Essential Norm of Operators into Weighted-Type Spaces on the Unit Ball

and Applied Analysis 3 see 10 and in terms of the weights. For a weight v the associated weight ṽ is defined as follows ṽ z : ( sup {∣∣f z ∣∣ : f ∈ H∞ v ,∥∥f∥∥H∞ v ≤ 1 })−1 . 1.7 For a typical weight v the associated weight ṽ is also typical. Furthermore, for each z ∈ B there is an fz ∈ H∞ v , ‖fz‖H∞ v ≤ 1, such that fz z 1/ṽ z , and the same holds for the space H0 v. It is known that H ∞ ṽ ∼ H∞ v and H0 ṽ ∼ H0 v, that is, they are isometrically isometric 10 . We say that a weight v satisfies condition L1 if it is radial and inf k∈N v ( 1 − 2− k 1 ) v ( 1 − 2−k) > 0. L1 For radial weights v satisfying condition L1 , we have that v and ṽ are equivalent, that is, there is a C ≥ 1 such that v ≤ ṽ ≤ Cv. Recently Lusky and Taskinen 12 have shown, among other results, that H0 α is isomorphic to c0. Since the closed unit ball BH∞ v is a compact subset of H B , τ , a result of Dixmier-Ng 13 gives that the subspace of H∞ v ∗ Gv : { l ∈ H∞ v ∗ : l | BH∞ v is τ-continuous } 1.8 is a predual of H∞ v , that is, G ∞ v ∗ ∼ H∞ v . Clearly the evaluation functional at z ∈ B, defined by δz f f z , belongs to Gv . The norm of δz is denoted by ‖δz‖v. Moreover, the set {δz : z ∈ B} is a total set, that is, its linear span is norm dense in Gv . More precisely, the next isomorphism result is due to Bierstedt and Summers. Lemma 1.1 see 11 . The map f → l → 〈l, f〉 is an onto isometric isomorphism between H∞ v and Gv ∗ and the restriction map l → l|H0 v gives rise to an isometric isomorphism between Gv and H0 v ∗. Similarly to the corresponding result in the one variable 14 , one can prove the following. Lemma 1.2. Suppose β > −1 and γ > 0. Then Aβ ∗ is isomorphic toH0 γ 〈 f, g 〉 β,γ ∫ B f z g z cβ γυβ γ z dV z , f ∈ H∞ γ , g ∈ Aβ. 1.9 Moreover, under the pairing 〈f, g〉β,γ with g ∈ H0 γ and f ∈ Aβ, we also have that H0 γ ∗ is isomorphic to Aβ. For z,w ∈ B, let K β z w 1 1 − 〈w, z〉 n β 1 . 1.10 4 Abstract and Applied Analysis Then the kernel function K γ z clearly belongs to H0 γ and to A 1 β . It has the reproducing property