Abstract
We characterize the conditions under which approximately multiplicative functionals are near multiplicative functionals on weighted Hardy spaces.
1. Introduction
Let be a commutative Banach algebra and the set of all its characters, that is, the nonzero multiplicative linear functionals on . If is a linear functional on , then define
for all . We say that is -multiplicative if .
For each define We say that is an algebra in which approximately multiplicative functionals are near multiplicative functionals or is AMNM for short if, for each , there is such that whenever is a -multiplicative linear functional.
We deal with an algebra in which every approximately multiplicative functional is near a multiplicative functional (AMNM algebra). The question whether an almost multiplicative map is close to a multiplicative, constitutes an interesting problem. Johnson has shown that various Banach algebras are AMNM and some of them fail to be AMNM [1–3]. Also, this property is still unknown for some Banach algebras such as , Douglas algebras, and where is a compact subset of . Here, we want to investigate conditions under which a weighted Hardy space is to be AMNM. For some sources on these topics one can refer to [1–8].
Let be a sequence of positive numbers with and . We consider the space of sequences such that The notation will be used whether or not the series converges for any value of . These are called formal power series or weighted Hardy spaces. Let denote the space of all such formal power series. These are reflexive Banach spaces with norm . Also, the dual of is , where and (see [9]). Let . So , and then is a basis such that for all . For some sources one can see [9–21].
2. Main Results
In this section we investigate the AMNM property of the spaces of formal power series. For the proof of our main theorem we need the following lemma.
Lemma 2.1. Let and . Then, , where .
Proof. Define by , where Then, Thus, is an isometry. It is also surjective because, if then , where Hence, and are norm isomorphic. Since , the proof is complete.
In the proof of the following theorem, our technique is similar to B. E. Johnson’s technique in [2].
Theorem 2.2. Let and . Then, with multiplication is a commutative Banach algebra that is AMNM.
Proof. First note that clearly is a commutative Banach algebra. To prove that it is AMNM, let and put . Suppose that and , where . It is sufficient to show that . Since , if , then . So suppose that . For each subset of , let
where
For any subsets and of we have that
Hence,
for all . Also if , then, by considering with support, respectively, in and , we get that and so
By taking supremum over all such and with norm one, we see that
So either or whenever .
For all we have that
Thus, we get that
Since , as we saw earlier, it should be or and equivalently it should be or for all .
Note that, if with for , then
Thus, the relation is not true and so it should be
Since , clearly there exists a positive integer such that for all , where
for all . Now, let for . Since and , . By continuing this manner we get that , which is a contradiction. Hence there exists such that . On the other hand, since , or . But , and so it should be .
Remember that for all . Now we have that
Therefore,
and so
But , and thus
Define . Then , and we have that
Thus, indeed , and so the proof is complete.
Disclosure
This is a part of the second author’s Doctoral thesis written under the direction of the first author.