Norms and essential norms of composition operators from H ∞ to general weighted Bloch spaces in the polydisk
© Wang and Wu; licensee Springer. 2011
Received: 11 December 2010
Accepted: 2 September 2011
Published: 2 September 2011
Let U n be the unit polydisk of and φ a holomorphic self-map of U n . H∞(U n ) and denote the space of bounded holomorphic functions and the space of general weighted Bloch functions defined on U n , respectively, where α > 0. This paper gives some estimates of the norm and essential norm of the composition operator C φ induced by φ from H ∞ (U n ) to . As applications, some characterizations of the boundedness and compactness of C φ from H ∞ (U n ) to are obtained. Moreover, we also characterize the weak compactness of the composition operator C φ .
Primary 47B33; Secondary 32A37
is called the composition operator with symbol φ. The study of composition operators is fundamental in the study of Banach and Hilbert spaces of holomorphic functions. We refer to the books  and  for an overview of some classical results on the theory of composition operators.
. By fixing a base point z0 ∈ D, the Bloch space of all Bloch functions on D is a Banach space under the norm . For convenience, we assume the bounded homogeneous domain D to contain the origin and take z0 = 0. In , Timoney proved that the space H∞(D) of bounded holomorphic functions on a bounded homogeneous domain D is a subspace of and for each , where C D is a constant depending only on the domain D and .
Composition operators on various Bloch-type spaces have been studied extensively by many authors. For the unit disk , Madigan and Matheson  proved that C φ is always bounded on . They also gave some sufficient and necessary conditions that C φ is compact on . Since then, there were many authors generalizing the results in  to the unit ball, polydisk and other classical symmetric domains, see, for example, [7–17]. At the same time, there were also many papers dealing with the composition operators between Bloch-type spaces and bounded holomorphic function spaces, refer to [18, 19] and the references therein for the details. Specially, Li and Liu  stated and proved the corresponding boundedness and compactness characterizations for C φ from H ∞ (U n ) to . But there is a little gap in the proof [, line 17, p. 1637]. In this paper, we apply methods developed by Montes-Rodriguez  to give some estimates of the norm and essential norm of C φ from H∞(U n ) to . Recall that the essential norm ||T|| e of a bounded operator T between Banach spaces X and Y is defined as the distance from T to the space of compact operators from X to Y. Notice that ||T|| e = 0 if and only if T is compact, so that estimates on || T || e lead to conditions for T to be compact. For convenience, we define || T|| e = || T|| = ∞ for any unbounded linear operator T. As an application of our estimates, we obtain the main results in  with new proofs. In addition, we also show the equivalence of the compactness and weak compactness of .
Throughout the remainder of this paper, C will denote a positive constant, the exact value of which will vary from one occurrence to the others.
2 The norm of C φ
In this section, we give the following estimate of the norm of .
Here and in the sequel, the symbol A ≲ B(or B ≳ A) means that A ≤ CB for some positive constant C independent of A and B. A ~ B means that A ≲ B and B ≲ A.
where is used in the last line above.
which completes the proof.
The following corollary is obtained immediately from Theorem 1.
3 The essential norm of C φ
This section mainly gives the following estimate of the essential norm of C φ from H∞(U n ) to .
Again ε is arbitrary, and we obtain the desired lower estimate.
Where and .
Thence, J2 → 0 (as m → ∞).
Similarly, we know that
The proof is complete.
As an application, we have the following corollary.
is weakly compact.
the inequalities in the third and fourth lines above follow from (5), and the last line is due to. This contradicts again , which completes the proof.
This work was supported by the National Natural Science Foundation of China (No.10901158,11071190).
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