Laplace Transform in Commutative Banach Algebras Citation Dean, Robert Yost (1952) Laplace Transform in Commutative Banach Algebras. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/R8JF-GT33. https://resolver.caltech.edu/CaltechTHESIS:11202017-144107731 Abstract After a brief review of the general theory of commutative complex Banach algebras in Section I, Section II introduces and discusses some important facts about the generalized Laplace-Stieltjes integral. Section III consists of an investigation of the regions of ordinary and absolute convergence of the Laplace-Stieltjes integral, and is followed by specializations to Dirichlet and power series in Section IV. Section V contains a consideration of the analyticity of functions defined by Laplace-Stieltjes integrals, while Section VI concludes the thesis with some remarks on the existence and distribution of singularities of such functions. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: (Mathematics and Physics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Minor Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Bohnenblust, Henri Frederic Thesis Committee: Unknown, Unknown Defense Date: 1 January 1952 Record Number: CaltechTHESIS:11202017-144107731 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:11202017-144107731 DOI: 10.7907/R8JF-GT33 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 10561 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 20 Nov 2017 23:03 Last Modified: 10 May 2023 23:31 Thesis Files Preview PDF - Final Version See Usage Policy. 18MB Repository Staff Only: item control page

LAPLACE TRANSFORM IN COMMUTATIVE BANACH ALGEBRAS Thesis by Robert Y. Dean In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena , California 19.52 ACKNOWLEDGMENT The author wishes to express his sincerest thanks to Professor H. F. Bohnenblust for his most valuable suggestions and criticisms, for his patience and understanding, and for the generous assistance he has given during the development of this thesis. ABSTRACT After a brief reviei.v of the general theory of commutative complex Banach algebras in Section I, Section II introduces and discusses some important facts about the generalized Laplace-Stieltjes integral. Section III consists of an investigation of the regions of ordinary and absolute convergence of the Laplace-Stieltjes integral, and is followed by specializations to Dirichlet and p~Ner series in Section IV. Section V contains a consideration af' the analyticity of functions defined by Laplace-Stieltjes integrals, while Section VI concludes the thesis with some remarks on the existence and distribution of singularities of such functions$ TABIE OF 001'J'TENTS PAGE SECTION I INTRODUCTION 1 II THE LA.PIACE-STIELTJES IlITEGRA.L 10 III REGIONS OF CONVERGENCE OF THE LAPIACE-STIELTJES INTEGR..\L 19 IV DIRICHLET AND PO~VER SERIES 41 v THE LA.PLACE-STIELTJES INTEGRAL AS AN ANALYTIC FUNCTION 44 VI SINGULARITIES ON THE BOIDIDARY OF THE REGION REFERENCES (Jc 57 65 -1- Section I INTRODUCTI ON During the past decade and a half increasingly more and more mathematical attention has been concentrated on the theory of those mathematical systems which have become commonly known as normed linear rings or Banach algebras. As early as 1934, in a generalization of Fredholm integral equation theory, Michal and Martin[1] laid down the postulates and gave several infinite dimensional examples of what they called "a special linear vector space S • • • having additional properties abstracted from those of a space of linear transformations". Their postulates are identical with those now used for a Banach algebra with unit element. Later, in 1936, Nagumo [2] studied the properties of the group of regular elements and the generalized exponential function in an abstract system which he cal led a linear metric ring. The major i..mpetus to the study of this new discipline appears however to have begun wi. th Gelfand in Russia who in the period from 1939 to 1941 published a series of abstracts on the subject culminating with his now famous papers [ 3] and [4] on Normierte Ringe. Since that time the theory has flourished under the contributions of a host of authors writing on the algebraic, topological, and functiontheoretic aspects of such systems. The bibliography contained herein is by no means complete or exhaustive. references may be found in ( A fairly comprehensive list of 51 and [ 6] • The properties of Banach algebras are such that they lend themselves -2- most readily to generalizations of techniques and theories of classical analysis. Among these generalizations, has been the extension of the scope of analytic function theory. Here the development of the theory possesses remarkable similarities to that taken by the c.lassical course. Nevertheless, there arise new and interesting phenomena which in the classical case either do not exist or are trivial. One of these peculiarities occurs in connection with power series, where in general the region of convergence is no longer a sphere, and as a consequence the distribution of singularities of the function which the series represents appears to be of an extremely complex nature. It was the latter problem - that of determining the existence and distribution of singularities on the boundary of the region of convergence of a power series - that led to the present study. In the case of the complex plane it has long been known that many statements concerning the occurrence and location of singularities on the circle of convergence of a power series are specializations of broader results known for Dirichlet series, or more generally, LaplaceStieltjes integrals. Such statements are embodied in a class of results usual]¥ referred to as gap and density theorems. Motivated by this knowledge, it seemed appropriate to attack the same question for connnutative Banach algebras in an analogous manner. Section II of this W"Ork therefore begins with the definition of the generalized LaplaceStieltjes integral and concludes with several results of a general nature concerning such integrals. Prerequisite to studying the occurrence of singularities on the -3- boundary of the region of convergence of a Laplace-Stieltjes integral, however, is the necessity of having an adequate description of the region itself. arises. Here a marked deviation from the classical situation The familiar half-plane of convergence is replaced by a set considerably more complex in nature. In fact, as yet a complete description of this set is not known. Nevertheless, a partial descrip- tion can be given, and for several classes of Banach algebras a full treatment is possible. It is to these matters that we devote Section III. Section IV consists of specializations of the results of Sections II and III to generalized Dirichlet and Power series. In Section V we consider functions defined by the Laplace-Stieltjes integral and show that they are analytic in a sense to be defined at that time. The thesis concludes with Section VI, in which some remarks are made concerning the occurrence and location of singularities on the boundary of the region of convergence. A special case is treated and a suggested plan of attack for further research along this line is given. In the interest of more or less completeness, the remaining paragraphs of this section will consist of the definition, examples, and a brief resume of soi.re of the well-known and fundamental results on Banach algebras to which it will be necessary to refer in the main part of this thesis. -4Although not at all times essential, it will be convenient to restrict our considerations to a special class of Banach algebras ~ those characterized by the fact that they are commutative, complex, and possess a unit element. It is to this special class that all ensuing cormnents apply. DEFINITION 1 • 1 • A set of elements ti (always denoted hereafter by capital letters s, T, ••• , X, Y, Z) is said to be a commutative complex Banach algebra with unit i f (a) 1ii is a Banach Space ( 7] • (b) t;L is a commutative algebra over the field of complex numbers, the elements of which will always be denoted by small case letters a, b, c, ••• , x, y, z. in 1?l_, In short, this means that a multiplication is defined Which satisfies the properties: (i) XY = YX (ii) X(IZ) = (XY)Z (iii) (aX)Y = a(XY) (iv) X(aY + bZ) = aXY + bXZ. ( c) -a has a unit element I satisfying IX= X. ( d) II XYll ~ 1\ XII II YI\ ' II I I\ = 1. For illustrative purposes we include the following set of examples which have been taken from the literature, and are among those most frequently encountered by the analyst. -5EXAMPLES 1. The field of complex numbers ·w ith the norm taken as the absolute value. 2. The set of all n by n diagonal matrices of complex numbers (a .. ) , l.J max \a .. \ • 1 ~ i,j ~ n 1 J The set of all n by n matrices of compl ex numbers (a .. ) such that with the norm taken as 3. 1J aij = 0 for j < i, and ai_1 j-1 = aij for j ~ i. The norm may be taken as in example 2. 4. The ring of complex-valued functions f(x), continuous on a compact space 5. Gf , with II f l\ = SUP. I f(x) I • x d~J The ring of complex-valued functions f (x) of bounded variation on the interval a ~ x ~ b, with multiplication defined point-~-ise and ll f ll = 6. sup a ~ x ~ b \ f(x)\ + v , a,b,f(x). 1..: ~ The ring of functions analytic over a bounded domain D in the complex plane and continuous over the closure D of D, with 11 r II 7. =sup I r(z) I • co IT The 1ing of absolutely convergent Fourier series with ll f l\ 8. - f(x) = [ co = L Ian \ • an einx, The ring generated by any bounded linear operator T defined over an arbitrary Banach space B. 9. The ring generated by a one-parameter group or semi-group of linear transformations over a Banach space. 1o. The ring of complex-valued. functions f(x) defined aver the interval a ~ x ~ b, possessing n continuous derivatives, with II f ll =~a ~ s~~ b If(m)(x) / /m! 9 -6- For the f ollowing list of definitions and theorems we refer the reader t o [ 5 ] and (6) where proofs and references to the ori ginal paper s are to be found. DEFINITION 1. 2. ~ is an element X An element X E ti is said to be regular if there . ~ called the inverse of X s uch that XX A non-regular element is called singular. =X~ X = I., The resolvent set the s et of all complex numbers z for 1mich (zI - X) is regular. spectrum o-(X) of Xis the complement in the complex plane of ('(X) is The ~ (X), and is a closed non-vacuous point set. THEOREM 1.1. The set of regular elements in (not necessarily connected) set DEFINITION 1.3. (a) X, Y E. d ~ form an open CY , and x-1 is continuous on CY • The set ci. c ti is called a non-trivial ideal if , A, B t__ --1;3, imply AX +BY e. cl. • ( b) cl, ;if ti. An ideal is said to be ~mal i f it is not a proper subset of another (non-trivial) ideal. THIDREM 1. 2. (a) An ideal contains no regular elements. ( b) The closure of an ideal is an ideal. ( c) A maximal ideal is closed. (d) Every ideal is contained in a maximal ideal. (e) An element Xis contained in a maximal ideal if and onl y if it has no inverse. -7DEFINITION 1.4. -fi and X - dl with Y E d • --J!l the set of all elements in The classes X - ~ Let The quotient algebra. be an ideal in of the form X - Y form an algebra denoted by ~~ cJ according to the definiti ons (i) a(X - cQ ) = aX - ~ • (ii) ( X - d/ ) + (Y - ~ ) (iii) (X - ~ )(Y - ~ ) = XY - ~ THEOREM 1.3. If c:J = X + Y - <:::JI is a closed ideal in a commutative Banach algebra. under the norm THEOREM 1.4. fi then lllX - cJ 11\ i3;~ is =inf l\ X - Y\\ • Y t~ If a commutative complex Banach algebra is a field then it is isometrically isomorphic t o the field of complex numbers. THEOREM 1.5. If 1fil is a comnmtative complex Banach algebra with unit element and if the norm satisfies the c ondition for all X, Y £. fl , then THEOREM 1. 6. Let 73 ~ is isomorphic t o the complex field. be a closed ideal in is the complex number system if and only if DEFINITION 1.5. then for every X £ f3 Let ¥t{_, ll XY ll = llXll llYll =.SI 1fi • be the set of maximal ideals in and '77( c. Yfb f3 /cl), is maximal. £i , the re is a uniquely determined complex number z such that = zI - /7( • The function X( /J1.. ) on Ytl is defined by X( YJ1. ) = z. X - /Jl THEOREM 1• 7. Then The function X( irl. ) has the proper ti es: -8- =X(/?'( ) + Y( m_ ) (a) (X + Y)( 7J7_ ) (b) (XY)( 77() = X( Jn )Y( lJ( ) (c) (I)( 7J( ) (d) (aX)( m. ) = aX( m ) ( e) lxc m )I ~ =1 ll XII xcm ) = 0 if and only if x m (g) If n; -:j. m 2 there exists an x such that X( m 1) F X( /11_2) (f) f: (h) X(OlG ) = <r(X) (i) If z E. cr- (X) there exists and )'Y{E.n(, such that X("rr{) = z (j) If for given x and m ' xcm ) F o, then there exists a y t 13_ such that Y( YY/. ) = 1/X(-m_ ). THEOREM 1 • 8. For a fixed maximal ideal /J1 t: Yfl and X ranging over 'Poi._ , the values X( /f( ) define a linear bounded multiplicative ti • and if l(_ functional F-:i-(X; 11'l) on functional on ti XCJ?1. ) = o, then 7(_ e ~ Conversely, if F*(X) "=/= 0 is such a is the set of points in t3.- and F*(X) where = X( /?._ ) for all X. Henceforth we shall employ the symbol Ml~ to denote either a maximal ideal or its corresponding multiplicative functional. No con- fusion will arise since the context will always make it clear which interpretation is to be used. DEFINITION 1.6. bound of 1z r denoted by The spectral radius of X is the least upper as z varies over the spectrum I cr(X) I • o- ( X) of X. It will be -9- THEOREM 1.9. ~ITION 1. 7. such that \ a-(X)\ = sup I~~(x) I = lim 1\rll M~l- f.'a'l, Il4 1 = lcr( X)I = o. n-+ co The set of all generalized ti . nilpotents is called the radical of The radical of a conunutative Banach algebra is the intersection of all maxi.mal ideals in THEOREM 1. 11. . CD A generalized nilpotent is an element X f. 1ii lim llXnl\ /n THED REM 1 • 1 o. 1/n · Let M-::- c. Yfl , 0 ti . E- > 0 and xi ['ti (i = 1,2, ••• ,n). The set of elements l.,{(M-'~; E ;~,x2, .... ,xn) = ~l'i*En'll l:tvHl- (Xi) - ~f'~(Xi) \< E ' i = 1,2, ••• ,~ is called a neighborhood of M~~. 0 With such a neighborhood system m becomes a compact Hausdorff space and the functions M*(X) are continuous for all I~~ £ btl, • THEOREM 1.12. Let 'ft be a corrunutative Banach algebra and ¥'il the compact Hausdorff space of maxi.mal ideals in 1ii • Let be the Banach algebra of continuous functions on 'ftl . Then the mapping X-+ ~~~(X) is a homomorphic mapping of an isomorphic mapping if and only i f 11. into ' ('a{_). ti. has no radical. t;, (Dfl) It will be -10- Section II THE L~PLACE-STIELTJES INTEGRAL We begin this section with a collection of fundamental definitions and results which we shall need later for the study of the LaplaceStieltjes integral J F( s) = (1 ) .., exp( -ts) dA ( t) • 0 A function A(t) of the real variable t defined DEFINITION 2.1. on the closed interval [a,bJ to the Banach algebra ti is said to be of strongly bounded variation in that interval if L \I A(t.)1 - A(t.1-1 )1\ < ""' sup where all possible partitions of l a,b] Let F( t) and A( t) be functions defined over the DEFINITION 2. 2. [ a,b] and let of =t 0 <. t [ a,b] lim b. .be a by the poin-ts If the limit (in the normed topology n-1 ~ -+ 0 Let 1 <. ••• < t n = b , S = max ( ti+1 - ti). -fi ) ii . to the Banach algebra subdivision of the interval a The supremu.m will [ S, page 39] • be called the strong total variation closed interval are allONed. Li=O F(u.)]. [ A(t. ") - A(t. )] J.+1 1 (i = 0, 1 , ••• , n-1 ) where -11- exists independently of the manner of subdivision and of the choice of the numbers u ., then the limit is called the Riemann- Stieltjes l. integr al of F(t) with respect to A(t) from a to b and will be denoted J by ( 2) F( t)dA( t) . a If F(t) is strongly continuous (i. e., continuous THIDREM 2. 1 • in the normed topology of tfL ) and A( t) is of strongly b:mnded variation, then the Riemann-Stieltjes integral of F(t) with respect to A( t) from a to b exists. Further, if F-~ is an arbitrary bounded linear f unctional belonging to the conjugate space ti_* f ' then t~ = F-i~ [ F(t)dA(t)] • a (3) F* [ J b F( t)dA( a - The proof follows standard patterns , examples of which can be found in [ 5, page 51] DEFINITION 2.3 . and [ 8, theorem 11] • If A(t) is of strongly bounded variation in a ~ t ~ b , it is said to be normalized there if A(a) = 0 A ( t) = A ( t+) + A ( t- ) (a < t < b) . 2 THEOREM 2. 2. A function of strongly bounded variation has right and left hand limits everywhere and is strongly continuous except for a countable set of discontinuities of t he first kind , and hence may be normalized. See [ 5, page 203] • -12- [ 9, page 14] As in the classical case if A(t) is not normalized, we may do s o by defining the function B(a) = 0 B(t) =A(t+) +A(t-) (4) -A(a) (a <- t <. b) 2 B(b) = A(b) - A(a). Since the replacement of A(t) in (2) by the function B(t) defined in (4) leaves the value of the integral unchanged, we shall assume henceforth that our functions of strongly bounded variation are always normalized. DEFINITION 2.4. Let A( t) be a function on [ o,C)) to ti, and let A(t) be of strongly bounded variation over every finite interval [o,b] • Then stnce exp(-tS) is strongly continuous the integral s b F(S;b) = (5) exp(-tS)dA(t) 0 exists for finite positive values of b. If, for a particular S ( -tJ.._, , lim F(S;b) exists in the nonned topology of b-+ C) limit by 11. , we will denote the C) (1) F(S) = ~ exp(-tS)dA(t), 0 and will call F(S) the LapJace-Stieltjes transform of A(t). Further- more, the set of s E. 1fj_ f or which (1) converges will be denoted by and -will be called the~ of convergence of (1 ) . of the set ~ will be denoted by the symbol the region £f_ convergence of (1). ro ' The open interior crc ' and will be called -13- <: C CTc• This statement is in general no longer valid for Banach algebras. Example J. 2 discussed below can In the classical case easily be specialized to give an example where CTc is void, and C:, is not void. In preparation for a discussion of the region CJc in S.ection II we now include some results of a general nature concerning the integral (1). For a given integral it will soon become apparent that it is the spectral properties of the element S rather than its nonu which plays the dominant role in detennining whether or not it belongs to the region of convergence. For this reason it will frequently be necessary to make use of THEOREM 2.3. Let 1l, be a Banach algebra with unit element and let S be an arbitrary element belonging to R1 (S) - (6) ti , then the expression 1n II exp(-tS) ll t-+0> t - lim exists and is equal to (7) min R1(z) z t u( S) Proof: = min M~..i- E)){_, R1M*(S) • Let S be a fixed element belonging to and consider the real-valued function of t defined by ( 8) h( t) = ln l\ exp(-tS) II (0 ~ t < cxo). -14(9) Equation (9) together with the fact that for arbitrary t [ 5, pages 135, shows that h( t) is a finite-subadditive function of t 136] over any interval (10) 0 ~ t < co and hence = lim ln II exp(-ts) ll lim f(t)/t t-+cio llexp( -tS) ll > 0 exists. t t-+co Letting t -+ co through integral values and comparing with theorem 1 • 9 we have lim ln lle.xp( -nS) Jl n-+ oo n = lim ln I\ [exp( -s)J nu = ln l max I M7~[exp(-s)J\) n -+ oo n M->-t- t YI(_ = ln{ max e-Rll\11-l~(s)l J Mi:-em = 1 { -min RllS~(s)} n ~)l-t:m = - min RlM*(S) M*em 1 from which the conclusion of the theorem is evident. The importance of theorem 2.3 lies in the fact that for large values of t, II exp(-ts)ll is governed by Rl(S) . In this respect vre have the foll01Ning COROLLARY: a t 0 such that II exp( -ts) I\ <- ( 11 ) THIDREM 2.4. (1) Let Rl(S) = a, then for arbitrary b ) a there exists e-bt If the integral s 0 exp(-tS)dA(t) for all t > t 0 • -15converges for S = S0 (i.e., S0 L <;; ) then it converges for any S such that RlM*(S) > RlM*(S ) (12) for all Ml~ t Yi/., • 0 Proof: J q q exp(-ts)dA(t) = S p exp [ -t( S-S 0 +S 0 D dA ( t) = p where s q p J exp [ -t(S-S 0 ~d.B(t), t B(t) = exp(-u3 0 )dA(u). 0 Note that B(O) = o, B(®) exists and since B(t) is strongly continuous there exists a constant m > 0 such that /I B( t) ll <. m for all t. An integration by parts yields j q exp(-tS)dA(t) = B(q)exp [ -q(S-S 0 tl p q + (S-S 0 ) ~ exp(-tS)B(t)dt. p By hypothesis RlM*(S-S ) > 0 for all Ml\- f.."'i'/l.. , hence by the Corollary 0 to Theorem 2.3 there exists an r > 0 and a t II exp [ -t( s-s 0 ~ JI < 0 such that e-rt Then I\ s exp(-tS)dA(t) ll <; 2me-rp + p and thus given arbitrary J q II S-S 0 II m e-rtdt (t 0 < p < q)' p E ) o, a simple computation shows that if -16- q>p > E- r ma.x{t,lln o r 2m llS-S II 0 then \J < exp(-tS)dA(t) and ( 1 ) converges. p If S COROIJ..ARY: ~: 0 f. ~ and s 1 is any e lanent such that c This follows from the fact that the spectrum of an element (See Theorem 1.7(e).) is a continuous function of the norm. S0 e ~ implies there exists an r > 0 such that an open sphere of radius r about S 0 also belongs to (J • c In particular the element s 2 = s 0 - 2r/3 I e. (jc , and RlM'.t-(S ) = RlMl:-(S ) 2 0 (14) - 2r/3 for every M"'A- E. ~. Now let S be any element contained in an open sphere of radius r/3 about s • 1 Then for all M* E Of(_, RlM•'t-(S) > Rll?A-(S1 ) - r/3 ~ Rll?A-(S 0 ) - r/3 = RJ.Ml:-(S 2 ) + r/3, and therefore (1 5) implies an entire sphere about s1 belongs to the region C, , and O'. c THEOREM 2.5. (1) Let (jc be the region of convergence of the integral ~~ exp(-tS)dA(t). 0 -17Then if S f CT and R is an arbitrary element belonging to the radical 0 of fiL c , (S 0 +R) e <J'. c Proof: This follows immediately from the corollary to theorem 2.4 and the fact that for all M'A- t. 0/0 M*(S 0 + R) = M*(S 0 ). In close analogy to a standard result of classical Laplace transf orrn theory [10] we have THEDRErv1 2.6. If the integral J "° exp(-tS)dA( t) (1) 0 converges f or S = S , then it converges uniformly with respect to z for 0 all S of the form S =S Proof: 0 + zI, where \ arg( z) I ~ e < 1i/2. Define B(t) ~ ~~ exp(-uS 0 )dA(u). t By hypothesis, for any E: > 0 there exists a t l\ B(t) I\< E Now and an integration by parts gives 0 = t 0 ( f ) such that -18- Consequently t2 j\t exp [ -t(S. + 0 zI~ dA( t) ~ 1 j t2 + \Z) e-tRl( z ) ll B( t) ll dt. t1 <E- e- t 2Rl(z) + E- e -t1Rl(z) 5 00 + El z \ e-tfil(z)dt t1 -t 2Rl(z) = E{e -t1Rl(z) +e ~ But for larg(z) I ~ 8 < 11 /2, Rl( z) > o, and hence e -t 2Rl(z) \ Z\ 1 e ~ 1, and --- , from which we get Rl(z) cos(e) ~t exp ( 2 \ t 1 -t(S + ° zI)] dA(t) l < Ee (2 + 1 ), cos (e ) theorem follows immediately from this inequality. -t1Rl(z 1 Jz l +-e Rl(z) -t Rl(z) 1 ~ 1, The conclusion of the • -19- Section III REGIONS OF CONVERGENCE OF THE LA.PLACE-STIELTJES INTEGRAL It is well lmown in the classical case where 1fi is the complex plane, that if the function A(t) is prescribed in advance for the integral (1 ), then it is possible to give specific criteria [ 9] with which to decide whether or not a given S lies in the region of convergence ~· Furthermore, the criteria provide (if it exists) as an open half- plane. a description of (jc Since it will be necessary to refer to these criteria we include THEORE1'1 3.1. For the integral (16) there exists t1~o real numbers c and a such that the integral is convergent for Rl(s) > c, but not for any s with Rl(s) < c, and it is absolutely convergent for Rl( s) > a, but not for any s with Rl(s) <. a. We have and (17) (18) c = lim t ~co a= lim t ~ ClO ln \ a( t) - a(°') I t , ln I v( t) - v( ro) I ' t where a(~) = lim a(t) or 0 according as the limit exists or not, and t ~ a; v(ro) is defined similarly for the function v(t) which denotes the total variation of a( t) in the interval (o, t ] • -20- Attempts to find analogous criteria for an arbitrary Banach algebra have not as yet been completely successful. Several special cases can be treated in full however, and for certain classes of Banach algebras some interesting results concerning the relation between the general shape of the region space "O/L Lfc and functions over the of maximal ideals can be given. As a preliminary to discussing the region of convergence of (1) we make the f ollowing definition. Let F* '/= 0 be any bounded linear functional belong- DEFINITION 3.1. ing to ~~, the conjugate space of 1/i , and let c be any real number. The set = { S t:'ti I RlF*(S) > c} . will be called the half-space determined by F* and c. (19) ~ M (F~;c) In particular if the functional is multiplicative, i.e., one of those functionals associated with some maximal ideal tf~ f. 'Dfl , then 7f(M* ;c) will be called a distinguished half-space. Because of the rontinui ty and linearity of F*, an open convex set in id • rt(F~'" ;c ) is clearly In fact it is easily verified that the set ( 20) defines a "hyperplane" which divides into two disjoint convex sets. Judging from the situation in the complex plane it might be -21- supposed at this point that the region of convergence of (1) would consist of a distinguished half-space. That this is not the case is adequately demonstrated by a simple exampl e. EXl\J\iPLE J.1. of order two. Let 'ti be the set of all complex diagonal ma trices One finds that there are but two maximal ideals in ~ . Letting :J' ( u(t) A(t) ~ \i where u(t) and v(t) are complex-valued functions of bounded variation, then w~(S) = z, M~(S) =w, and "" 0 j exp(-tS)dA(t ) = ( 21) • 0 0 It is apparent that if (21) is to converge it is necessary and sufficient that ~ ( 22) M; r~ exp( -tS )dA ( e-ztdu(t) and 0 L1 Q) exp(-ts)dA( e-wtdv(t) 0 0 converge in the ordinary sense. J Rl(z) t~ = J"' t)] = S > c l_u(t)] ( 23) L Rl(w) I This implies that and c [v( t)] . Stating conditions ( 23) in the style of definition 3.1 we have ( 24) Rlll~(S) > c ( M1(A( t) ~ (i = 1, 2) • -22- This shows that the region of convergence consists of the intersection of two distinguished half-spaces. Motivated by this example we make the following definition. DEFINITION 3.2. M~~ t. To every bounded, linear, multiplicative functional 'otG let th ere correspond a real number c(M~-). The set { st i;t I RlM'\s) > c(M-1'.. ), ( 25) all M*f. will be cal led a cylindrical quadrant. "ftL} Furthermore, if the numbers * c(M ) are all equal we shall say the cylindrical quadrant is diagonal. Ramrk: It is to be noted that the numbers c(M-l$) in the above definition define a function over the space m of maximal ideals, and consequently the decision as to whether a given S 0 belongs to the cylindrical quadrant or not depends upon the ordering of the two functions Rlli*(s ) and c(]t°). This does not mean however, that at all 0 .>< times that the determining function c(Mn) for a given quadrant is unique. Examples of Banach algebras can be given for which the function .)~ c (M" ) can be altered by decreasing its value at one point to form a new function d0f) -/; c(M*), and yet from the fact that for a given S the fuLC ti on Rlll (S) is continuous over m. Several important properties of cylindrical quadrants are given by THEOREM 3. 2. Let Q.. be a cylindrical quadrant contcrl,.ned in ti , -23- then ( i) Q__ is a convex set. (ii) If S E. Q and S is any element belonging to 0 1J. such that for every 1f E Of( (iii) If S f. 0 Q_ and B. is an arbitrary element belonging to the radical of ~ , then S Proof. 0 +R Q. E. (i) is obvious since distinguished half-spaces. ({_ is the intersection of convex Let Q_ (ii). = { s lR11{(S) > c(M·:~), all M*tDJZ.} , S ~ ~ ,and S be an element satisfying (ii) of the theorem. 0 Rlll(s) ~ RU{(s ) > c(M*), and hence S t'\_ . all M* t:. YfL 0 Then for (iii) follows from the fact that for all ~i\. 41(_ and R. belonging to the radical RlM*(s 0 + R) = "n:ul(s 0 ) > c(11.t). Example 3.1 is somewhat misleading in that it may have given the impression that (1) will converge if for each M* t. Yfl ( 26) M* [ 5 j 0 0 the integral co ru<p(-tS)dA(t )] = converges in the ordinary sense. e-tM*(s)dl,t [ A(t)] At the risk of belaboring a point the following example is cited to illustrate that this is not the case. EXAMPLE 3. 2. Let of the form s= -a be the set of two by two complex matrices G:) . Here there is but one maximal ideal and M~ (S) = z. ~~ Letting -24A(t) = v(t)) ( :(t) u( t) we have (27) j e-tzdu(t) J"° e-tzdv(t) + w J (-t )e-tz du(t) co <lO 5 ClO exp(-tS)dA( t) = 0 0 0 "" 0 50 e-tzdu(t) 0 Here M* [ j , exp(-ts)dA( t)] 0 and hence for convergence it is necessary that Rl(z) = ro..M'~(S) > c [ u(t)] = c LM~~(A(t))] • A glance at ( 27) shows however that it is also necessary to have (28) Rl(z) = RlM*(s) > c ( v(t)] , a condition that is apparently not detectable by the use of :rlA-. The region of convergence can now be described as (29) ~ = { S t iL ) Rll.ir*(s) > max [ c(u(t)), c(v(t) )] which is plainly a distinguished half-space. By choosing the functions u(t) and v(t) in ( 27) properly it is easily seen that it is possible for M* [ sexp(-tS)dA(t~ 0 to converge for all S f.. ti and yet have ( 27) convergent for no S E. ti . Despite the fact that the second condition (28) of the last example - 25was not predicted by use of the multiplicative functional, it is significant that (j_c is nevertheless a cylindrical quadrant. There is c onsiderable evidence to support the view that this is the general situation. Although it appears impossible to give a specific formula (J for an arbitrary Banach algebra, c t hat will define the region it is believed that it is always a cylindrical quadrant, and as such is defined by an upper seni-continuous function over the space of maximal ideals. We state the above opinion in the form of a CONJECTURE. the set of S (.ti for which the i;: Denote by integral (1) converges, and by (Jc the interior of ~ • Then ~ '-'c is a cylindrical quadrant in the sense of definition 3. 2. The following paragraphs will be devoted to the discussion of a special case of the integral (1) for which it is possible to give a precise description of the region ~c• This is followed by examples of several classes of Banach algebras for which t he conjecture can be verified. I n any case h owever, it follows from the corollary to theorem 2.4 and theorem 2.5, that dc possesses properties (ii) and (iii) given in theorem 3.2 for a cylindrical quadrant. Before proceeding to the special cases we state THEOREM 3.3. of convergence as (JO) If for any particular Banach algebra the regi on dcis- a cylindrical quadrant, then it can be described -26- Proof: Let and define We show that (i) Let S0 t. '(c(Ii.f)' then RJJl(s0 ) > c(M->.:-) ~ d(M*), hence S0 t ~ = '\)dOl) and Qc(M-l'~) C'\_)d(M-r.-). Let S0 £. ~ = (ii) ~ > 0 such that s particular. i f <. (31) S 0 Rl(z) +s E.. Since <. (j0 is open there exists a crc for all s for whi ch + zI E e'. c o, then for al l M~~ f.. bit I z I .t:.. '2>' , " -~ 0 9d(M*r then S 0 ll S II . : :. ~ • In Choos e z such that > Rllx*(s 0 + zI) = Rlll(s 0 ) + Rl(z) > d(Tuf"). But Rl.M-ll-(S ) 0 + zI (. cY and hence (3 2) c Rllt'°(s 0 + zI) ~ inf RlM*(s) = c(M-1~). The inequalities (31) and (32) show that for every M* €. ¥tl Rllt(s 0 ) > cOt), heme s 0 t. ~c(M*) and qd(M~) C. 'tc cM7~). We now discuss three sp ecial cases when the region of c onvergence (]_ can be identified as a cylindrical quadrant. c Case 1: For an arbitrary commutative Banach algebra with unit element if the function A(t) is a multiple of the identity I, (33) A(t) = a(t)I, -27where a( t ) is a complex-valued function of bounded variation, t hen t h e integral (1) reduces to t he f orm 00 ~ exp(-tS)da(t). (34) 0 Under these conditions t he region of convergence of (34) is a diagonal cylindrical quadrant, and is described in THEOREM 3 .h. For the integral co ~ (34) exp(-tS)da( t) 0 there exists a real number - co ~ c .::; (17) c = lim oo given by the expression ln la(t) - a(«>) I t t.+<» where a(«>) = lim a(t) or 0 according as the limit exists or not, such t-+ 00 that (34) is convergent f or Rl(S) > c, and divergent for Rl(S) < c. This describes d0 as the diagonal cylindrical quadrant 0:, =~ 'H (M*;c) ={ Sd3 j RJM*(s) > c, all M*tm} , Proof: Bearing in mind the convention established for the meaning of the term a(«>), let (35) lim t-+ °" _....._ ____________ ln la(t) - a(«>) I = c. t Equation (35) implies that for arbitrary a > o, there exists a t ( '$ ) 0 such that (36) la(t) - a(0) I < e(c+ <i> )t Secondly, if Rl(S) = d > b > c , then by the corollary to theorem 2.3 -28- there exists a t 1 (37) 11 exp(-ts) II < such that e-bt Now q J q j exp(-tS)d [ a(t) - a(<»)] exp(-tS)da(t) = p p hence on integrating by parts and employing the bounds we have for arbitrary \j (36) and (37), E- > 0 exp(-tS)da(t) \I < E , provided p q > p .:> max { t , t , 0 1 1 r E: ( c+ ~ -b)ll inc*), 1 -b) n L II S 11 ·j J ( c + <i) (c+ ~ -b) 1 This shmrn the convergence of (34) for Rl(S) > c. On the other hand, assume that the integral (34) converges for some S0 such that Rl(S 0 ) = h < c. one rvr* E. Yfl By theorem 1.7(i) there exists at least such that 0 RlM*(s ) = h < c. 0 0 By theorem 2.1 the strong convergence of (34) implies weak convergence, .)I... .)(. hence f or every bounded linear functional F" E. J 'ta:. the integral co (38) F* [ exp(-tS )] 0 da( t) 0 is convergent. ~(­ In particular for M 0 Q) j M~ ( exp(-tS 0 )] 0 Q) da(t) = J e -tM* (S ) o o da(t) 0 ~-!. converges., which by theoremJ.1 is impossible since RJ.M'(S) =h 0 0 <. c. -29- This completes the proof of the theorem. Note that in theorem 3.4, as in the classical case no information is obtained about the convergence of the integral for those S for ·which = c. Rl(S) In case 1 it was a specialization of the form of the integral for which the conjecture was verified. We now consider two cases in which it is the Banach algebra itself which is specialized. Case 2: If the Banach algebra is finite dimensional it is possible to verify the conjecture. The proof of this fact depends upon the following lemma. If ~ is fin.i te dimensional then there exist LEMM.A J.1. X. J satisfying the conditions (39) (i) ~tcx.) = (ii) x.x. =0 • 1 J Proof: 1 J a1J.. There are only a finite numb er of maximal ideals M~, and 1 since they are linearly independent there exist Uj such that (40) Clearly the elements U.U. 1 M' \ U.l.UJ• ) = 0 J (i ~ j) ( i :j: j ) for al 1 M~~. belong to the radical because But U.U. l. J implies there exists an integer n. . such that l.J n .. (U. U.) l.J = 0 l. J Let n 0 = max(n .. ) • • l.J i, J (i =} j). (i :j: j), then ( i =J: j ) in the radical -30- (41) (iij). Setting n x.J = u.J 0 ( 42) we have GM.n ( U.)J -),~ (i) ~ no 1 = ( ~ .. ) no = 1J n n Let 1fi be finite dimensional. ~ij' x.1 xJ. = u.1 0 uJ.0 = (u.1 uJ.) 0 = 0' (ii) as was to be shown. THEOREM 3 • .5. (1) the region Then for the integral ~ is the cylindrical quadrant { s E. t3 I RL.·£\s) > c(J\{) = Sinf RlM.t-(s), all M·\ rte1 . t O' J 11 c Proof: elements Let S0 t.. 'fc(Il)' then there exists a finite number of s1, s2, ••• ,sn such that ~ (i) S. £. (ii) RlM-l:-(s, ) (44 ) J. i=1,2, ••• ,n c 0 > min 1 ~ i ::: n RJJt(s.) all i~E. 1 "dL • Since there are but a finite number of ma.xi.raal ideal~' (k the S. may be relabelled in such a manner that we may write 1 k = 1, 2, ••• ,p. (45) Now by lemma 3. 1 determine X. J (46a) M~(Xj) = 'b kj ( 46b) xixj = o (i f; j). (j =1 , 2,.., ,p) such that o Uext choose an a such that for all i,j = 1,2, ••• ,p =1,2,.u,P) -31- RlM.(S.) +a <. O, (47) J l. and construct the elements k = 1,2, ••• ,p. (48) A simple computation and the use of (47) shows RlM. (Zk) ~ Rllf. ( Sk) (49) l. l. i ~ = 1,2, ••• ,p, Zk E. ~' k then by the corollary to theorem 2.4 = 1,2, .... ,p. Finally define p z = (50) L [ Xk(sk + ar)1 - aI. k=1 ' By property (46b) for the ~ s an easy computation yields b p b ja exp(-tZ)clA(t) = 2:: j exp(-tZk)clA(t) k=1 a (51) and consequently 5 b (52) \\ e:xp(-tZ)dA( t) I~ a But the Zk' s f. ~ ' and hence by choosing a and b sufficiently large the left side of (52) can be made less than arbitrary This at least implies that Z f. argument based on the fact that C, • Since the Zk' s t q', a standard ~ is open allo~vs one to conclude z (. O'. c From (50) we see that (53) Rl.M:(z) = Rl t{ Mi(X;;) [ M(Sk) + •1} k=1 and using (46a) and (47) it follows that (54) RJ..M.(Z) * l. ~ RllvL(S.), * l. l. e > o. i = 1'2, ••• ,p. -a, -32- But (54) together with (45) imply Rn~:cs ) ~ (55) J. Rn{(z) J. 0 i = 1,2, ••• ,p and the corollary to theorem 2.4 again implies that s 0 (. erc as was to be shown. On the other hand, let S that S 0 - rI €.. O". c (j . c E. Then there exists an r > 0 such Hence ~ RD((S ) > 0 Case J: 0 ~ RlM* (S - rI) ~ c(M'). 0 To discuss the second class of specialized Banach algebras for which it is possible to verify the conjecture we make the following Two elements X, Y E ti DEFINITION 3.3. will be said to be equi- valent at M~~ (-written X '""' Y at M-i<) if M*(X) = M~~(Y) for all M-r' in a. -----o 0 neighborhood of M~. (See theorem 1.11 in this respect.) We assume now that the algebra Ti satisfies the following tvvo propertj_es. Property 1: If S, s , 1 s2, ••• , Sn t 'fl , and if for each M* t. YI(; there exists an i such that S r-..J S. at M*, then J. n (56) 11 s 11~Z11 s. ll • • " J.=1 Property 2: J. If S e 1i 0 and if for some M~ e. ~ Rlli1~(S 0 ) > o, then there exists an s1 ~ t:L such that ( i) (57) { ( ii) R.lll (s1 ) > 0 for all M*~'IT{_ , s1 '"'-' s0 It is easily verified that those Banach algebras given in examples -33- 4, 5, 6, and 10 of Section I satisfy the above two properties. We proceed t o verify that the conjecture is true f or algebras satisf ying properties 1 and 2. LEMMA 3. 2. If for every M''l- e:. Yit , RlM-i:-(S.) min 1 ~ i~ n then there exi.st i 1 = 1, 2, ••• ,n elements T , T , ••• , Tm vr:i th t he properties 1 2 for each k = 1, 21 ••• ,m, there exists an i = 1, 21 ••• ,n such (i ) that for every M* ~ 'flt RJ.M-i1-(Tk) > R1M-r.· (s ), 1 (ii) f or each M* £ Y/t , there exists k such that * S ~ Tk at M. -r.· Choose an M~ Proof: RL-riiI-1(-(S) 0 f..4/l > RL"\r''l-(S - S.) 0 1 1 then there exi.sts an i such that Rl.M-i~(S.) or 0 > 1 0• ft:.Ol'l 1 By property 2, there exists an element U e. i;1, such that for all M A . Rll.C(U) ~ > 0 and u ~ (s - S.) at},['. 1 0 Set TM~'*"= U + S., then 1 TM-1~ "'-' S at M~- and for all M*i: ~ , RlM* (TM*)0 > RJ..M*(S. ). 0 0 0 By this construction a collection 1 { Trlf} is obtained such that (i) given M*, there exists an i for vrhich RJll ( T:i{ ) > RlM''l- (Si) all 1l~ 1 (ii) given Tu(, - Since T ,t r v S at M''v-. 1 'ttt is compact, the collection can be reduced to a finite collection which satisfies the conditions of the lemma. LEMMA 3.3. If for all~{" ~Yfl Rni~(S) > min 1~ i~n Rllvt(s.) and 1 -34- CY. c S. f. CJ then S c_ 1 c Determine the finite collection T , T , ••• ,Tm by lemma J.2. 1 2 Proof: = 1,2, ••• ,m and all Mi~ (¥fl , RJJA*(Tk) > Rll.t(si) Then since for each k for some i, it follows from theorem 2.4 that the Tk ~ ~ since the S. E. Lf. Furthermore if S rv Tk at M* then c 1 q J q exp(-tS)dA(t) ,...._, S exp(-tTk)dA(t) p p By property 1 \J -c-ts)dA( t)j p Tk f. (J , S t:. and since c THEOREI~I 3 .. 6. c(Ivt) CJc . For each 1r~ e. Ofl let = inf RJ.M'<(s), s €. O"c and let S be such that 0 RlM-{(-(S ) 0 then S 0 E. > cO.t), c::f • c Proof: We first note that c(Mh) is an upper semi-continuous function over ~ since it is the infimum of continuous functions. Consequently, at each point M~~e. Oil__ there exists an SM* E. ~ such that Rlll(S ) > RJ..M*(sM~). 0 But 'ft{_, is compact and hence there exist s1 , s 2, ••• , Sn t efc such that for all M-r;. t ¥t{_ -'~ Rll\( (S ) 0 > min . 1~ 1 ~ n Rlll(s.), 1 and an application of lemma 3.3 shows that s 0 €, erc . -35- THEOlIDJ! 3. 7. > RlM{(·(s) Proof: =inf RJ.M-:~ ( S), then S t. dc implies c(M-l~). S €. ~ implies there exists an r > 0 such that if \ ZI <. r, then S + zI e. O'c and RJJ.l (s + zI) = RJJl~(S) + Rl(z) >,.. c(M-:~), hence > c(tf). Rll.t.i-(S) THEOREM 3. 8. then Let c ( M...:-) If a Banach algebra satisfies properties 1 and 2 dc is a cylindrical quadrant. Proof: From theorems 3.6 and 3.7 it follows that the statement Rllrf~(S) > c(M*) all M* l. Yfl.. is equivalent to the statement S ~ crc is the cylindrical quadrant and thus = { s t.15 \ RlM-:~(S) > c(1l~) O'c = inf Rli'ii*(s) , all J,l e. Y'/ll. s t: 0 c ~J The discussion in the previous paragraphs has concerned itself with the question of ordinary convergence of the · integral ( 1) and a partial dc . description of the region of convergence The question now arises , as to what we shall mean by absolute conIt would be desirable of course to formulate a definition vergence. that would reduce to the already existing notion of absolute convergence when 11 is chosen as the complex field. that this is possible in several ways. It is ic'Illltediately apparent One might say that the integral (1) converges absolutely for a given S if any one of the followi ng three integrals converges: (58) S II 0 exp(-tS )dA( t) II ' -36- J 00 (59) \\exp(-tS) \\ \l dA(t) ll , 0 ~ exp(-ts l 11 dA( tl II • <6ol In the light of theorem 1.5 it is a priori evident that in general the regions of convergence of (58) and (59) will not coincide, and as a consequence there are several notions of absolute convergence. Before discussing integrals (58), notation,, ~ = (59), and (60) we introduce some Let l S \ integral ( 1 ) converges} (Jc = the interior of ~ , a = { s \ integral (58) converges1 eJa = the interior of CL , (\ = { S \ integral (59) converges } ~sa = the interior of as ' ?r = { S \ integral ( 60) converges c:Jv =the interior of o . 1 From the properties of the norm it is obvious that and hence ~)a ) ~ d~CT)() . c a sa As in the case of integral (1) no definite criteria have been established with which one may describe t he region CJa. We state without proof however THFDREM 3. 9: If S f. <:Y' and R is an arbitrary element belonging o a to the radical of ~ , then (S o + R) f. CTa . -37- As a first step toward establishing criteria for the description of the region efsa we quote [5, pg. 204J without proof the following theorem which is analogous to theorem 3.1. THEOREM 3. 1o. co For the integral 5 e-tzdil.( t) (61) 0 there exist two real numbers c and a such that the integral is convergent for Rl(z) > c, but not for any z with Rl(z) < c, and it is absolutely convergent f or Rl(z) > a, but not for arry z with IU(z) < a. We have and ln ll A(co) - A(t) ll t , ln I V(co) - V(t) I , (63) c = lim (64) a= lim where A(co) = lim A(t) or 0 according as the limit exists or not and t t-+ co V( co) is similarly defined for the function V( t) which denotes the strong total variation of A( t). We can now prove THEDREM 3.11. For the integral co (.59) · llexp(-ts) ll \l dA(t) ll j 0 there exists a real number - co :s. a ~ ln I V(t) - V(co) j , (64) a= lim co given by the expressi on where V(t) denotes the strong variation of A(t ) in the interval [ o,t] 1 -38- and V(oo) = lim V(t) or 0 according as the limit exists or not, such t~ 00 that (59) is convergent for lll ( S) > a, and is divergent for Rl(S) < a. dsa as the diagonal cylindrical quadrant ef = (\ 1'/-(M-l~ ;a) = { S e.13 \ RlM-i~(S) ) a, all M"l~t. Yi7 1. sa }(A-f..m_ uul.JJ This establishes the region Proof: This is an immediate consequenc e of theorem 3.10 and the corollary to theorem 2.3 concerning the relationship between Rl(S) and J\ exp(-tS) II • In the particular case where '£2 the rate of growth of is the complex field ~ = ~a· That this is not true in general is demonstrated by the following example which sh01us that for a gi.ven A(t) the integral (59) may diverge for a certain S and yet the integral (58) converges in an entire neighborhood of s. EXAMPLE: Let '1j_, be the Banach algebra of complex-valued continuous functions F(x), 0 :s. x :s. 1, with the usual norm (65) ll F(x) II = sup I F(x) I • o~ x~1 Integral (1) then assumes the form (66) i e-tS(x)dtA(x,t) • 0 Let ( 67) ( 68) A(x,t) = - 2a(x)e- t/ 2 , where a(x) = { :_4x 0 Then 0 ~ x ~ 1/4 1/4 ~ x ~ 1/2 1/2 ~ x ~ 1 • -39dtA(x, t ) = a(x)e- t/2 dt. (69) Choose -1 /4 = S (x) (70) 0 0 ~ x ~ 1/2 5/4 - Jx 1/2~ x ~ 3/4 3/4 :s.x - 1 ~ 1 • Computation shows = e- t/4 dt, (71) \\ e-ts o(x) dtA(x, t) \\ (72) 2 I\ dtA(x, t) II = e-t/ dt, and (73) II e- tSo (x) \\ lldtA(x,t) l\ = et/ 2 dt. From (71) and (73) it follows that integral (58) converges while integral (59) diverges for S 0 (x). Now define a neighborhood of S0 (x) by (74) 7(_(S 0 (x)) = For any S(x) l (75) { S(x) t: t3 fl.CS 0 (x)) II e-tS(x)dtA(x,t) II !S I supx IS(x) - S (x) I < i} . 0 e- t/B dt and hence integral (58) converges. ' It may be pointed out however, that it is not possible for (58) to converge and (59) to diverge for every S(x) t ti , for the two integrals must agree at least for all S(x) which are a multiple of the identity I. Finally we come to the case of integral ( 60) for which we have (60) THEDREM 3. 12. For the integral J exp( - tS) II dA( t) II oo 0 -40there exists a real number - co ~ c ~ oo given by the e.xpression (64 ) c = lim t ~ Cl) ln I V(t) - V(co) I t ' where V(t) denotes the strong variation of A(t) in the interval [ o,t] , and V(oo) = lim V(t) t-+ or 0 according as the limit exists or not, such Cl() that (60) is convergent for Rl(S) > c, and is divergent for Rl(S) < c. This follows immediately from theorem 3.4 since Proof: is merely a :rrru.ltiple of the identity., and shows that (\Noi'(;;c). ~iI' t:'ftt . JI d.ti..( t) II (j = (f = v sa -41Section IV DIRICHLEl' AND PO~IER SERIES As in the case of the complex plane, a special case of the Laplace- Stieltjes integral is the generalized Dirichlet series ClO f(S) = ( 76) L An exp(-tnS) n=1 lim t n~ oo n = Q)• For i f A( t) is defined by the equations A( t) = A1 + A2 + • •• +A n (t < (t > 0) n t < tn-1) A(O) =O (77) A(t) =A(t+) + A(t-) 2 we have (78) l "° exp(-tS)dA(t) A exp(-t S) n n 0 whenever the integral or the series converges. In particular if t n == n,. and we set exp( -nS) == [ exp( -S) J n == zn then (78) reduces to Q) (79) ClO j exp(-tS)dA(t) = L A 0 nzn. n~ It must be pointed out however, that in general a power series cannot be represented as a Laplace-Stieltjes integral. from the fact that the equation exp(-S) This follows = Z has a solution for a given -42- Z if and only if it belongs t o that component of the regular elements of ~ which contains the identity I [5, page 451] • Lorch [11, 12] has shown that in a commutati ve Banach algebra the set of regular elements either has one ar infinitely many components. Inf act the number of components may well be non-denumerable. The failure of power series to become completely a speci al case of the Laplace-Stieltjes integral d oes not alter t he previous theorems, however. As is to be eJqJected it is no longer Rl( S), but the spectral radius 1/n \\zn l\ \cr (Z) \ = lim (80) = inf \ M*(z) \ M*'eTIL n-+ 00 that now plays the essential rol e in the oonvergence theory. Since only a. slight revis i on of techniques is required, 111e state without proofs the following t heorems. co TIIEO RE!I~ 4.1. If the series it converges for any Z such that THEOREM 4. 2. ~ Let q L0 Anzn converges for Z = Z , then 0 lcr( Z) I <. denote the interior of the set of elements 00 for which the series L AnZn converges. Then if Z t (j and R o c is an arbitrary eleme~t belonging to the radical of t;2 , ( Z + R) t:. Cf• . 0 THEOREM 4.3. number 0 ~ a ~ For the series c 0 co L \\A \\ \\Znl\ , there exists a real n:::O n + co given by the expression a = 1/ lim ll An II 1/n n~oo such that the series converges for all Z for which diverges for all Z for which la-(Z) \ > a. }o-(Z) \ < a, and -43- Finally, as a special case of a theorem which may be found in [.5, page ss] we have THEOREM: 4. 4. For tl1 e power series 00 L: (81) n=O there exists a real number a given by the expression 1 a =-------.1. . . lim llA II /n ' n-+ co n such that ( 81 ) converges absolutely for surf ace II Z II II Z I\ < a. On every spherical = r .> a there are points where the series di verges. Moreover the series is uniformly convergent for II Z II < (1- E: )a, and fails to converge uniformly on any spherical surface JI Z II E. >O, =r < a. The form of the c onjecture stated in Section III undergoes obvious appropriate changes and can be verified for the same classes of Banach algebras that were treated previously. -44Section V THE LAPLACE-STIELTJES INTEGRAL AS AN ANALYTIC FUNCTION The problem of extending analytic function theory has been ~< considered in one form or another by many authors.~ reader to We refer the [5, chapters 3,4, chapter 4, paragraphs 5.1 5-5.17, and chapter 22, paragraph 22.9 ] where numerous references are to be found. Particularly applicable to the case of a commutative Banach algebra with unit element, where one desires to study functions on the algebra to itself, is the following definition of differentiability and analyticity given by Lorch [ 11] • DEFINITION 5.1. with a unit element. and range Z ft H in (82) fJ A single-valued fun::tion F(Z) whose domain ~ are in = Z0 if for each with a be a conunutative complex Banach algebra Let E: > 1 tfJ... is said to have a derivative F (Z ) at 0 a ~ > 0 can be found such that for all 0 !I H I\<~ 1 II F (Z 0 + H) - F( z0 ) - HF z0 ) II < ( If F(Z) has a derivative everywhere in Lorch-analytic in E. \\HII • o6'-' , then it is said to be ff . The theory of analytic function in real and complex normed linear spaces was initiated by A. D. Michal and R. s. Martin in a seminar at the California Institute of Technology during the years 1931-1932. See footnote 1, page 2 of [13] • -h.5- The theory of Lorch-analytic functions closely parallels the classical course both in its methods and principal identities. In order to develop a Cauchy theoren Lorch defined the following analogue of the Rieillflnn integral. DEFINITION 5. 2: W LY be an open connected subset of fl . Let =F(Z) be a function on 13_ a rectifiable arc in f f • to itself ·with domain ,,()-' • By this it is meant that tP be -P is given * 1, where Z( t) is continuous and of by an equation Z = Z( t), 0 ~ t strongly bounded variation. Let Let We then define n j F(Z)dZ = lim L F [ z(un,k)) { Z(tn,k) - Z(tn,k-1)J . (83) f' where max k n _,. c.o k=1 (t n, k - t k n, - 1 ) _,. O. If F(Z) is continuous in AY , the existence of the integral is established in the usual manner and it has the properties of linearity and boundedness which are to be expected. In particular we have the f a llowing THEOREM 5.. 1 : (84) max l\ F(Z) ll ). (<f'), f where _--f. (-P) is the length of -P , that is, the strong total variation of Z(t) in the interval [o,11. A function F(Z) Lorch-analytic in a region to be continuous and Frechet differentiable in ff can be shown jj • for terminology and references to the literature). (See [5, page 72] As such, it is but -46a matter of computation to show that F(Z) possesses derivatives of all orders in ctY which are themselves Lorch-analytic functions. Furthermore if A denotes an interior point of jJ one can establish the formulas F(n) (A) = (85) 1 21r i j 1mere z describes acy curve 0 F(A + zI)dz z (n = o,1,2, ••• ), n+1 surrounding the origin in the complex plane once in the positive sense and in such a manner that 1z1 remains small enough to assure that (A + zI) remains in the domain of definition of F(Z). For the justification of the above statements, and further details we refer to [ 5, Chapter 4, especially pages 113-114] • With the aid of (85) it is now possible to represent F(Z) and its derivatives in a form more closely analogous to the classical formulas. In fact, we ba ve THEOREM 5.2. Let F(Z) be Lorch-analytic in an open connected region LY<. ~ , then the formulas (86) FCn\A) = 1 21l'i are valid for every path \ F(Z)(Z-A)-(n+1 )dZ (n = 0,1,2, ••• ) Jr r of the form Z =A + zI, where z again describes any simple, closed, positively oriented path the origin once in the complex plane, provided that ?f surrounding I ZI remains small enough to assure that (A + zI) remains in the domai..n of definition of F(Z). Proof: Letting Z =A + zI clearly reduces (86) to the form (85). -47Should the domain ff include that portion of the subspace zI (z varying over the complex numbers) which includes the spectrum of A considered as elements of the algebra, then an argument on the deformation of paths will allow a restatement of the above theorem. The path Z = zI may nOV\r be chosen, where z surrounds the spectrum of A in the complex plane onc e in the positive sense. In the latter form, formula (86) has been used by H. Poincare [14] , F. Riesz [15] , L. Fantappie [16] , and more recently by Dunford [ 17, 18] , Lorch [11] and A. E. Taylor (19] . To be brief, the usual pattern of fundamental theorems may be developed, culminating with the establishment of a Taylor expansion for a Lorch-analytic function into a power series which converges in the largest sphere with center at the point of development i n which the function is analytic. As has already been pointed out however, the series may converge for points outside of this sphere ~ for example, points in the radical of ti . Conversely, a power series of the form A (Z - A)n n F(Z) defines a Lorch-analytic function in the interior of its set of convergence. As a first step toward establishing the Lorch- analyticity of the function (1) F(S) = s exp(-tS)dA(t) 0 -48- dc , we prove the following analogue of the within the region Vi tali theorem. THEOREM 5.3. Let { Fn(s)} be a sequence of function such that (j , n = 1 , 2, • •• , (i) F ( S) is Lorch-analytic in an open region n (ii) Fn(S)-)- F(S) for each S f.. <Y , and (iii) for each S (_ Cf , there exists a closed sphere II Fn(s) fl ~ a constant m(S) such that m(S), n = 1,2, ••• , then the limit function F(S) is Lorch-anaJ,ytic within Proof: entirely in Cf , and a positive constant m, such that tl Fn (s) ll ~ m, 1 < s £ cf( so;r) o =1,2,3, ••• n d cs ;r1 ) = { s / Us By (i) and theorem 5. 2, for s Fn(S1) - Fn(S2) = 1 2 1l' i s r 1 -S and S 0 2 ll .$ r 1} . iJ(S ;r1) l 0 F (1li)(W-S1 )-1 dW n 1 = - - (S1- s 2) 2 rr i Jr 1 2 1f i 1 1 Fn (W)(w-s )- (W-S )- dW , 1 2 may be chosen as W • r 0 and form the sphere 0 r LY • By (iii) there d es 0 ;r0 ) of radius r 0 > O about S0 contained exists a sphere where lf . be an arbitrary point belonging to 0 Choose an r () ., It will be sufficient to verify that F( S) is Lorch- analytic in a neighborhood of each point S l Let S Q/(s) and = S0 + r 0 ei6 I (0 ~ 8 ~ 2 Tf ). -49We then have l1Fn(S1)- Fn(S2)lf~ from r to then n=1,2, ••• • Q/(s 0 ;r1 ). This states that the Fn(S) are equi-uniformly continuous in Novr choose r W < r 2 1 and define the path O .:s e ~ = S0 + rei 6 I, 21I' , r 2 P1 as < r < r • 1 r1 belongs to the sphere Q/( S ;r1 ) and hence By the choice of r, 0 by the preceding argument, given E > O there exists a S ( €: ) > 0 such that 11Fn(w1 ) - Fn(W2 )11 < E for any W1' w2 E. r 1 for which n = 1,2, ••• l! w1- w2 II < ~ ( E ). B-.r (ii) F (W) -+ F(W), and hence n JIF(W )-F(W ) \\< E 1 2 for Uw -w 11 < 1 2 ~(E; ). r1 is a compact set it is possible to choose a finite Since number of points W. 1 (i = 1, 2, ••• ,p) along ll wi-Wi+1 l\<~(E) and for arbitrary w {. ll w-w. 11 <~(E ) . r1such that (i=1,2, ••• ,p) (Wp+1 =W1), r 1 there exists at least one W.J such that J Again by (ii) since F (W) -+ F(W), we have the existence of p n integers n.(W.) 1 1 i = 1,2, ••• ,p, such that, given E: > 0 -5011 F cw.) n 1 - Fmcw.) n< ~ J. Let n = max [ n. (W. )] , 1 ~ i~ p 0 J. J. r1 then f or arbitrary w E.. l/ F (W)- F (W) ll ~ n m ll F (W)-F (W.) \I n ni + !I Fn (W.)-F (W.) l\ i mi provided n,m > n < 3E 0 + ll Fmi (W.)-F (W) I\ m and ll w - wi \\ < ~( E: ). wi is chosen so that Thus the Fn(W) converge uniformly to F(W) for W ~ r1 • Now restricting S to the sphere rd(S 0 ;r 2 ) F ( S) - F (S) n m = {s \ ll S-S 0 /I s. r 2} , =-2 1i1 - i rj { F (W)- F (W) } n m 1 (W-S)- cnv, 1 from which it follows II Fn (S )-Fm( S) II ~ But n o Er , n > n • d o is independent of S E s;J'(s ,r ), and hence the F (S) converge o 2 n uniformly to F(S) in this sphere and c onsequently to a Lorch analytic function. This completes the proof of the theorem. We next establish two lemmas. LEMNJA. 5. 1. If A( t) is a function of strongly bounded variation in every finite interval ( o,b 1 then the sequence of functions defined by (87) Fn(S) = ~ exp(-tS)dA(t) n=1,2, ••• 0 are Lorch analytic throughout --ra_ and their k-th derivatives are -51- given by the formula n F;k)(S) (88) J = (-t)k exp(-tS)dA(t ) n=1,2, •••• 0 Proof: The proof vr.i.11 be demonstrated for the case k = 1 only. From (87) and (88) we have II Fn ( S-l!l)-Fn( S )- HF' (S) II = \\ J exp(-ts) { exp(-tH) - I + tH} dA( t) II 0 llHll { l\H\I ~ 2 t et llHll II exp(-ts) II ll dA(t)I} E": > O,for all H such that Then given II Fn ( S + H) - Fn ( S) - HF' ( S) If < E- II H )I • = llHll { JIHll m(n)J • l\Hll< -5:_ we have m(n) Similar proofs may be devised for higher derivatives. LEMMA 5. 2. de denote the interior of the region of con- Let vergence of the integral (1 ) ~m exp(-tS)dA(t) , 0 and let n ( 87) = Fn(S) J exp(-tS)dA ( t) (n =1,2, ••• ). 0 Then if S0 is an arbitrary element belonging to the region exists a real number r(S ) > 0 the s phere ~ ' there o, and a constant m(S 0 ) , such that ·within d(r ;S ) = l\S-S \\ < r(S ) 0 0 \\ F ( S) I\ ~ m( S ) n o 0 (n = 1,2, ••• ). -52Proof: ~.. Let S be an arbitrary point belonging to 0 Since Cic is open, there exists an open sphere of radius S which also lies in o CJc •· In particular Now we restrict S to lie in the sphere We have s n ( 89) Fn(S) = n exp(-tS)dA ( t) = J exp [ -t(s-s1+ s1 )] dA(t) 0 0 n = j0 exp [ -t(s-s1 ) ] dB( t) , where t B(t) = (90) J exp(-uS)dA(u). 0 An integration by parts yields n fn(S) = exp [ -n(s-s1 )] B(n) + (s-s1 ) J [ -t(s-s )1 exp 1 B(t)dt, 0 and hence (91 ) II f n ( S) I\ ~ n II exp l -n( s-s1 5) I\ I\ B( n) II + II s-s1\\ J II exp [-t(s-s1 )] II 0 l\B( t)I\ dt. For each finite n, the right side of (91) exists giving a sequence of constants: The restriction II S - S0 I\~ f!./ 3 implies that Rl( s-s1. ) ~ t > ~ > 0, -53hence there exists an n 0 such that II exp [ -n(s-s1 )]I) < (92) (n > n ). 0 From (90), B( t) is a continuous function of t, and since s lies in 1 B(oo) exists, and therefore for some positive constant c II B( n) JI < c (93) for all n. Using (92) and (93) in (91) yields (94) llFn(s)ll < e -n <f/7 + 5c f /6 jno 0 0 =m Setting ' c:J(s0 ) = { s \ \IS..S0 1\ "- r:1 =r(S ( n = 1 , 2 ,3, ... ~ ) we have for S belonging t o 3 0 )} as was to be sho1m. From the last two lemmas we see that the functions n ( 87) ~ exp(-tS)dA(t) Fn(S) = n=1,2, ••• 0 satisfy the conditions of theorem 5.3 and hence we have established THEOREM 5.4. The function defined by the LapJace-Stieltjes integral s 00 (1) F(S) = exp(-tS)d.A.(t) 0 represents a Lorch-analytic function within the region C{. ~ -54A more general theory of analytic functions on one Banach Space to another may be found stmunarized in [ 5, Chapter 4] where adequate references t o an extensive literature on ·t,he subject is available. When couched in the language of Banach algebras the definition of an analytic function assumes the following f orm. DEFINITION 5. J. A function F(Z) on e;()-" is said to be analytic in domain -fJ._ to ~ , defined in the f f i f it is single- valued, locally bounded, and Gateaux-differentiable [ 5, page 72 ] in ff . The question arises if perhaps it is not possible for a Lorchanalytic function to be analytic in the sense of the above definition in a more extensi ve domain than that in which it is Lorch- analytic. That this is not possibl e is shown in THEOREM 5.5. Let F(Z) be analytic (in the s ense of definition 5.3) in an open connected set contained interior to CY<. 'fl. lf Let i n which F( Z) is Lorch- analytic . is a Lorch- analytic functi on throughout Proof : join Z 0 ~ be a i:phere, center at Z0 , (j . Let W be an arbi tra!'IJ point in to W by a path Then F(Z) d . By hypothesis we may r lyi ng entirely within <:f • To each Zr belonging to r ther e exists a sphere, center at Zr , with a non- zer o radius (.> (Zr ) in which F(Z) may be represented by an absolutely convergent F- power ser ies ( 9.5) -55By standard arguments ~(Zr zero function of Zr on the compact set r we By the compactness of points Z, 0 i r and hence attains > O. ~o its lovver bound ) can be shown to be a continuous non- r z1 , ••• ,zp =W along r = 0 , 1 p •• ,p-1 and cover may choose a finite number of II zi+1- z 1 l\ < ~/ 2 such that with the p spheres ~ J i = o, 1, ••• ,p. Furthermore, the cf.J. rJi = { Z \ II Z-Z i II < ~J.+14 is contained in ~ . have the property that the center Zi+1 of i~ i = 0,1, ••• ,p-1. Let z I be the first one of the zi (. r contained in ~ and 1 0- II Z - Z 11 <- fo includes a portion of for which the sphere 0L. xfL' F(Z) may be expanded in a · Lorch-power 1 Since Z is interior to series co (96) F(Z) =[ 1 F(n) (Z') (Z-Z )n I I I fo· II Z-Z II< ~(Z ). t n • 0 Simultaneously we have the F-power series L ~~cz, ;z-z ) n=O n. 00 (97) F(z) = I I II Z-Z I\ < By hypothesis, ( 96) and ( 97) agree in [5, page 72] such Z it may be shovm I II Z-Z II <. that t II Z-Z II < (98) I Now let Z -:j. Z be an arbitrary point in (99) t z = z + ~(Z ) II 2 ' (Z-Z ) II Z-Z 'II t ~(Z ) and for • t I ~( z ) • II Z-Z JI < ~ , and set 0 -56- Obviously Jl z"- z'll = ~~z ' ) < ~cz') " and hence (Z - Z ' ) satisfies (98), that is Substituting from (99) into (100) and making use of the linearity and homogeneity of the G- differential we have II Z-Z ' II< (101) ~o· This reqµires that 00 (102) \ ' ~ ~ ( Z ; Z-Z ') F(Z) = ~ 0 1 =F ( n \ Z n.r be absolutely convergent f o r all Z such that 1 ) ( Z- Z' ) n n.' II Z-Z 'II < \ o· Since the right side is an absolutely convergent series of Lorch-analytic functions for ' IJ Z-Z JI..:::: \ o and converges to F(Z) in that domain; then F(Z) is a Lorch analytic function outside of C{. Proceeding in an analogous manner we thus see it is possible to reach an arbitrary point W f. (] in a finite number of steps , at each step establishing the Lorch-analyticity of F(Z). -57Section VI ~ SINGULARITIES ON THE BOUNDARY OF THE REGION When 1fi is the complex plane it is well known that a function defined by a Laplace-Stieltjes integral may fail to have any singularities on the abscissa of convergence of the integral. Indeed, it is not difficult to construct exa.~ples for which the abscissa of convergence is finite and yet the function represented is entire. In the remaining paragraphs of this section the analogous situation is considered for an arbitrary Banach algebra. A special case is dis- cussed for which it can be asserted that there exists singularities on the boundary of the region dc . When the term singularity is used it will be in the following sense. DEFINITION 6.1 .. A point S 0 on the boundary of the region 0c s will be called a singularity of the function defined by (1) F(S) = exp( - tS)dA(t) 0 if, given any neighborhood /(( S ) of S , it is impossible to extend the definition of F(S) into ~( S ) in such a manner that it remains 0 analytic and agrees wi. th F(S) in 0 0 /?.. (S ) /) ( { . 0 It has already been pointed out that a function F(S) Lorch- analytic in a region in Cf may be expanded in a power series about any point S 0 r:J • Furthermore, the series converges not only in the largest sphere contained in the domain of analyticity, but for all points of that sphere translated in the direction of the radical of tfi . This -58- serves to extend the domain of analyticity into a cylindrical set parallel to the radical. THEOREM 6.1. Because of this fact we have If the Lorch-analytic fumtion F(S) defined by (1) has a singularity at the point S 0 and if R is an. arbitrary element belonging to the radical of then S ~' on the boundary of the region £1 , + R is a singularity of F(S). 0 Proof: S + R £ We first recall from theorem 2. 5 that if S t:. CJ'c then dc for arbitrary R in the radical, so that F(S) is defined and by theorem 5.4 is analytic in a cylinder in the direction of the radical. It also follows that if S0 is a boundary point of so is S + R. 0 Now if S 0 de then + R were not a singular point of F(S), it could then be included within the interior of the set of corwergence of some power series representing F(S) and agreeing with F(S) in de. But by the remarks made previous to the theorem this would imply that F(S) could be analytically extended back to inc lude the point S • 0 This is a contradiction. Yve now consider a case when the existence of singularities on the boundary of cJc can be assured. THEDREM 6. 2. Let the integral defining the function co ( 16) f(s) = S e-tsda(t) 0 have abscissa of convergence Rl( s ) = c, and let s f(s) on Rl(s) 0 be a singularity of = c. Then the function defined by the integral -59(34) F(S) = ~ exp(-tS)da(t) 0 possesses a si.ngulari ty at any point S which there exists an M~ t:. Proof . Let S 0 on the boundary of 0 o'c for O'll such that M~(S 0 ) = s 0 • be such a point. Assuming S 0 is not a singular point of (34), then for any r > 0 the point s1 = S 0 + rI t. ~ ' Rl(S ) > c, and F(S) may be expanded in a power series 1 00 (103) F(S) = .2::: A (S - s1 )n, n=O where n co ~ (-t)nexp(-ts1 )da(t) 1 = n ' (104) A n. The series ( 1 03) vvi 11 include the point S vergence. n=0,1, •••• 0 0 within its sphere of con- This implies however that z M~(An) 1{~(S) n oo (105) M: l F(s)] = [ - M:(s1 u n=O is a convergent power series which includes the point s within its circle of convergence. 0 = M*0 (S 0 ) Since n=o,1, ••• ' (1o6) we see that the power series (105) represent (16) at s • 0 a contradiction to the assumption that s 0 But this is was a singularity of (16). Theorem 6.2 implies that any of the known results concerning singularities of the function represented by (16) apply to integral (34) in the sense stated above. In particular we ref er to [ 9, page 58] , -60- [ 20, pages 89-92] , and [ 21 , Chapter 4] where many of these results are discussed. In conclusion we sketch the proof of a generalization of a theorem used in the theory of Dirichlet series by Cramer (?2] (23] • and Ostrowski To do so we first need LEM.MA 6.1. -t;j_ - valued entire function of the Let P(z) be a complex variable z of exponential type, i.e., there exists a nonnegative constant k such that for arbitrary (107) \\ P(z) \l < e(k+ E: ) l zl for \z I E: > 0 sufficiently large. Let co ~I P(z) = L (108) n=O Il Cz n be the power series expansion of P( z) and define the function co Q(z) = (109) L n: n n;:O 1 C /zn+ constructed by using the same C as in (108). n Then P(z) and Q(z) are related by the following formulas: Se-wzP(z)dz Q(w) = (110) 0 P(z) = - (111) 1 - 2 Ti' i S \W I ezwQ(w)dw. =k + E The proof follov;s mutatis mutandis that of the classical case and may be found in Now let defined by J• [ 1 O, pages 61-65 Oc be the open region of convergence of the function -61- 5 co F(S) = (1) exp(-tS)dA( t), 0 and let P( z) be an entire function of the complex variable z satisfying the conditions of lemma 6.1. by the integral C(S) = (112) f Introduce the new function C(S) defined exp(-tS)P(t)d.A(t) • .0 By use of (107) it ccm. be verified that the integral (112) has an open region of convergence which at least includes the set ~ = {s ca I s = s0 + ( 113 ) k I , so.me s0 E, (jc 1. From theorem 2.4 it follows that l;, Now let 1?i_ be an open set contained in ~ n CJk is non- void. in ~ and such that Further, let G(S) be an analytic function defined and such that (114) G(S) =F(S) for S e Define the region (115) Since CJc and cJk are open sets it is easily seen that S t:.. O'k implies that S + ]> I e. (jc for all Moreover, it follows that S f. Construct within ( 116) -;..;., ~ such that It l ~ k. E" C \ implies S + 'f, I E. 1./- for I bl ~ k. the set 1f k = { s [ rl I s + ~ I f If ' I t I -:S k } • -62- Certainly Furthermore, 7-.fk is non-void since it contains at least ~ n ~· 'Nk is an open set. This follovrs from the fact that the minimum distance from the compact disk S + complement of 7? 1t I, ( I ti~ k) to the is definitely positive. ~ Now let /y\ be that component of :1/k which is connected with "-' Again, 7->lk is non-void since ~ f\ C\ C: /Y-k ~ C /Y-k. r"J 11 Define the function (117) 2 1i' i where S 1 J(S) = - - f WI = k G(S-wI) Q (w)dw , + €: Q (w) represents the analytic continuation of the function defined by (11 O). If S is restricted to remain in 50 can be chosen sufficiently small 1that as w traverses the path l wl = k + E- , S - wI remains in the domain of analyticity of G(S), and hence by a slight modification of a stan::l.ard theorem of complex variable theory 99 ] [24, page it can be shown that J(S) defines a Lorch analytic function of S ,......., Moreover, since S is restricted to lie in 1v k' then by ( 114) J(S) may be written as (118) J(S) =-1- F(S-wI)Q (w)dw. 2 11 i lw l = k + E. ~ If S is even further restricted to lie in equation (1) as a representation of F(S). (118) we have (119) 1 J(S) = 211 i j (~ Jw l = k +E- 1-f k n CJk we may use Substituting from (1) into exp [ -t(S-wI)] dA(t0 Q (w)dw. -63- ""' n ~ The restriction S E Uk and the choice of 0 sufficiently small makes the inner integral in (119) uniformly convergent with respect to w (see theorem 2. 6), and an interchange of order of integration is justifiedo J(S) = (120) Carrying out this process yields 5 exp(-tS) ( 0 ~: \ i j \WI etwQ(w)dw) dA( t) , = l{ +c which by (111) becomes (121) J(S) = ~ exp(-ts)P(t )dA(t) = C(S). 0 We thus have THEOREM 6.3. Let F(S), P(z) , and G(S) be the functions described above, and define the function C(S) as in (1 12) . Then J(S) as defined in (117) is an analytic extension of C(S) into 7yk. AJ It is to be noted that J(S) need not be equal to C(S) at all points where both are defined, since this region may not be simply connected. This multi-valuedness of the extension can even be realized in the classical case where t3. is the complex plane. The importance of the last theor6Jl lies in the fact that in the classical case it plays an essenti al role in gap and density theorems . The function P( z) is an entire function vmich is associated with the sequence of exponents of a Dirichlet series, and aids in a discussion of the density of that sequence. Its function is essentially that of modifying the given Dirichlet series to one with all positive coefficients. Once the coefficients are positive the existence of -64singularities on the abscissa of convergence c a..~ then be assured. It appears reasonable that a similar type of analysis could be extended to those special Banach algebras called Banach algebras in which there exists a notion of positiveness. It is the intention of the author to carry out such an investigation in the near future. -65REFERENCES 1. Michal, A. D., and Martin, R. s., Some expansions in vector space. Journal de Mathematiques Pures et Appliqutes (1934), 13, 69-91. 2. Nagumo, M., Einige analytische Untersuchungen in linearen metrischen Ringen. Japanese Journal of Mathematics (1936), 13, 61-80. Rec. Math. N. s. (1941 ), 9, 3-24. 3. Gelfand, I., Normierte Ringe. 4. Gelfand, I., Ideale und Primarideale in Normierte Ringe. N. S. 5. Rec. Math. (1941), 9, 41-48. Hille, E., Functional Analysis and Semi-Groups. Coll. Pub. Amer. Math. Soc., (1948) , 31. 6. Proceedings of the Symposium on Spectral Theory and Differential Problems. Math. Dept., Oklahoma Agriculture and Mechanical College, Stillwater, Oklahoma. (1951 ). 7. Banach, s., Theorie des Operations Line'aires. 8. Dunford, N., Uniformity in linear spaces. Warszawa (1932). Trans. Amer. Math. Soc. (1938), 44, 305-356. 9. Widder, D. v., The Laplace Transform. 10. Doetsch, G., Theorie und Anwendung der Laplace-Transformation. Princeton Univ. Press (1946). J. Springer, Berlin (1937). 11 . Lorch, E. R., The theory of analytic fun::tions in normed abelian 12. vector rings. Trans. Amer. Math. Soc. 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