Abstract

In this paper a class L of entire functions represented by Dirichlet series in n variables has been considered whose coefficients belong to the set of complex numbers C and is further proved to be a Banach Algebra. Also characterization of continuous linear functional is done for the set L.

Subject Classification. 30B50, 46J15, 47A10, 46A11, 54D65.

1. Introduction

Let

(1.1)

be a n-tuple Dirichlet series where

and . Also

To simplify the form of -tuple Dirichlet series, we have the following notations

and

Thus, the series (1.1) can be written as

(1.2)

Janusauskas in ¹ showed that if there exists a tuple such that

(1.3)

then the domain of absolute convergence of (1.2) coincides with its domain of convergence. Sarkar in ² proved that the necessary and sufficient condition for series (1.2) satisfying (1.3) to be entire is that

(1.4)

Consider L as the set of series (1.2) satisfying (1.3) and (1.4) for which

is bounded. Thus there exists a such that

This implies

Then every element of represents an entire function. Define the binary operations in as

The norm in is defined as

(1.5)

During the last two decades a lot of research has been carried out in the field of Dirichlet series and many important results have been proved where few of them may be found in ^{3, 4}. Kumar and Manocha in ^{5, 6} considered the condition of weighted norm for a Dirichlet series in one variable and established some results on it. Until now a lot work has been done for the Dirichlet series in one variable. The purpose of this paper is to give a broader view to the study of Dirichlet series in -variables.

2. Main Results

In this section main results have been proved. For the definitions of terms used, refer ^{7, 8}.

Theorem 1. L is a commutative Banach algebra with identity.

Proof. To prove the theorem we first show that is complete under the norm defined by (1.5). Define a metric on as and let be a Cauchy sequence in For each let be the m-th tail of sequence and be twice the diameter of . Also let be the closed ball centered at of radius . Then

Since the sequence is Cauchy therefore Now let be arbitrary. Therefore there exists such that

Suppose then

Therefore and hence .

In the like manner we construct a nested sequence of the closed balls . Then from hypothesis it is known that a space is complete if and only if every nested sequence of closed balls whose radii tends to zero has a non empty intersection say . Let be a Cauchy sequence in where

Then for given we can find such that

that is

Clearly forms a Cauchy sequence in for all values of .

Hence

Letting ,

Thus that is as Also

Hence . Thus is complete under the norm defined by (1.4). If then

The identity element in is

This completes the proof of the theorem.

Theorem 2. The function is invertible in if and only if

is a bounded sequence where is inverse of .

Proof. Let be invertible and

be its inverse. Then Therefore

which implies

This further implies

Equivalently

and is thus a bounded sequence since Conversely suppose

be a bounded sequence. Define such that

Further

Hence the theorem.

Theorem 3. A necessary and a sufficient condition that an element to be a topological zero divisor is that

Proof. Let the given condition holds. Construct a sequence such that

Thus, for all , and Now

Therefore

As ,

Thus is a topological zero divisor.

Conversely, suppose the given condition is not true that is

Then, given with we can find integers such that for all ,

hold true. Also since is a topological zero divisor, there exists a sequence of elements in with unit norm such that for all one has

where

Next, for such that there exist integers and subsequences of sequence of indices such that

for all This implies

for all Therefore

which is a contradiction. Hence the theorem.

Theorem 4. is not a Division Algebra.

Proof. Let

Clearly and does not possess an inverse in . Let if possible

be its inverse. Hence This implies

which does not belong to L. This completes the proof of the theorem.

Theorem 5. Every continuous linear functional is of the form

where

and is a bounded sequence in .

Proof. Let us first assume that be a continuous linear functional. Since is continuous,

where

Let us define a sequence as

Therefore,

Since is a linear functional therefore

We now show that is a bounded sequence in ,

and which further implies

Thus is a bounded sequence in . This proves the theorem.

References