 is given in Theorem 4.3. However, for −1 < α < 1, an obvious necessary condition for wC_{ϕ} to be bounded on D_{α} is that w = wC_{ϕ}1 ∈ D_{α}. In the following, we characterize the boundedness of such operators.

Theorem 5.1. Suppose w ∈ D_{α}. Then wC_{ϕ} is bounded on D_{α} if the measures μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} are α-Carleson measures, where dμ_{α} = |w′|^{2}dλ_{α} and dν_{α} = |w|^{2}|ϕ′|^{2}dλ_{α}.

Proof. Assume μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} are α-Carleson measures. Then, for every f in D_{α} we have f′ ∈ A_{α}^{2} ⊂ L^{2}(ν_{α} ∘ ϕ^{−1}) by Theorem 4.2. For every f in D_{α} we have (wC_{ϕ}f)′ = w′f ∘ ϕ + w(f ∘ ϕ)′. We now have

∥w(f ∘ ϕ)′∥_{2,α}^{2} = ∫ |w|^{2}|ϕ′|^{2}|f′ ∘ ϕ|^{2} dλ_{α} =
= ∫ |f′ ∘ ϕ|^{2} dν_{α} =
= ∫ |f′|^{2} dν_{α} ∘ ϕ^{−1} < ∞,

therefore, w(f ∘ ϕ)′ ∈ A_{α}^{2}. Note also that

∫ |w′|^{2}|f ∘ ϕ|^{2}dλ_{α} = ∫ |f ∘ ϕ|^{2}dμ_{α} = ∫ |f|^{2}dμ_{α} ∘ ϕ^{−1}.

Since f ∈ D_{α} ⊂ A_{α}^{2} ⊂ L^{2}(μ_{α} ∘ ϕ^{−1}), we have ∫ |w′|^{2}|f ∘ ϕ|^{2}dλ_{α} < ∞. Combining these two observations we conclude that (wC_{ϕ}f)′ ∈ A_{α}^{2} for every f in D_{α}. Therefore wC_{ϕ}f ∈ D_{α} and wC_{ϕ} is bounded on D_{α}.

6. Compactness on Dirichlet Spaces

The main result of this section concerns sufficient conditions for the compactness of weighted composition operators on Dirichlet spaces D_{α}. We would like to investigate whether an analogue of Theorem 4.3, the Carleson measure characterization of compact weighted composition operators, holds for Dirichlet spaces.

Theorem 6.1. If μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} are compact α-Carleson measures, where dμ_{α} = |w′|^{2}dλ_{α} and dν_{α} = |w|^{2}|ϕ′|^{2}dλ_{α}, then wC_{ϕ} is compact on D_{α} for α > −1.

Proof. Let X denote the space D_{α} taken in the metric induced by ∥.∥_{1} defined by

∥f∥_{1}^{2} = (∥f∥_{2} + ∥f∥_{3})^{2} + |w(0)f ∘ ϕ(0)|^{2},

where ∥f∥_{2}^{2} = ∫_{D} |f|^{2}dμ_{α} ∘ ϕ^{−1} and ∥f∥_{3}^{2} = ∫_{D} |f′|^{2}dν_{α} ∘ ϕ^{−1} (f ∈ D_{α}). Let I : D_{α} ⟶ X be the identity map and define S : X ⟶ D_{α} by Sf = wf ∘ ϕ. So wC_{ϕ} = SI. To show that S is a bounded operator let f ∈ X. Then

∥Sf∥_{D_{α}}^{2} = ∫_{D} |(wf ∘ ϕ)′|^{2}dλ_{α} + |w(0)f ∘ ϕ(0)|^{2} ≤

≤ (∥w′f ∘ ϕ∥_{2,α} + ∥wϕ′(f′ ∘ ϕ)∥_{2,α})^{2} + |w(0)f ∘ ϕ(0)|^{2}.

We use the change of variable formula to get

∫_{D} |w′|^{2}|f ∘ ϕ|^{2}dλ_{α} = ∫_{D} |f|^{2}dμ_{α} ∘ ϕ^{−1} = ∥f∥_{2}^{2}

and

∫_{D} |w|^{2}|ϕ′|^{2}|f′ ∘ ϕ|^{2}dλ_{α} = ∫_{D} |f′|^{2}dν_{α} ∘ ϕ^{−1} = ∥f∥_{3}^{2}.

Thus we have

∥Sf∥_{D_{α}}^{2} ≤ (∥f∥_{2} + ∥f∥_{3})^{2} + |w(0)f ∘ ϕ(0)|^{2} = ∥f∥_{1}^{2}.

Hence ∥S∥ ≤ 1 and S is bounded. If we show that I is compact, then wC_{ϕ} = SI is compact and the proof is complete.

Now, we use the idea of [5, Theorem 4.3] to prove that I is compact. It is enough to show that each sequence (f_{n}) in D_{α} that converges uniformly to zero on compact subsets of D must be norm convergent to zero in X. Fix 0 < δ < 1 and let μ_{δ} and ν_{δ} be the restriction of the measures μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} to the annulus 1 − δ < |z| < 1. Observe that the α-Carleson norm of μ_{δ} and ν_{δ} satisfy

∥μ_{δ}∥_{α} ≤ c_{1} sup μ_{α} ∘ ϕ^{−1}(S(ζ, r))/r^{α+2},

and

∥ν_{δ}∥_{α} ≤ c_{2} sup ν_{α} ∘ ϕ^{−1}(S(ζ, r))/r^{α+2},

where the supremum is taken over all 0 < r < δ and ζ ∈ ∂D, and c_{1} , c_{2} are positive constants which depend only on α. Since μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} are compact α-Carleson measures, the right-hand sides of the above two inequalities, which we denote by ϵ_{1}(δ) and ϵ_{2}(δ), respectively, tend to zero as δ → 0. So we have

∥f_{n}∥_{2}^{2} = ∫_{|z|<1−δ} |f_{n}|^{2}dμ_{α} ∘ ϕ^{−1} + ∫_{D} |f_{n}|^{2}dμ_{δ} ≤

≤ o(1) + k_{1}ϵ_{1}(δ)∥f_{n}∥_{2,α}^{2},

and in the same manner

∥f_{n}∥_{3}^{2} ≤ o(1) + k_{2}ϵ_{2}(δ)∥fn^{′}∥_{2,α}^{2},

where k_{1} and k_{2} are constants depending only on α. We recall that the estimate of the first terms comes from the uniform convergence of (f_{n}) to zero on |z| ≤ 1 − δ, and the estimate of the second terms comes from the first part of [6, Theorem 4.3]. Since ϵ_{i}(δ) → 0 as δ → 0, i = 1, 2, and w(0)f_{n} ∘ ϕ(0) → 0, we have ∥f_{n}∥_{1} → 0, which completes the proof.

We now characterize compact weighted composition operators in terms of an angular derivative of ϕ and angular limit of w,w′.

Theorem 6.2. If w′ has a zero angular limit at any point of ∂D at which ϕ has an angular derivative, then μ_{α} ∘ ϕ^{−1} is a compact α-Carleson measure. Here dμ_{α} = |w′|^{2}dλ_{α}.

Proof. Suppose w′ has a zero angular limit at those points of ∂D at which ϕ has an angular derivative. Choose 0 < γ < α with r = 2 − (α − γ) > 0. For 0 < δ < 2 define

ϵ(δ) = sup \{\frac{(1 − |z|^{2})|w′(z)|}{1 − |ϕ(z)|^{2}} : 1 − |z| ≤ δ\}.

Since w′ has a zero angular limit at those points of ∂D at which ϕ has an angular derivative we have lim_{δ→0} ϵ(δ) = 0. With no loss of generality assume that ϕ(0) = 0. Fix S = S(ζ, δ). By the Schwartz Lemma and definition of ϵ(δ) we have

|w′(z)|(1 − |z|^{2}) ≤ (1 − |ϕ(z)|^{2})ϵ(δ) ≤ 2δϵ(δ)

whenever ϕ(z) ∈ S. So we have

μ_{α} ∘ ϕ^{−1}(S) = ∫_{ϕ^{−1}(S)} |w′(z)|^{2}(1 − |z|^{2})^{α}dλ(z) ≤

≤ (2δϵ(δ))^{α−γ} ∫_{ϕ^{−1}(S)} |w′(z)|^{r}(1 − |z|^{2})^{γ}dλ(z) ×

×(2ϵ(δ))^{α−γ}δ^{α−γ}μ_{r,γ} ∘ ϕ^{−1}(S).

Here dμ_{r,γ}(z) = |w′(z)|^{r}dλ_{γ}(z). Now by Proposition 1.1 and Theorem 4.2, μ_{r,γ} ∘ ϕ^{−1} is a γ-Carleson measure. Thus there exists a constant k independent of ζ, δ such that μ_{r,γ} ∘ ϕ^{−1}(S) ≤ kδ^{γ+2}.

Hence μ_{α} ∘ϕ^{−1}(S) ≤ k(2ϵ(δ))^{α−γ}δ^{α+2}. Since ϵ(δ) → 0 as δ → 0, μ_{α} ∘ϕ^{−1} is therefore a compact α-Carleson measure and the proof is complete.

To state our main result we need

Theorem 6.3. Suppose w has a zero angular limit at any point of ∂D at which ϕ has an angular derivative. If, in addition, for some −1 < γ < α, the measure η_{γ} ∘ ϕ^{−1} is a γ-Carleson measure, where dη_{γ} = |w|^{2−α+γ}|ϕ′|^{2}dλ_{γ}, then ν_{α} ∘ ϕ^{−1} is a compact α-Carleson measure (dν_{α} =|w|^{2}|ϕ′|^{2}dλ_{α}).

Proof. For 0 < δ < 2 define

ρ(δ) = sup \{\frac{(1 − |z|^{2})|w(z)|}{1 − |ϕ(z)|^{2}} : 1 − |z| ≤ δ\}.

By the argument of the proof of Theorem 6.2 lim_{δ→0} ρ(δ) = 0. Also, we have |w(z)|(1 − |z|^{2}) ≤ (1 − |ϕ(z)|^{2})ρ(δ) ≤ 2δρ(δ), whenever ϕ(z) ∈ S(ζ, δ). Thus

ν_{α} ∘ ϕ^{−1}(S) = ∫_{ϕ^{−1}(S)} |w|^{2}|ϕ′(z)|^{2}(1 − |z|^{2})^{α}dλ(z) ≤

≤(2δρ(δ))^{α−γ} ∫_{ϕ^{−1}(S)}|w|^{2−α+γ}|ϕ′(z)|^{2}(1−|z|^{2})^{γ}dλ(z)=(2ρ(δ))^{α−γ}η_{γ} ∘ϕ^{−1}(S).

Now we use the hypothesis that η_{γ} ∘ ϕ^{−1} is a γ-Carleson measure; so there exists a constant k independent of ζ and δ such that η_{γ} ∘ ϕ^{−1}(S) ≤ kδ^{γ+2}.

Thus ν_{α} ∘ ϕ^{−1}(S) ≤ k(2ρ(δ)^{α−γ})δ^{α+2}. Since ρ(δ) → 0 as δ → 0, ν_{α} ∘ ϕ^{−1} is therefore a compact α-Carleson measure, and the proof is complete.

Now we state the main theorem.

Theorem 6.4. Let w′ ∈ H^{∞} and ϕ ∈ D_{α}. Assume w and w′ have a zero angular limit at any point of ∂D at which ϕ has an angular derivative. If, in addition, for some −1 < γ < α the measure η_{γ} ∘ ϕ^{−1} is a γ-Carleson measure, then wC_{ϕ} is compact on D_{α}. Here dη_{γ} = |w|^{2−α+γ}|ϕ′|^{2}dλ_{γ}.

Proof. By Theorems 6.2 and 6.3, the measures μ_{α} ∘ ϕ^{−1} and ν_{α} ∘ ϕ^{−1} are compact α-Carleson measures. Thus Theorem 6.1 shows that wC_{ϕ} is compact on D_{α}.

Acknowledgement

This research was partially supported by a Grant (No. 69-SC-591-317) from Shiraz University Research Council.

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(Received 09.08.1995)

Authors’ address:

Department of Mathematics

Shiraz University

Shiraz, Iran