On the Norm of Certain Weighted Composition Operators on the Hardy Space
We obtain a representation for the norm of certain compact weighted composition operator on the Hardy space , whenever and . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on and . Moreover, we characterize the norm and essential norm of such operators in a special case.
Let denote the open unit disk in the complex plane. The Hardy space is the space of analytic functions on whose Taylor coefficients, in the expansion about the origin, are square summable. Also we recall that is the space of all bounded analytic function defined on . For , the reproducing kernel at for is defined by An easy computation shows that whenever . For any analytic self-map of , the composition operator on is defined by the rule . Every composition operator is bounded, with
(see ). We see from expression (1.1) that whenever . There are few other cases for which the exact value of the norm has been known for many years. For example, the norm of was obtained by Nordgren in , whenever is an inner function. In  this norm was determined, when , with and if and the norm was found in  for .
In 2003, Hammond  obtained exact values for the norms of composition operators for certain linear fractional maps . In , Bourdon et al. determined the norm of for a large class of linear-fractional maps, including those of the form , where The connection between the norm of certain composition operators with linear-fractional symbol acting on the Hardy space and the roots of associated hypergeometric functions was first made by Basor and Retsek . It was later refined by Hammond . In  Effinger-Dean et al. computed the norms of composition operators with rational symbols that satisfy certain properties. Their work is based on the initial work of Hammond . Some other recent results regarding the calculation of the operator norm of some composition operators on the other spaces can be found in [10–14].
If is a bounded analytic function on and is an analytic map from into itself, the weighted composition operator is defined by . The map is called the composition map and is called the weight. If is a bounded analytic function on , then the operator can be rewritten as where is a multiplication operator and is a composition operator. Recall that if is an analytic self-map of , then the composition operator on is bounded, hence in this case is bounded, but in general every weighted composition operator on is not bounded. If is bounded, then belongs to . These operators come up naturally. In 1964, Forelli  showed that every isometry on for and is a weighted composition operator. Recently there has been a great interest in studying weighted composition operators in the unit disk, polydisk, or the unit ball; see [12, 16–27], and the references therein. In this paper we investigate the norm of certain bounded weighted composition operators on .
2. Norm Calculation
In this section we obtain a representation for the norm of a class of compact weighted composition operators on the Hardy space , whenever , , and . Also we give the norm and essential norm inequality for a class of noncompact weighted composition operators on when , for some , and is a bounded analytic map on such that the radial limit of at one of the th roots of is the supremum of on . Also, when we obtain the norm and essential norm of such operators.
Lemma 2.1. Let be the reproducing kernel at . Then In the next proposition we generalize the lower bound in (1.1).
Proposition 2.2. Let be a nonconstant analytic self-map of , and let be a nonzero analytic map on . If is the smallest nonnegative integer such that , then
Proof. We note that if is in , then for every we have . Hence we have
Let be a bounded operator on a Hilbert space . We recall that , the essential norm of , is the norm of its equivalence class in the Calkin algebra. Since the spectral radius of the operator equals , we study the spectrum of when trying to determine . We say that the operator is norm-attaining if there is a nonzero such that We know that if and only if . Moreover, if , then the operator is norm-attaining and so the quantity equals the largest eigenvalue of ; see  for more details. If , , and , then the operator is compact (see the proof of Proposition 2.5). Hence and so is norm-attaining.
Now our goal is to find a functional equation that relates an eigenvalue of to the values of its eigenfunctions at particular points in the disk. In what follows we use the techniques used in [5, 6, 30] and present some results that help us to obtain the norm of .
Let be an analytic self-map of and let be a bounded analytic map on . Then
where , and
From now on, unless otherwise stated, we assume that and . Since is the backward shift on , we see that for all in not equal to 0, where
In particular, if is an eigenfunction for corresponding to an eigenvalue , then Formula (2.8) is essentially identical to [5, Formula (3.3)]. Using (2.8) we can find a set of conditions under which we determine . In the trivial case we have Also if , then and if , then . Therefore we assume that are nonzero.
Throughout this paper, we write to denote the th iterate of that is, is the identity map on and
By a similar argument as in the proof of [5, Proposition 5.1], we have the following lemma.
Lemma 2.3. Let be an eigenfunction for corresponding to an eigenvalue , and for each nonnegative integer , . Then one has where one takes
Lemma 2.4. For each , , where is strictly increasing sequence such that for each . Also
Proof. (By induction) Since and , the claim holds for . Assume the claim holds for . We will prove it for . We have Now if we set , then . But by hypothesis , so which implies that also Hence the proof is complete.
Proposition 2.5. Let , and let . If and , then for each with the property that for every nonnegative integer , one has
Proof. Since , it is easy to see that if and only if and . By assumption , so . Therefore is compact and, since , the operator is compact. Now according to the paragraph after Proposition 2.2, there is function in such that . Let and for each integer , . By Lemma 2.3, we have
Now if is the Denjoy-Wolff point of it suffices to show that
Suppose the above inequality holds. Then we conclude that there is and such that for we have . Now we break the proof into two parts.
The Denjoy-Wolff point of lies inside , then converges to . Hence
The Denjoy-Wolff point of lies on , then by [31, Lemma 5.1] must be parabolic and by [6, Lemma 3.3] there is a constant such that
Thus it follows that Hence Now we show that . Since and , we see that By , we have Applying the assumptions and , an easy computation shows that Also by using Proposition 2.2, and by Lemma 2.4, there is such that . Therefore
Proposition 2.6. Let , , and . Then satisfies the equation
Proof. Since for every integer , , in Proposition 2.5 we set , then we have Since , we see that But , because otherwise Proposition 2.5 would dictate that the function is identically 0. Thus eigenfunction must have the property that . Hence we have
Proposition 2.7. Let , , and . Then has the following properties.(a)The power series that defines has radius of convergence larger than . (b) is non-negative real number for all in the interval . (c) for all in the interval .
Proof. (a) By Lemma 2.4, for each positive integer there is such that , then . Also in the proof of Proposition 2.5 we have , hence there is and such that if , then
Now let and . Then if we have
Therefore there is a constant such that
By Lemma 2.4, there is strictly increasing sequence such that , and by hypothesis , hence . Also we have , so we conclude that . Therefore
Also it is obvious that Hence the proof of part (b) is complete.
(c) Every coefficient of is positive and so for all in the interval .
Now we find an equation that involves the norm of .
Theorem 2.8. Let , , and . Then is the unique positive real solution of the equation
Corollary 2.9. In Theorem 2.8 if one replaces with and with such that , and , then norm of does not change.
Example 2.10. Let and , where and . Then we have For positive integer , let denote the positive solution of Now by using numerical methods, we have Hence we see that
The hypotheses of Theorem 2.8 restrict us to considering the norms of compact operators. In the remainder of this section we investigate the norm and essential norm of a class of noncompact weighted composition operators.
Theorem 2.11. Let , for some , where , ,let be one of the th roots of such that has radial limit at and let attains its supremum on at . Then
Corollary 2.12. In Theorem 2.11 if , then
Example 2.13. (1) If and , then
(2) If and , then
(3) If and , then
The authors would like to thank the referee for his valuable comments and suggestions.
C. C. Cowen and B. D. MacCluer, Composition Operators on Spaces of Analytic Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1995.View at: MathSciNet
G. Gunatillake, Weighted composition operators, Ph.D. thesis, Purdue University, West Lafayette, Ind, USA, 2005.