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F a S C I C U L I M a T H E M a T I FASCICULIMATHEMATICI Nr 35 2005 Mushtaq Shaker Abdul-Hussein and G.S. Srivastava A STUDY OF BORNOLOGICAL PROPERTIES OF THE SPACE OF ENTIRE FUNCTIONS REPRESENTED BY MULTIPLE DIRICHLET SERIES Abstract: The space of entire functions represented by Dirichlet series of several complex variables has been studied by S. Dauod [1]. M.D. Patwardhan [6] studied the bornological properties of the space of entire functions represented by power series. In this work we study the bornological aspect of the space ¡ of entire functions represented by Dirichlet series of several complex vari- ables. By ¡ we denote the space of all analytic functions P1 ®(s1; s2) = am;n exp(¸ms1 + ¹ns2), having ¯nite abscissa of m;n=1 convergence. We introduce bornologies on¡ and ¡, and prove that ¡is a convex bornological vector space which is the completion of the convex bornological vector space ¡. Key words: convex bornological vector space, Dirichlet series. 1. Introduction Let C be the ordinary complex plane equipped with its usual topology and ¡ be the space of entire functions represented by Dirichlet series of two complex variables (s1; s2). (We consider the case of two variables for the sake of simplicity, though our results can be easily extended to any ¯nite number of variables). Let X1 (1.1) ®(s1; s2) = am;n exp(¸ms1 + ¹ns2) m;n=1 2 where am;n 2 C, s1; s2 2 C , s! = σ! +it!, ! = 1; 2, 0 < ¸1 < ::: < ¸m ! 1 with m, 0 < ¹1 < ::: < ¹n ! 1 with n and further (see [1]) m + n (1.2) lim sup = D < +1: m+n!1 ¸m + ¹n 136 M.S. Abdul-Hussein, G.S. Srivastava Such a series is called a Dirichlet series of two complex variables. If ¸m = log m; ¹n = log n, then ®(s1; s2)is a power series in (s1; s2). F.I. Geche [2] considered the general case when (¸m)m¸1; (¹n)n¸1 are two increasing sequences of real numbers whose limits are in¯nity, he determined the nature of the region of convergence of the series (1.1) and the formula for the system of associated abscissas of convergenceP which depends substantially on whether A < 1 or A = 1, whereA = j ai;j j. For example, V.P. Gromov i;j m+n [3] proved that if lim sup ¸ +¹ = 0 , then the formulae for the system of m+n!1 m n associated abscissa of convergence depend on j am;n j. It is well known that the function ®(s1; s2) is an analytic function in the hyperplane σ! < ¿(¡1 < ¿ < 1) where log j a j¡1 (1.3) ¿ = lim sup mn : m;n!1 ¸m + ¹n For each entire function ® " ¡ we associate a real number k ® k de¯ned by 1 k ® k= supfj am;n j (¸m+¹n) ; m; n ¸ 1g; satisfying for all ®; ¯ " ¡ (a) k 0 k = 0; (b) k ® k ¸ 0; (c) k ® k = 0 () ® ´ 0; (d) k ¡® k = k ® k; (e) k ® + ¯ k · k ® k + k ¯ k: Thusk ® k is a total paranorm on ¡. 2. De¯nitions In this section we give some de¯nitions. We have 2.1 A bornology on a set X is a family B of subsets of X satisfying the following axioms: S (i) B is a covering of X , i.e. X = B" BB; (ii) B is hereditary under inclusion, i.e. if A 2 B and B is a subset of X contained in A, then B 2 B; (iii) B is stable under ¯nite union. A pair (X; B) consisting of a set X and a bornology B on X is called a bornological space, and the elements of B are called the bounded subsets of X. A study of bornological properties of the space . 137 2.2 A base of a bornology B on X is any subfamily B0 of B such that every element of B is contained in an element of B0. A family B0 of subsets of X is a base for a bornology on X if and only if B0 covers X and every ¯nite union of elements of B0 is contained in a member of B0. The collection of those subsets of X which are contained in an element of B0 de¯nes a bornology B on X having B0 as a base. A bornology is said to be a bornology with a countable base if it possesses a base consisting of a sequence of bounded sets. Such a sequence can always be assumed to be increasing. For further de¯nitions and notations, we shall refer to [4] and [5]. 2.3 Let E be a vector space over the complex ¯eld C. A bornology B on E is said to be a vector bornology on E, if B is stable under vector addition, homothetic transformations andS the formation of circled hulls or, in other words, if the sets A+B, ¸A, j´j·1 ´A belong to B, whenever A and B belong to B and ¸ 2 C. Any pair (E; B) consisting of vector space E and a vector bornology B on E is called a bornological vector space. 2.4 A vector bornology on a vector space E is called a convex vector bornology if it is stable under the formation of convex hulls. Such a bornology is also stable under the formation of disked hulls, since the convex hull of a circled set is circled.A bornological vector space (E; B) whose bornology B is convex is called a convex bornological vector space. 2.5 A separated bornological vector space (E; B) (or a separated bornology B) is the one in which f0g is the only bounded vector subspace of E. 2.6 A set P is said to be bornivorous if for every bounded set B there exists a t 2 C, t 6= 0 such that ¹B ½ P for all ¹ 2 C for which j ¹ j · j t j. 2.7 Let E be a bornological vector space. A sequence fxng in E is said to be Mackey-convergent to a point x 2 E if there exists a decreasing sequence ft g of positive real number tending to zero such that the sequence f xn¡x g n tn is bounded. 2.8 Let E be a vector space and let B be a disk in E not necessarily absorbent Sin E. We denoteS by EB the vector space spanned by B, i.e. the space ¸>0 ¸B = ¸2K ¸B, where K be the real ¯eld R or the complex ¯eld C. 138 M.S. Abdul-Hussein, G.S. Srivastava 2.9 Let E be a separated convex bornological space . A sequencefxng in E is said to be a bornologcal Cauchy sequence (or a Mackey-Cauchy sequence) in E if there exists a bounded disk B ½ E such that fxng is a Cauchy sequence in EB. 3. The Space ¡ Let C denote the complex plane, and I be the set of positive integers. We write for n 2 I, n C = f(z1; z2; :::; zn); zi 2 C; 1 · i · ng: We are concerned here with the space of entire functions from Cn to C under the usual pointwise addition and scalar multiplication. We shall denote by ¡ the space of all entire functions ® : C2 ! C, where P1 ®(s1; s2) = am;n exp(¸ms1 + ¹ns2). m;n=1 We de¯ne a bornology on ¡ with the help of jj : jj de¯ned in sec- tion 1. We denote by Bk the set f® " ¡:k ® k· kg. Then the family B0 = fBk : k = 1; 2; :::g forms a base for a bornology B on ¡. We now prove Theorem 3.1. (¡; B) is a separated convex bornological vector space with a countable base. Proof. Since the vector bornology B on the vector space ¡ is stable under the formation of the convex hulls, it is a convex vector bornology. Since the convex hull of a circled set is circled, B is stable under the formation of disked hulls, and hence the bornological vector space (¡; B) is a convex bornological vector space. Now to show thatf0g is the only bounded vector subspaces of ¡, we must show that contains no bounded open set. Let U(") denote the set of all ® " ¡ such that k ® k < " . To prove the result stated, it is enough to show that no U(") is bounded. That is, given U("), we have to show that there exists U(´) for which there is no c > 0 such " that U(") ½ cU(´). For this purpose, take ´ = 4 . Given c, we can ¯nd 1 su±ciently large positive numbers ¸m and ¹n such that j c j (¸m+¹n) < 2. " (¸m+¹n) " Let ® = ( 2 ) exp(¸ms1 + ¹ns2). Then k ® k = 2 ; so ® " U(") and ¡1 ® " (¸m+¹n) " ¡1 k c k= supm;n¸1( 2 j c j ) > 4 = ´, so that c ® does not belong to U(´) i.e. ® does not belong to cU(´). This shows that U(") is not bounded. Thus f0g is the only bounded vector subspace of ¡, and hence (¡; B) is separated. Since B possesses a base consisting of increasing sequence of bounded sets, B is a bornology with a countable base. Thus (¡; B) is a A study of bornological properties of the space . 139 separated convex bornological vector space with countable base. This proves Theorem 3.1. ¥ Theorem 3.2. B contains no bornivorous set. Proof. Suppose B contains a bornivorous set A. Then there exists a set Bi 2 B such that A ½ Bi and consequently Bi is also bornivorous. We now assert that if i1 > i, then tBi1 * Bi for any t 2 C which leads to a contradic- tion.
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