Approximately Multiplicative Functionals on the Spaces of ... · g n zn ∞ n 0 f n g n zn 2.5 is a commutative Banach algebra that is AMNM. Proof. First note that clearly Hp β is

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2011, Article ID 874949, 6 pagesdoi:10.1155/2011/874949

Research ArticleApproximately Multiplicative Functionals onthe Spaces of Formal Power Series

F. Ershad and S. H. Petroudi

Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran 19569, Iran

Correspondence should be addressed to F. Ershad, [email protected]

Received 20 February 2011; Accepted 5 May 2011

Academic Editor: John Rassias

Copyright q 2011 F. Ershad and S. H. Petroudi. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We characterize the conditions under which approximately multiplicative functionals are nearmultiplicative functionals on weighted Hardy spaces.

1. Introduction

LetA be a commutative Banach algebra and A the set of all its characters, that is, the nonzeromultiplicative linear functionals onA. If ϕ is a linear functional onA, then define

ϕ(a, b) = ϕ(ab) − ϕ(a)ϕ(b) (1.1)

for all a, b ∈ A. We say that ϕ is δ-multiplicative if ‖ϕ‖ ≤ δ.For each ϕ ∈ A� define

d(

ϕ)

= inf{

∥

∥ϕ − ψ∥∥ : ψ ∈ A ∪ {0}}

. (1.2)

We say that A is an algebra in which approximately multiplicative functionals are near mul-tiplicative functionals or A is AMNM for short if, for each ε > 0, there is δ > 0 such thatd(ϕ) < ε whenever ϕ is a δ-multiplicative linear functional.

We deal with an algebra in which every approximately multiplicative functional isnear a multiplicative functional (AMNM algebra). The question whether an almost multi-plicative map is close to a multiplicative, constitutes an interesting problem. Johnson hasshown 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 asH∞, Douglas algebras,

2 Abstract and Applied Analysis

and R(K) where K is a compact subset of C. Here, we want to investigate conditions underwhich a weighted Hardy space is to be AMNM. For some sources on these topics one canrefer to [1–8].

Let {β(n)}∞n=0 be a sequence of positive numbers with β(0) = 1 and 1 < p < ∞. Weconsider the space of sequences f = { f(n)}∞n=0 such that

∥

∥f∥

∥

p =∥

∥f∥

∥

p

β =∞∑

n=0

∣

∣

∣

f(n)∣

∣

∣

pβp(n) <∞. (1.3)

The notation f(z) =∑∞

n=0f(n)zn will be used whether or not the series converges for any

value of z. These are called formal power series or weighted Hardy spaces. LetHp(β) denotethe space of all such formal power series. These are reflexive Banach spaces with norm ‖ · ‖β.Also, the dual of Hp(β) is Hq(βp/q), where 1/p + 1/q = 1 and βp/q = {β(n)p/q}∞n=0 (see [9]).Let fk(n) = δk(n). So fk(z) = zk, and then {fk}∞k=0 is a basis such that ‖fk‖ = β(k) for all k. Forsome sources one can see [9–21].

2. Main Results

In this section we investigate the AMNM property of the spaces of formal power series. Forthe proof of our main theorem we need the following lemma.

Lemma 2.1. Let 1 < p <∞ and 1/p+1/q = 1. Then,Hp(β)� = Hq(β−1), where β−1 = {β−1(n)}∞n=0.

Proof. Define L : Hq(βp/q) → Hq(β−1) by L(f) = F, where

f(z) =∞∑

n=0

f(n)zn,

F(z) =∞∑

n=0

f(n)βp(n)zn.

(2.1)

Then,

‖F‖qHq(β−1) =

∞∑

n=0

∣

∣

∣

f(n)∣

∣

∣

q(

βpq(n)βq(n)

)

=∞∑

n=0

∣

∣

∣

f(n)∣

∣

∣

qβp(n)

=∥

∥f∥

∥

q

Hp(βp/q ).

(2.2)

Thus, L is an isometry. It is also surjective because, if

F(z) =∞∑

n=0

F(n)zn ∈ Hq(

β−1)

, (2.3)

Abstract and Applied Analysis 3

then L(f) = F, where

f(z) =∞∑

n=0

(

F(n)βp(n)

)

zn. (2.4)

Hence, Hq(βp/q) and Hq(β−1) are norm isomorphic. Since Hp(β)� = Hq(βp/q), the proof iscomplete.

In the proof of the following theorem, our technique is similar to B. E. Johnson’s tech-nique in [2].

Theorem 2.2. Let lim inf β(n) > 1 and 1 < p <∞. Then,Hp(β) with multiplication

( ∞∑

n=0

f(n)zn)( ∞

∑

n=0

g(n)zn)

=∞∑

n=0

f(n)g(n)zn (2.5)

is a commutative Banach algebra that is AMNM.

Proof. First note that clearly Hp(β) is a commutative Banach algebra. To prove that it isAMNM, let 0 < ε < 1 and put δ = ε2/16. Suppose that ϕ ∈ Hq(β−1) and ‖ϕ‖ ≤ δ, where1/p + 1/q = 1. It is sufficient to show that d(ϕ) < ε. Since d(ϕ) ≤ ‖ϕ‖, if ‖ϕ‖ < ε, thend(ϕ) ≤ ‖ϕ‖. So suppose that ‖φ‖ ≥ ε. For each subset E of N0(= N ∪ {0}), let

nϕ(E) =

⎛

⎝

∑

j∈E

∣

∣ϕ(

j)∣

∣

qβ−q(

j)

⎞

⎠

1/q

, (2.6)

where

ϕ(z) =∞∑

j=0

ϕ(

j)

zj . (2.7)

For any subsets E1 and E2 of N0 we have that

nqϕ(E1 ∪ E2) =

∑

j∈E1∪E2

∣

∣ϕ(

j)∣

∣

qβ−q(

j)

≤∑

j∈E1

∣

∣ϕ(

j)∣

∣

qβ−q(

j)

+∑

j∈E2

∣

∣ϕ(

j)∣

∣

qβ−q(

j)

= nqϕ(E1) + nqϕ(E2)

≤ (nϕ(E1) + nϕ(E2))q.

(2.8)

Hence,

nϕ(E1 ∪ E2) ≤ nϕ(E1) + nϕ(E2) (2.9)


for all E1, E2 ⊆ N0. Also if E1 ∩ E2 = ∅, then, by considering f, g with support, respectively, inE1 and E2, we get that fg = 0 and so

∣

∣ϕ(

f)∣

∣

∣

∣ϕ(

g)∣

∣ =∣

∣ϕ(

f, g)∣

∣ ≤ δ∥∥f∥∥∥∥g∥∥. (2.10)

By taking supremum over all such f and g with norm one, we see that

nϕ(E1) · nϕ(E2) ≤ δ. (2.11)

So either nϕ(E1) ≤ ε/4 or nϕ(E2) ≤ ε/4 whenever E1 ∩ E2 = ∅.For all E ⊆ N0 we have that

ε ≤ ∥∥ϕ∥∥ = nϕ(N0) ≤ nϕ(E) + nϕ(N0 \ E). (2.12)

Thus, we get that

nϕ(N0 \ E) ≥ ε − nϕ(E). (2.13)

Since (N0 \ E) ∩ E = ∅, as we saw earlier, it should be nϕ(E) ≤ ε/4 or nϕ(N0 \ E) ≤ ε/4and equivalently it should be nϕ(E) ≤ ε/4 or nϕ(E) ≥ 3ε/4 for all E ⊆ N0.

Note that, if E1, E2 ⊆ N0 with nϕ(Ei) ≤ ε/4 for i = 1, 2, then

nϕ(E1 ∪ E2) ≤ nϕ(E1) + nϕ(E2) ≤ ε

2. (2.14)

Thus, the relation nϕ(E1 ∪ E2) ≥ 3ε/4 is not true and so it should be

nϕ(E1 ∪ E2) ≤ ε

4. (2.15)

Since ‖ϕ‖ > ε, clearly there exists a positive integer n0 such that nϕ(Sj) > ε for all j ≥ n0,where

Sj ={

i ∈ N0 : i ≤ j}

(2.16)

for all j ∈ N0. Now, let nϕ({i}) ≤ ε/4 for i = 0, 1, 2, . . . , n0. Since nϕ(S0) ≤ ε/4 andnϕ({1}) ≤ ε/4, nϕ(S1) ≤ ε/4. By continuing this manner we get that nϕ(Sn0) ≤ ε/4, whichis a contradiction. Hence there existsm0 ∈ Sn0 such that nϕ({m0}) ≥ 3ε/4. On the other hand,since (N0 \ {m0}) ∩ {m0} = ∅, nϕ({m0}) ≤ ε/4 or nϕ(N0 \ {m0}) ≤ ε/4. But nϕ({m0}) ≥ 3ε/4,and so it should be nϕ(N0 \ {m0}) ≤ ε/4.

Abstract and Applied Analysis 5

Remember that fj(z) = zj for all j ∈ N0. Now we have that

∣

∣ϕ(

fm0 , fm0

)∣

∣ =∣

∣

∣ϕ(

f2m0

)

− ϕ(fm0

)

ϕ(

fm0

)

∣

∣

∣

=∣

∣

∣ϕ(

fm0

) − ϕ2(fm0

)

∣

∣

∣

=∣

∣ϕ(

fm0

)∣

∣

∣

∣1 − ϕ(fm0

)∣

∣

=∣

∣ϕ(

fm0

)∣

∣

∣

∣1 − ϕ(fm0

)∣

∣

≤ δβ2(m0).

(2.17)

Therefore,

∣

∣ϕ(m0)∣

∣β−1(m0)(

β−1(m0)∣

∣1 − ϕ(m0)∣

∣

)

≤ ε2

16, (2.18)

and so

nϕ({m0})(

β−1(m0)∣

∣1 − ϕ(m0)∣

∣

)

≤ ε2

16. (2.19)

But nϕ({m0}) ≥ 3ε/4, and thus

β−1(m0)∣

∣1 − ϕ(m0)∣

∣ ≤ ε

12. (2.20)

Define ψ(z) = zm0 . Then ψ ∈ Hp(β), and we have that

∥

∥ϕ − ψ∥∥ =

∥

∥

∥

∥

∥

∑

n/=m0

ϕ(n)zn +(

ϕ(m0) − 1)

zm0

∥

∥

∥

∥

∥

=

(

∑

n/=m0

∣

∣ϕ(n)∣

∣

qβ−q(n)

)1/q

+ β−1(m0)∣

∣1 − ϕ(m0)∣

∣

= nϕ(N0 \ {m0}) + β−1(m0)∣

∣1 − ϕ(m0)∣

∣

≤ ε

4+ε

12< ε.

(2.21)

Thus, indeed d(ϕ) ≤ ε, and so the proof is complete.

Disclosure

This is a part of the second author’s Doctoral thesis written under the direction of the firstauthor.


References

[1] K. Jarosz, “Almost multiplicative functionals,” Studia Mathematica, vol. 124, no. 1, pp. 37–58, 1997.[2] B. E. Johnson, “Approximately multiplicative functionals,” Journal of the London Mathematical Society,

vol. 34, no. 3, pp. 489–510, 1986.[3] T. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,” Journal of

Mathematical Analysis and Applications, vol. 246, no. 2, pp. 352–378, 2000.[4] R. Howey, “Approximately multiplicative functionals on algebras of smooth functions,” Journal of the

London Mathematical Society, vol. 68, no. 3, pp. 739–752, 2003.[5] K. Jarosz, Perturbations of Banach algebras, vol. 1120 of Lecture Notes in Mathematics, Springer, Berlin,

Germany, 1985.[6] B. E. Johnson, “Approximately multiplicative maps between Banach algebras,” Journal of the London

Mathematical Society, vol. 37, no. 2, pp. 294–316, 1988.[7] F. Cabello Sanchez, “Pseudo-characters and almost multiplicative functionals,” Journal of Mathematical

Analysis and Applications, vol. 248, no. 1, pp. 275–289, 2000.[8] S. J. Sidney, “Are all uniform algebras AMNM?” The Bulletin of the LondonMathematical Society, vol. 29,

no. 3, pp. 327–330, 1997.[9] B. Yousefi, “On the space lp(β),” Rendiconti del Circolo Matematico di Palermo, vol. 49, no. 1, pp. 115–120,

2000.[10] K. Seddighi and B. Yousefi, “On the reflexivity of operators on function spaces,” Proceedings of the

American Mathematical Society, vol. 116, no. 1, pp. 45–52, 1992.[11] A. L. Shields, “Weighted shift operators and analytic function theory,” in Topics in Operator Theory,

vol. 13, pp. 49–128, American Mathematical Society of Providence, Providence, RI, USA, 1974.[12] B. Yousefi, “Unicellularity of the multiplication operator on Banach spaces of formal power series,”

Studia Mathematica, vol. 147, no. 3, pp. 201–209, 2001.[13] B. Yousefi, “Bounded analytic structure of the Banach space of formal power series,” Rendiconti del

Circolo Matematico di Palermo. Serie II, vol. 51, no. 3, pp. 403–410, 2002.[14] B. Yousefi and S. Jahedi, “Composition operators on Banach spaces of formal power series,” Bollettino

della Unione Matematica Italiana. B, vol. 8, no. 6, pp. 481–487, 2003.[15] B. Yousefi, “Strictly cyclic algebra of operators acting on Banach spaces Hp(β),” Czechoslovak Mathe-

matical Journal, vol. 54, no. 129, pp. 261–266, 2004.[16] B. Yousefi and R. Soltani, “On the Hilbert space of formal power series,” Honam Mathematical Journal,

vol. 26, no. 3, pp. 299–308, 2004.[17] B. Yousefi, “Composition operators on weighted Hardy spaces,” Kyungpook Mathematical Journal,

vol. 44, no. 3, pp. 319–324, 2004.[18] B. Yousefi and Y. N. Dehghan, “Reflexivity on weighted Hardy spaces,” Southeast Asian Bulletin of

Mathematics, vol. 28, no. 3, pp. 587–593, 2004.[19] B. Yousefi, “On the eighteenth question of Allen Shields,” International Journal of Mathematics, vol. 16,

no. 1, pp. 37–42, 2005.[20] B. Yousefi and A. I. Kashkuli, “Cyclicity and unicellularity of the differentiation operator on Banach

spaces of formal power series,” Mathematical Proceedings of the Royal Irish Academy, vol. 105, no. 1, pp.1–7, 2005.

[21] B. Yousefi and A. Farrokhinia, “On the hereditarily hypercyclic operators,” Journal of the Korean Math-ematical Society, vol. 43, no. 6, pp. 1219–1229, 2006.

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Approximately Multiplicative Functionals on the Spaces of ... · g n zn ∞ n 0 f n g n zn 2.5 is a commutative Banach algebra that is AMNM. Proof. First note that clearly Hp β is