Rev.Acad.Canar.cienc., XI (Núms. 1-2), 57-70 (1999)
TWO PARANORM ANO THE SET OF ALL ENTIRE
DIRICHLET SERIES
D. K. Bhattacharya and S. Manna.
ABSTRACT : In this paper, the notion of two-paranorm is introduced, and it is shown that the
set of all entire Dirichlet series with different exponents possesses such two-paranorm structure.
Next, the concept of Saks space of a two-paranorm space is introduced, and it is proved that
the aforesaid two-paranorm space possesses a Saks space. Further, the importance of this
Saks space is highlighted in the form of an interesting proposition. Lastly, the definition of a
two-paranormed algebra is given and the existence of different types of two-paraalgebraic
structures on n is investigated under suitable compositions and paranorms.
1. INTRODUCTION : The study of the set of entire Dirichlet series with different exponents
was made for the first time by R. K. Srivastava (1990). He studied it as a two norm space and
as a two norm algebra. As he considered only those Dirichlet series f = ( ª" , "-n ), where
~
f(s)= ¿anexp(A.ns), S=cr+it , O<A., < A.2 < 00
·; "-n ~ oo, as n~ oo and
where D = lim (logn/ I ") < oo , so the most interesting type of Dirichlet series for which D = + oo
n~ ~
[Gopala Krishna, 1980) could not be accomodated in the discussion. Moreover, dueto the
special choice D < oo , the two norms considered by Srivastava were always found to be
homogeneous. Further, the norms were algebra norms also.
In the general case when D ;t O for f = ( ª" , "-n} en , one or both the norms eould be
nonhomogeneous. Such nonhomogeneous norm was called F-norm by W.Orlicz, 1950 and
paranorm by Chaudhury and Nanda, 1991 . Collatz, 1966 showed that a paranórm could always
induce a supermetric on a linear space, and that the concept of derivative of an operator could
be generalized upto a linear space equipped with a supermetric. So it was realised that, in the
general case when D ;t O, the study of n w~uld remain interesting from the application point of
view, even if it could be shown that n was at least a paranorm space. This was highlighted by
Bhattacharya and Manna, 1997. Again, the importance of a two norm space and its Saks
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space was shown by mathematicians like A. Alexiewicz, 1954 and sorne others. Therefore,
the study of nas a two-paranorm space and as a two-paranorm algebra might deserve special
consideration.
Keeping this in mind, we study and consider its following subclasses :
(i) W0 ={te v/t,la.I exists} (ii) n , ={ten./ D = ~~ (logn A.0 ) = + oo}
where for f = (a •• A.n) En ' D '#-o ' "'·E E •. Card (En)= 1. (a fixed integer) :
further if x. = ,':'!i~"(A.n) and if {a., / \n} is convergent, then
Moreover, f = (a., A.n) , g = (bn, μn) , f,g e n are taken equivalent (f = g)
if (a. /A..) = (b. / μ0 ) , \;/ n . Lastly, n is considered as the set of ali equivalent
classes n, = {g / g = f}
2. rwo PARANORMED SPACE ;
Definition 2.1 Let X be a linear space over e
A functi9n p: X---> 1 IR.U{O} is called a paranorm on X, if the following holds :
(i) p(x) = 0 iff X= 9 (ii) p(-X) = p(x) (iii) p(x -f- y)~ P(x) + p(y) , X , y E X
(iv) lf {A..} e C and A. e C be such that ¡1.." -A.I ~O as n ~ oo and
if{x.}cX andx eXbesuchthat p(x. - x)~o asn~ oo. then
p(t... x. - A. x) ~o as n ~ oo.
Definition 2.2. Let X be a linear space equipped with two paranorms p and p* where p* is
weaker than p. A sequence {x.} e X is said to be y -convergent to X0 e X if there exists a real
number K >O such that sup p(xn) ~ K < oo and p'(xn - x) ~O as n ~ oo .
n
The linear space (X,p,p*) is said to be a two-paranormed space if convergence of a sequence
of X means its y - convergence.
Definjtjon 2.3. A sequence {x"} of a two-paranormed space (X,p,p*) is called.a Cauchy sequence
if there exists a K > O such that sup p(xn) ~ K < oo and corresponding to e> o, arbitrarily
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small, there exists a positive integer m such that p'(x, - x,) <e , for i , j ~ m .
Definition 2.4. A two-paranormed space (X,p,p*) is called y -complete if every Cauchy sequence
is y -convergent in (X,p,p*).
A y -complete two-paranormed space is always complete with respect to the weaker paranorm.
Definition 2.5. A two paranormed space (X,p,p*) is said to be normal if for every
{ xn} e X , xi: X · for which ~i~ p'(xn - x) =O, it is implied that p(x) < ~i~ inf p(xn).
Tbeorem 2.1. Let f = (an ,t...n)' g = (bn ,μn) belong to n. and z E e .
Let f + g, zf be defined as
Let p, p* : W0 ~ 1 R·u{o} be defined as
lanl • (f} lan A.ni
p(f) = sup - , p = sup (I I)
n ~ n J+ ~ ~
Then ( n0 , p , p') is a normal y -complete two paranormed space.
frQQf : lt follows from [ Theorem 3.1. Bhattacharya and Manna, 1997) that n0 is a complex
linear space. Again p is obviously a norm on n0 and p* is weaker than p. But p* is not
homogeneous. We now show that p* is a paranorm. In fact,
p*(zf; - zf) = p'(z(f; -t)) :5 I~ p'(f; -f) , I~ ~ 1 where {f;} e n0 , f e n0
So p'(f; -f) ~o=> p'(z f, - z t) ~o as i ~ ~ , v z e e if I~ ~ 1
Again p'(z t) < p'(f)' V z f, e if I~ < l.
Hencep' (f;- f)~O => p'(zf; - zf)~oasi~oo. VzeC if l~<l
Thus p' (f; - f) ~ o=> p' ( z f, - z t) ~ o as i ~ oo, V z e e
Next let z, ~ z as i --+ CXl where { Z¡} e e and z E e . We show that
p· ( z, f - z t)-+ o as i -+ro , v f e n •.
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So p'(z; f - z t) ~o as i ~ oo for each f En •. Thus p* is a paranorm.
Now we show that (n. , p , p·) is y-complete. Let {t¡}={(ain', A.in)}cn0 be a Cauchy
sequence. Then sup ( s~p lainl/ A.in) ~ K < oo , K > O.
Further, corresponding to E> o , arbitrarily small, there exists sorne m > O such that
p'(fp - t.) < e, V p , q ;:>: m
i.e. s~p 1 (apn / A.pn)-(aqn /A. qn) 1 < E' , E'= E' (E) , p,q ;:>: m (2.1)
Sol (apn / A.pn)-(aqn / Aqn)1 .< E', p,q ;:>: m andeachn=1 , 2,3,. ....
Thus {(ai,,./ A.in)} is a Cauchy sequence of complex numbers for n = 1, 2, .....
Let (ajn/A.jn )~an asJ~ oo. n = 1,2,3, .... . ....
We want to show that f = (an , xn) E Q0 such that fi ~fas i ~ oo.
Hence corresponding to (l/an) (a> 1 integer) , there exists a positive integer in such that
(2.3)
(2.4)
(2.5)
< I 1 R(ain) I · Vi;:>: in [by suitable choice of a]
1
As f 1 R(ajn) 1 converges for each i. so I 1 R(an) 1 converges.
1 1
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~
lt may be similarly shown that ~ 1 I(an) 1 converges. Hence ~ lanl converges
(2.6)
log 1 tak 1 log 1f(a;k, 1.;k)-(o;k, ak) 1
Now lim k-n = lim -~k·_n -----~ (2.7)
.- log 1 t.a¡k 1
:5 h m [ for suitable choice of a )
n-+ xo xn
r 1 ~ 1 ]
1og Lªik
= - oo as xn = m.in '-;n and lim k·n = - oo
1 n ..... ~ A~
(2.7)
lt follows from (2.6) and (2.7) that f = (an. x~) e n0 • We show that f; ~fas i ~ oo . As
f; = (a;n/f";n) so f; -f = (((a;n/'-;n)-(an/xn)) xn. xn) . Hence
. ! (a¡n l1n) -(anxn)I
p (t1 - t) = sup~----~
n 1 + 1 (a Jn 1 Jn} - ( an Xn) 1
Since (a;n /'-;n) ~ an as i ~ oo, so corresponding to e> O , arbitrarily small, there exists sorne
positive integer in such that i >in' n = 1 , 2, .....
(2.8)
so it follows from (2.8) and the given assumptions on {(a;n/'-;n)} that
1 (a;n/A.,n ) - (an / xn) 1 < &3 , V j;:~ in· n = 1,2,3, ....
Thus p· (f; - t) ~O as i ~ oo. So n0 is y - complete.
Now we show that ( n," p , p') is normal. Let p*(f¡) ~ p*(f) as i ~ oo . Then
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1 (a., , A.,)-(a0 1 A0 )1 .
sup --+ O as 1 --+ oo
" 1 + 1 (a., A.,) - (a. A0 ) 1
(2.9)
Now 1 (a10 / A10 ) 1~1 (a0 /A0 ) 1-1 (a10 / A,,)-(a./A0 ) ¡ , V n
So by (2 .1) \~'"!_! inf (a10 /A10 )~ 1 (a0 / A0 ) ¡. V n .
Thus p· (f, - f)--+ O as i--+ oo :::) p(f) < !~~ inf p(f,)
Hence(n11,p ,p") is normal. This completes the proof of theorem 2.1 .
Theorem 2.2. Let f = (a.,1.) , g; (b. m.) belong to n 11 and z e C;
let f + g, zf be defined as in theorem 2.1. Let p, p· :n 0 --+ IR 'U{O} be defined as
< X
P(f) = L la.I • p'(f) = L la.l/A • .
n=I
Then (n11 ,p ,p") is a y ·complete two paranormed space.
Proof: In this case p and p* may be preved to be norms (and hence paranorms). The
rest of the proof follows as in theorem 2.1.
Theorem 2.3. Let f = (a.,1.) , g ; (b. m.) belong to n, ; let f+g, zf be defined as in
theorem 2.1. Let p, p· :n,--+ IR'U{O} be defined as
1 / 1
• ( ) la.A. ni
p(f) = sup a0A0 n , p f = sup I( I
" " 1 + a n An n)
Then (n,,p ,p") is a y-complete two-paranormed space.
frQQf: As in theorem 2.1. n, may be shown tobe a linear space over C.
Now pis well defined as L: la.I is convergent and D = + oo:::) (A./logn) < 1 .
1
So (A0 / n) < < 1. Now p(f) = O iff f = e. Again p(f+g) ~ p(f) + p(g), as
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p(f+g) = sunp 1 ((an / "-n) + (bnf μn))(xn' / n) 1
!> s~p 1 (a0/A.0 )(x02 /n)1+s~p1 (b0 / μ0 )(x0
2 / n)1
!> s~p 1 (a0 x0 /n) 1 + s~p 1 b0 x0 /n 1 ~ s~p 1 (á0 A.0 / n) 1 + sup 1 (b0 μnfn) 1
So p(f + g) !> p(f) + p(g) . Also pis found to be a norm on n, .
Next p* is well defined and moreover p'(f) < P(f) , V f E n,. Again p* (f) =O
if and only if f =e . Moreover, it follows from standard inequality that p' (f + g) !> p· (f) + p' (g).
But p is not homogeneous. In fact,
1 (a0 l0 n) 1
Hence p'(zf) = lzlsup I I !> lzl p(f) if lzl ~ 1
n 1+ (anln n)
(2 .10)
(2.11)
So p'(zf) ;t I ~ p'(f) ' z E e
We now verify that p* is actually a paranorm on n, .
By(2.2) p'(zf; -zf)!>lzl p'(f;- f) when 1~~1
By (2.3) p'{zf; - zf) < p'{f; - f) when lzl < 1.
So p • ( f, - f) ~ o as i ~ oo => p • ( zf; - zf) ~ o as i ~ oo , v z E e .
So z; ~zas i ~ oo => p(z;f - zf) ~o as i ~ oo , v f E n,.
Hence pis a paranorm on n, . Proceeding as in theorem 2.1., it may be s~own that ( n,. p , p')
is also y -complete. So (n,.p ,p') is a y-complete two-paranormed space over C.
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3. SAKS SPACE
Definition 3.1. Let (X,p,p*) be a two-paranormed space. Let X, be a closed unit ball in (X,p,p*)
with respect top, i.e. X,= {x e X/p{x) :51}.
Further, Let X, be complete with respect to p*. Then (X,p,p*) is called a Saks space.
Proposjtjon 3.1. Let f, g e 0 0 , z e C ;let f+g, zf be defined as theorem 2.1.
Then (n0 ,p ,p') contains a Saks space.
By definition 0 0, is a closed unit ball of 0 0 -. Also it is known from theorem 2.1. that ( n0 , p , p·)
is a normal y -complete two-paranormed space. Now normality of (n0,p ,p') is equivalent to
closedness of n 0, in weaker paranorm p* and y -completeness of (n0 ,p ,p') implies the
completeness of n •• with respect to the metric d(f,g) = p*(f-g). Hence (n11 ,p ,p') is a Saks
space.
Proposjtjon 3.2. Let f,g, f+g, zf, p, p* be defined as in Proposition 3.1 . Let K(f;r) denote the
open ball with respect to the paranorm p* with centre f and radius r ; i.e .K(f,r)
= {g e n 0 / pº (g-f) < r} . Given f0 = ( ª"·, A.n, ) e 0 0 , and p >O, it is always possible to determine
8 > O such that every element of !10, íl K( 8_ , 8) [ 8 = (O , x") en0, . null element of 0 0 ] . can
be represented as the difference of two elements of K(t0 ; p) .
.PrQQf: Let us construct a sequence {(bn / μn)} as follows :
(bn / μn) = (an) A.n, )-(8/1-8) foreach n, where 28 <p.
Let g = ( (b" x" / μ") , x") , then
Hence ge K{t0 ; p) , let f e n 0, ílK(e ; p) and let us take the Dirichlet series h = f + g.
We now show that h e K{t11 ; p). Now
p* (h -f0) = p'(f +g-f11 ) :5 p"(f)+p"(g-f11 ) <8 +8 = 28 < p . So h e K{f0 ; P) .
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Hence f is expressed as the difference of h and g of K(f0 ; p ).
Remark 3.1. lt is clear from Proposition 3.2 that 8 can be chosen in infinitely many
ways. A stronger result also holds for n 0 , viz.
Given fil En •• and p > o . every element o_f n •• n K(0 ; o). where o< 28 < p can be
represented as the difference of two elements of K(f0 ; p ).
Remark 3.2. For every f0 E n 0 , , there exists a 8 > O such that p*(f) < 8 implies that
f0 + f E Ü 05 , for every f E Ü 0 , •
4. PARAALGEBRAIC STRUCTURE :
Definition 4.1. An algebra X over a field F is called a paranormed paraalgebra over F if
X is a paranormed space over F with paranorm p and if multiplication is a continuous
operation ; i.e. if
(i){xn}cX, XEX, p(xn-x)~O=>P(xny -.:... xy)~oasn~oo . '</ yi;X
(ii) {Yn}cX, y E X, P(Yn -y)~O=>p(xyn-xy)~o as n~ oo, '</ x E X
Definition 4.2. Let X be an algebra over F. X is called a paranormed algebra over F if X
is a paranormed space over F and if
p(xy) ~ p(x) P(Y) , '</ X , y E X .
Example 4.1 . Let f = (an,A.n). g = (bn,μn) belong to n. and z E C. Let f+g, zf be defined
1 I
r, M
as in theorem 2.1 . Let p(f) = s~p (an/A.n) ,rn > O; let
Then (X,p) is a complete paranormed algebra over C .
Verification : As in theorem 2.1. it is shown that n 0 is a linear space over C. Now p(f) = O
if f = 0. Also
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p(-f) = p(f). But pis not homogeneous. We now show that pis a paranorm.
In fact, for {f,} e n 0 , f, = (a;n• '- ;n) p(zf; - zf) = lzl '· M p(f; - f) .
So p( f, - f) ~ o ~ p( zf, - zf) ~ o as i ~ oo , v z i: e
Nextif { z,} e e z E e su ch that z, - z ~ o as i ~ 00 ' then
~(z, - z) t) = 1 z, - z ¡ '· M P(f) ~O as i ~ oo for each f i: 0 0 •
Thus pis a paranorm on n0 . To show that n0 is complete. let {f.} e n0 ,
f; = (a,,, i .. ," ) be a Cauchy sequence in n0 • Then given i: < O, arbitrarily small, there exists
an integer m > O such that p(f;-f¡) < i: , 'd i , j ~ m
So (a," ¡ i .. ,") is a Cauchy sequence of complex numbers. Now proceeding as in theorem
2.1. it may be shown that n0 is complete. So ( n0 ,p ) is a complete paranormed space.
We now show that fg is entire if f,g are so.
In fact, lim log J ~ (a,b, x, ) (J .. ,μ, ) 1
J(n
log : ~ (a.b,x.2) (i .. ,μ. ) 1
s lim -------~
log 1 k (a,x, ) /,,¡ log 1 ~ {b,x.) m,¡
s tim + :5lim =- oo
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= p(f) p(g)
So n 0 is a complete paranormed algebra over C
Obviously n 0 is a complete paranormed paraalgebra over C.
Example 4.2. Let f, g, E ºº' z E e ; let f+g, zf. fg be defined as in
Example 4.1. Let p(f) = . s~p 1 (a, , >..,) 1 / [1+1 (a, >..,) 1 J.
Then n 0 is a complete paranormed paraalgebra which is not a paranormed algebra.
Verification : From theorem 2.1 and example 4.1 it follows that n 0 is an algebra over C which
is also a paranormed space. Now we show that multiplication is continuous with respect to p.
Let {f;} en. where
f, = (a,n /A.,,) and fe n 0 where f = (an / A.n) such that P(f, -f) ~O as i ~ oo;
then we show that P(f,g- fg) ~o as i ~ co , 'r:I g e n 0 .
as i~ co.
~1(a,n/A.,n)-(anfA.n)1~ 0 as i~co
~ 1 (a,n/A.,n)-(an /A.n) (bnxn / μn) 1 ~o as i ~<X)
[ as(xn/111,) < 1 an~bnl ~O when n ~ co, ~Jbnlbeing convergent]
So p(f,g -fg) = P((t, -f)g)~o asi~co. 'r:lgen0 •
Similarly it may be shown that
p(f, - t) ~o as i ~ co ~ P(gf, - gf) ~o as i ~ co , 'r:I ge n0 •
So multiplication is continuous with respect to p. Thus ( n 0 ,p ) is a complete paranormed
paraalgebra over C.
We now show that n 0 is not a paranormed algebra. In fact, for f = (ª",A.")
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So p(fg) $ p(f) p(g) does not hold for all f, g i: n • .
Hence ( n 11 • p ) is not a paranormed algebra.
Example 4.2. Let f = (a0 ,A.0 ) and g = (b0 ,μ0 ) belong ton, . Let f,g,zf, f+g be defined as
1 (anA.n) n 1
in example 4.1. Let p(f) = s~p 1+1 (a"A.") n 1
Then n , is a complete paranormed paraalgebra which is not a paranormed algebra
over e .
Proof follows as in example 4.1 and example 4.2.
5. TWO PARAALGEBRAIC STRUCTURE
Definition 5.1. Let X be a complex algebra which is also a two-paranormed space with
respect to each of the paranorms p and p* where p* is weaker than p. Then (X,p,p*) is
called a two-paranormed paraalgebra over C, if the operation of multiplication is
continuous with respect to both p and p*.
Definition 5.2. A two-paranormed paraalgebra (X,p,p*) is said to be of type 1,1 1,111
respectively according as p and p* satisfy the following additional properties :
(i) p(xy) $ p(x) p(y)
(ii) p*(xy) $ p*(x) p*(y)
(iii) p(xy) $ p(x) p(y) ; p*(xy) ~ p*(x) p*(y)
A two-paranormed paraalgebra of type III is also called a two-paranormed algebra.
Example 5.1. Let f, g E n •. z E e ; let f+g, zf, p, p* be defined as in
theorem 2.1 ; let fg = ((a0b0 x0 )/(l.. 0 μ0 ) , x.) , then (n0 ,p,p') is a y-complete two-paranormed
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paraalgebra of type J.
Verificatjon: As in theorem 2.1. it may be shown that (n0 ,p,p·) is a y-completetwo-paranormed
space. Also from example 4.1. it follows that fg is entire if f and g are so. lt is ultimately found
that ( n0 , p, p·) is an algebra witti'. the given compositions. lt may be easily verified that p(fg)
s P(f) P(g) , 'd f, g 1: n 0 . Also it ·follows from example 4.2 that multiplication is continuous with
respect to p*. Hence ( n, .. p, p·) is a y-complete two-paranormed paraalgebra of type I.
Example 5.2. Let f,g 1: n 0 , z i; C ; f+g, zf, p, p* be defined as in theorem 2.2 and
let fg = ( ( a"b"x" ) {f .. "μ") , x") . Then ( n0 , p, p·) is a y-complete two paranormed algebra over C.
Example 5.3. Let f,g, be defined as in theorem 2.3 and let fg = ((anbnxn)/ (A.nμn) , xn) . Then
(n,, p,p' ) is ag -complet~ two-paranormed paraalgebra over C.
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Department of Pure Mathematics
UNIVERSITY OF CALCUTT A
35, Ballygunge Circular Road
Calcutta - 700 019 .
.INDIA.
Email : telemail@Cal.vsnl.net.in
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