Solution of operator equation and the space of entire Dirichlet series

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Rev.Acad.Canar.Cienc., VIII (Núm. 1), 31-39 (1996)
SOLUTION OF OPERATOR EQUATION AND THE
SPACE OF ENTIRE DIRICHLET SERIES
D. Bhattacharya
and
S. Manna
Abstract
Let n be the set of ali complex Dirichlet series ofthe form f(s) = E an exp( -.X.ns),
n=l
an E C, O ~ A¡ < .X.2 < .. . .X.n-> oo as n-> oo, An E En, Card(En) = ln (a fuced integer),
D = limn~00 (log lanl/ .X.n) # O. It has been proved that nis a nontrivial supermetric space
under suitable compositions. Further, conditions have been obtained for the existence of
solution of the operator equation T(f) = (), where T is a Frechet differentiable operator
on n and () is the additive identity of n.
1 Introduction
It is known that if A : R -+ R is a differentiable function, then an approximate solution of
A(X) = O is a obtained by Newton's method/simplified Newton's method. At the very outset,
it appears that Newton's method can be developed for an operator equation A(X) = B, provided
A : X -+ X is a differentiable operator on X. In fact, Newton's method for the s01Ution of
operator equation on a Banach space is well known- details are available in L. V. Kantarovich and
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G.P. Akilov, 1982. It may be remarked that although it is known that idea of differentiability of
operators is available even on a supermetric space, still Newton's method can not be developed
on a supermetric space, in general, because only differentiability of operators is not enough
to discuss convergence of a iterative sequences in Newton's method- Mean valuable inequality
involving the differentiable operators is another essential requirement for the purpose. As Mean
valuable inequality can be proved only when the supermetric space is a L-supermetric space,
so Collatz, 1966 could develop Newton's method on a L-supermetric space. Our purpose,
in this paper, is to use the results of Collatz to discuss solution of the equation Tf = B,
T: 0---> O; f E O.
Study of Dirichlet series with different exponents as normed linear spaces, was made, for
the first time, by R.K. Srivastava, 1990. He considered Dirichlet series of the form f(s) =
f a.,,exp( ->-ns), abbreviated as
n=l
f = (a.,,,>-n), D = lirun-->OO(log 1an1 / >-n) < oo. Naturally, the Dirichlet series for which
D = oo cloud not be considered. But it may be noted that de most interesting type of Dirichlet
series are those for which D = + oo, because only this type ofDirichlet can be convergent there.
For example,
f(s) = f ((-l)n/n) exp(- log logns), for which O"c = -oo, O"a = 1; where O"c and O"a are
n=2
the abscissa of the ordinary ánd absolute convergence.
In this paper, f = (a,,, >-n) and g = (bni μn) of n has been called equivalent
(J ~ g) if (a.,,/>-n) = (bn/ μn), \:/n. Further, O has been taken as the set of ali equiv-alence
classes O¡ = {g E O/g ~ J}.We have denoted by f, the equivalence class 0 1 whose
representative is f.
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2 Sorne known definitions:
Definition 2.1 The Dirichlet series f = (a,,,, An) is said to be entire, if
(i) f a,,, is convergent
n=l
~
log 1 L; ª• 1
(ii) lim •= = -oo
n-oo An
Definition 2.2 A metric space (X, p) which is also a linear space is called a supermetric space
if the linear operations are continuous with respect to the topology induced by the metric p and
if p satisfies an additional property p(x,y+z) = p(x -z,y) ,'ix,y,z E X.
Definition 2.3 A paranorm pon a linear space X is the function p : X----> R+ U {O} such that
(i) p(x) = Ü ijf X=(}
(ii) p(-x) = p(x)
(iii) p(x +y) :::; p(x) + p(y)
(iv) If {>-n} e C,.A E C such that l>-n - .Al----> O, as
n----> oo and {xn} C X, x E X, be such that p(xn - x)----> O as n----> oo, then
Remark 2.1 A supermetricspace/linear metric space/ metric linear space (X,p) is a para-normed
space (X, p) where p(x) = p(x,B),B is the additive identity of X. A paranormed space
(X,p) is always a supermetric space (X,p) where p(x, y)= p(x - y), x, y E X.
Definition 2.4 A supermetric space (X, p) is called a L-supermetric space if for every element
x E X, there exists a bounded linear functional L on X such that 11 L 11= 1, Lx = p(x,B), where
11 L 11= inf K(> O) such that \l(Lx,Ly) :::; Kp(x,y), 'ix,y E X; O' being the metric on the
scalar field.
Obviously, every Banach space is a L-Banach space. But there may be L-spaces which are
not L-Banach spaces. An example of such space is given by theorem 3.1.
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Definition 2.5 Let ( R, p) and ( S, O") be two complete linear supermetric spaces and let T be
an operator with a domain D in R anda range in S.
The operator T is called Frechet differentiable at the point f E D if there exists a bounded
linear operator H with a domain in D anda range in S anda real valued function s(ó) with the
property o-(Th - tf, H(h - f)) ~ p(h, f)s(p(h, !)) for a certain neighbourhood p(h, f) < K0 ,
where f~ s( 6) = O. The operator H = Tcn is called the Ftechet derivative of T at the point f.
Theorem 2.1 Let (R,p) be a L-supermetric space. Let D be a convex domain of R. Let
T : D ---+ be Prechet differentiable having T as the Préchet derivative on D. Let the itera-tive
sequence {Un} of approximate values far the solution of the equation Tu = B be given by
simplified Newton's method as un+I =Un -T,,;:1un,where T,;;1exists at u0 E D.
If p = 11 T,,;:1 11 sup 11 T -Y'¡ 11 < 1 and if with first two members u0 , u1 of {un} ,the sphere
JED
K = {v E D/p(v,u1 ) ~ (p/1- p)p(Uo,u1)} C D, then there exists in Da solution of the
equation Tu= B, where u lies also in K, Un remains in D and Un---+ u as n---+ oo.
3 L-supermetric and the space of entire Dirichlet series.
Theorem 3.1 Let !1 be the set of all entire Dirichlet series far which D =f. O. Then !1 is a
L - supermetric space {!&, p) if far f = (a.,,, >..n) and g = (bn, μn) of !&, z E C, f + g and zf are
and pis defined as p(f,g) =~~E <I>(e) iYiW€l~~(~ll where <1> is a bounded function on C with
<l>o = max <1>(0 = <1> (ea).
€EC
Proof: We first show that f + g is entire if f and g are so. In fact,
lim log lk~n ((ak / >..k) + (bk/μk))xk l ~
n-oo Xn
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if l~(akj,\k)J 2 l~(bk/μk)I
= -oo, as (a,,j>..n) ~ ((~/>..n)xn,Xn)·
Again if l~(bk/ μk) I > lt,,(ak />..k) ¡, then also the same result follows. Hence f +gis entire
if f and g are so. Next zf is entire, since
By standard inequality it follows that p(f + g) :::; p(f) + p(g), p(x) = p(x,B). p(f) 2 O,
p(f) =O iff f =B. But p(zf) =1 lzlp(f). Hence pis not homogeneous. In order to prove that p
is a supermetric, it therefore, remains to show that scalar operation is continuous with respect
to p. Let {ji} en, f En such that Íi ---t fas i ---t OO. We show that zfi ---t zf for each z E c.
Now
p(zf) p(zf,B) = sup <I>(s) lzf(s)l /(1 + lzf(s)I)
s
lzl sup<I>(s)lf(s)l/(1 + lzf(s)I)
s
:S: lzl p(f), lzl 2 l.
So, p(z(fi - f)) :S: lzl p(fi - f), lzl 2 l.
Hence zfi ---> zf when Íi ---> f, as i ---> oo, for each z E C.
When lzl < 1, p(z(fi - f)) < p(f; - f). Hence, for Íi ---> f, zf; ---> zf, as i ---> oo.
Thus f; ---> f '* zfi ---> zf as i ---> oo, for each z E C.
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Next we show that if {z;} e e, z E e be such that Z; -> z as i -> oo, then zd -> zf as
i-. oo for each f E O. For JzJ 2: 1, the result is obvious, as p((z; - z)J) ::::; Jz; - zJp(f) when
Jzl 2: l. When Jzl < 1,
p(zf) sup<l> (s)Jzf(s) J/ (1 + lzf(s)J) = lzJsup<l>(s)IJ(s)J/(1 + lzf(s) J)
s s
::::; JzJsup{<l>(s) JJ(s)J} ::::; JzJsup<l>(s)supJf(s)J = JzJ <I> (eo)sup Jf(s) J
s 8 s s
So p(( z; - z)J)::::; JzJ <I>(eo) sup. lf(s)J . Hence zd-> zf if z;-> z, as i-. oo.
This proves that pis a paranorm and so pis a supermetric on O. Now we prove that nis
complete with respect to p. Let {!;} = { a;m .\n} be a Cauchy sequence in O. Then
sup<l>(s) lf;(s) - f;(s)I < e, i, j 2 N (3.1)
s
where e > O is arbitrary and N is a positive integer depending on c. So
lf;(s) - fi(s) J < e' , for i, j 2: N (3.2)
Hence {f;(s)} is a Cauchy sequence in C. Let f;(s) -. f(s) in C, as i -. oo, where f =
(a,,, >-n), say. We show that f;-> f as i-> oo and that f E O.
Now keepingi fixed and talkingj-> oo in (3.2), Wehavesup8 lf;(s) - f(s)I < e', ifi 2 N.So
ultimately we have p(f;, f) -> O as i -> oo. Thus f; -> f as i -> oo. So
sup<l>(s) J(f; -f)(s)J < éo, i 2 M (3.3)
s
where c0 is arbitrary and the positive integer M depends on c0 • Now
So it follows from (3.3) that l~((a;n/A;n) - (a,,/ >.n))xnexp(-xns) I < e", i 2 M, say. Thus
J ( a¡n/ A;n) - (a,,/ >-n) J -> O as i -> oo. Hence, (a,,/ >-n) = ( a;n/ A;n) + D;m D;n E C , J8;nJ -. O as
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Now putting a1 = a2 =O and writing J(x,y,z) = La,b,m,n(x)ymzn in (2.5), we get
· L a,b,m,n ((X +a-6 -y+ -a3-by) ( 1 + -a4 +a-5-bz)) y m Zn .
a z a y y
But
(3.2)
nL+r Loo Loo Loo ( -a6 )' a5r ( -a4 )k ( -a3 )P
-------(n + l)r(-n - p-m + l )k ·
s! r! k! p! s=O r=O k=O p=O
. La,b,m+p-k-r+s,n+r-s( X )ym+p-k-r+s zn+r-s .
Equating the two results (3.1) and (3.2), we get
(3.3) ( ( a6 y a3 by) ( a4 a5 bz ) \ ·La b m n X+ -- + -- 1 + - + -- } =
" ' a z a y y
nL+r Loo Loo Loo ( -a6 )' a5r ( -a4 )k ( -a3 )P
-------(n + l )r (-n - p - m + l )k ·
· s! r! k! p! s=O r=O k=O p=O
· La,b,m+p-k-r+s,n+r-s(x )yp- k-r+s z r- s.
The above generating function <loes not seem to appear before. We can get a large number
of generating functions from (3.3) by attributing different values to a; ( i = 3, 4, 5, 6).
Before discussing the particular cases of (3.3), it may be pointed out that if at least one
of the all possible pairs of the operators A; ( i = 3, 4, 5, 6) be non-commutative then
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Theorem 3.2 Let n be the set of all entire Dirichlet series of the form f = (Cln, fn) where
D = lim n-=(logn/>.n) =f O. Let for g = (bn,μn) E O, z E C and a bounded function
<P on C, f + g, zf, p(f,g) be defined as in the theorem 3.1. Let D be a convex domain
of n. Let T : D --> n be Prechet differentiable having T as the Prechet derivative on D.
Let the itemtive sequence of approximate values for the solution of the equation Tf = B be
given by simplified Newton's method as fn+1 = fn - T¡;1 f n, where r'¡;:1 exists at f 0 E D. lf
P = llT¡;:1 ll sup/ED llr.'.0 -r;.11 < 1 and if with f 0 , f¡ as the first two members of Un} , the
sphere K = {g En / p(g, f 1) < (p/1 - p)p(f0 , f 1)} e D, then there exists in D a solution of
the equation Tf = B, where f lies also in K, fn remains in D and fn--> f as n--> oo.
Concluding Remarks: Although a Banach space is a L-Banach space, but the construc­tion
of the functional L may not be possible, as its existence is proved by Hahn Banach theorem
which depends on 2'.orn's lemma. Even when the Banach space is separable, it is very difficult
to find out L. To find out L corresponding to h =f B adds further difficulty. In fact, its existence
does not follow directly from Hahn Banach theorem. In case of a supermetric space existence
of Lis not assured as Hahn Banach theorem is not available. However, construction of L may
be tried directly. Even when the supermetric space is not separable, it may be impossible to
find out L.
References
[1] Bhattacharya, D.K.& Manna, S. "On the space of entire Dirichlet series'', Bull. Cal.
Math. Society, 86, 21-26 (1994).
[2] Bhattacharya, D.K.& Manna, S. "Supermetric and paranorm on the set of ali entire
Dirichlet series with different exponents", accepted for publication in GUMA, India.
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[3] Chaudhury, B. & Nanda, S. "Functional analysis with applications". Wiley Eastern
Ltd. New Delhi, Bombay, Calcutta, 1991.
[.i] Collatz, L. "Functional analysis and numerical mathematics" . Academic Press, New York
and London, 1966.
[5] Kantarovich, L.V. & Akilov, G .P. "Functional Analysis", Pergamon Pres.s, Oxford,
New York, Toronto, Sydney, París, Frankfurt, 1982.
[6] Srivastava, R.K. "Functional analytic structures for various classes of entire Dirichlet
series with variable sequences of exponents", Bulletin of the Institute of Math. Academic
Sínica, Vol. 18, nº 3, June 1990.
[7] Gopala Kr ishna, J. "Sorne basic results on general Dirichlet series and applications",
lndian Journal of Mathematics, Vol. 22, 3 (1980).
[8] Widder, D. V. "An introduction to transform theory", Academic Pres.s, New York, London
(1971).
Department of Pure Mathematics
University of Calcutta
35, Ballygunge Circular Road
Calcutta - 700 019
INDIA
39
Recibido: 24 Mayo 1996