#### 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.