ON THE REPRESENTATION THEOREM BY THE LAPLACE TRANSFORMATION OF VECTOR-VALUED FUNCTIONS ISAO MlYADERA (Received February 18, 1956) 1. Introduction. The theory of the Laplace integral of vector-valued functions has been developed by E. Hille [2]. But, there is no theorem giving conditions that a vector-valued function is represented as the Laplace transformation of a function in B p ([, ); X). P. G. Rney [4] has recently developed this representation theory in terms of an inversion operator given by the formula (1. 1) *,«[/] = (ke* k /πt)fs-ϊcos (2ksbf(k(s + l)/t) ds, where the integral is Bochner integral. Since Rney's method is quite general, his argument can be used for Widder's operator given by the formula where / (fc) (O denotes the k-th strong derivative of /(/). In his argument, the basic space X is a reflexive Banach space for 1 < p < and is a uniformly convex Banach space for p =. The main purpose of the present paper is to give a necessary and sufficient condition in order that a function f(s) is the Laplace transformation of a function in #([, ) X) where X is a reflexive Banach space. The representation theorems are stated in terms of the Widder operator (1. 2) and we can obtain the theorems similarly as in the numerically-valued case. 2. Preliminary theorems. Let X be a Banach space. DEFINITION. A vector-valued function /(s) on (, ) into X is said to belong to B p ([, ) X) (l<i/>< ) if /(s) is Bochner measurable and :) \\ p ds <. Similarly /(s) is said to belong to Z?([, ); X) if it is Bochner measurable in (, ) and \\f(s) is bounded except in a null set. It is obvious that the class B P ([Q, ); X) becomes a Banach space under the norm
ON THE REPRESENTATION THEOREM II/( OH* = {/ll/w U p d S γ /P ίl^p < co), /C )Hoc = ess sup /(s). The following inversion formula has been proved by E. Hille [2, Theorem 1.3.4]: THEOREM 1. // <p(t) is in #i([, ω]; X) for each ω >, and if the integral o converges for some s, then /(s) = Je- St φ(t) dt i«/λ^ Lebesgue set of <p(t). The following theorem which is fundamental in the representation theory, is proved similarly as in the numerically-valued case, so that we omit the details. THEOREM 2. // for each positive integer k then /() exists and / lim Γ^- sί L fc, f [/J^-/(5) -/() fc / (<s<). 3. Representation of vector-valued functions by Laplace transformations of functions in B p ([, ); X), 1 <p<. THEOREM 3. // X is a reflexive Banach space, then a necessary and sufficient condition that f(s) can be expressed in the form f(s) =Je- st φ(t) dt (s>), o where φ(t) e B p ([, ); X), p fixed, 1 < p <, is that (i) /(s) has strong derivatives of all orders in < s < and /() =, (ii) there exists a constant M such that (/ll I*..Γ/JP Λ) VP^ M (A = 1,2,...). PROOF OF NECESSITY. Suppose f(s) = fe- st φ(t) dt (5>), and <^(O e β p ([, ); X). Then using Holder's inequality we have
172 I. MIYADERA l/(s) 11^ JV S ΊI <P(t)\\dt^(f\\ φ(t) ψ where \/p + \/q = 1, so that (i) is necessary. inequality gives I*,. [/] * = Wfe-* u/tj M (y)%(ιθ du Another application of Holder's 4/ / b \ so that oc J ϊ== J \J k Hence (ii) is necessary. =j\\<p(u)\\"du. PROOF OF SUFFICIENCY. By (ii) and Holder's inequality 1 J I*,.Γ/] Λ^^ for every s > 1 and for each positive integer k. Thus we obtain by Theorem 2 and (i) (3. 1) lim fe- st Ljc,t[fldt=f(s) ( < s < ). Since X is a reflexive Banach space, B P ([Q, ) X) f 1 < p <, is reflexive (see S. Bochner and A.E. Taylor [1] and B.J. Pettis [3]). Therefore ^([, ) X) is locally weakly compact. Since (J\\Lt,t[f]\\*dt) l/p^m, J o there exists an element φ(t) of B p ([, ) X) and an increasing sequence {&} of positive integers such that for every y* in B ([, ) X) Let ** be an arbitrary element of X*. element of B p ([, ) X), Then if #(/) is an arbitrary
ON THE REPRESENTATION THEOREM 173 oc x*{fe- st g{t) dt)=yf(g( )) defines an element in *( [, ) X) for each s >. Thus we have oc y*(<p( )) = x*( fe- st φ(t)dt) = lim #(I*. [/]) = lim* ( ΐe- s 'L ki, t [f]dt), so that by (3. 1) Hence and the theorem is proved. **(/(s)) =x*(fe~ st φa) dt). oc /(s) - je- st <P(t) dt ( < s < ), 4. Representation of vector-valued functions by Laplace transformations of functions in ft, ([; ) X). The following Lemma is due to P. G. Rney [4]: LEMMA. // {T σ \ < a < } is a set of bounded linear operators on a separable Banach space X into a reflexive Banach space Y, and if T σ <J M independently of a for all a >, then there exists an increasing unbounded sequence {>i} and a linear operator T on X into Y with \\T\\^M f such that lim y"(t σ< (*))«/ (TOO) for every x in X and every y* in Y*. THEOREM 4. // X is a reflexive Banach space, then a necessary and sufficient condition that f(s) can be expressed in the form oc /(s) = fe- st <PU) dt J o where <p(t)^b(lo, ) X), is that (i') /(s) has strong derivatives of all orders in < s <, (if) there exists a constant M such that for < s < (s>), ^ ll/ (fe) (s) ^ M (k =,1,2,...). PROOF OF NECESSITY. Suppose
174 I MIYADERA f(s) = fe- st φ(t)dt o and φ(t) is in #([, ) X). Then (i') is obvious. Since (ii') is necessary. PROOF OF SUFFICIENCY. By (i') and (ii') ' 11/( I ^M, and for < / < so that /() = and «- Λ ess sup II *,«[/] II^M (* = 1,2,...), *,.[/]<«I = O(s) (s->). Thus we have by Theorem 2 (4. 1) lim f e- st L kf t [/] dt^f(s) ( < s < ). Let <KO be in Li(, ). Define It is obvious that {T fc } is a set of bounded linear operators on a separable Banach space Li(, ) into a reflexive Banach space X and T fc <I M. Thus, by the preceding lemma, there exists an increasing sequence {ki} of positive integers and a bounded linear operator T on Li(, ) into X with [ T <J M, such that for every x* in X* and every in Li(, ), (4. 2) lim **OΓ fc.(</) = x(t(φ)). Let ω be an arbitrary positive integer. and Since B 2 ( [, ω] X) is reflexive there exists an element <p<»( of J? 2 ([, ω]; X) and a sequence such that for every j> ω) in *([,α>]; X) lim y?-)cl*ί» Let Λ:* be an arbitrary element of X*, Then if ψ(t) is an arbitrary
ON THE REPRESENTATION THEOREM 175 Lebesgue measurable function such that f\ ψ (f) 2 dt < and ψ(t) = for t > ω, and if g (O is an arbitrary element of Z? 2 ([, ω]: X), then j defines an element in B*([, ω~ ; X). Therefore we have ω lim W Γ*(OL fcί, ί [/]Λ) = lim x*( f <Kt)L kut [f]dt) = W f Ψ(t)φ<> ω Kt)dty On the other hand, since such (/) belongs to Lj.(, ), lim x*(t H (ψ)) = lim * ( f φ{t)l H, t[f]dt) = x*(t(ψ)). Thus so that **/ Γ Ψ(t)φ( ω Ht) dή = X*(T(ψ)), ϋ (4. 3) Γ() =fψ(t)<p^(t) dt for every ( such that ( e L 2 (, ω) and (t) = for / > ω. It is easy that if ω' > ω, then φ^(t) =-- φ<. ω ')(t) ίor almost all / in (, ω). We now define a function φ(t) on (, ) into X by ίor ω -l^t <φ (ω = 1, 2, 3,....) From the definition of φ(t), it is obvious that φ(t) = ^^ω) (/) for almost all / in (, ω), <p(o is Bochner measurable and <KO II 2 is integrable in any finite interval. Hence (4. 3) may be written as follows: (4. 4) T(φ) = fψ(t)φ(t) dt i for every ψ(t) such that ψ(t)el«(, ω) and ψ(t) = for t > ω. Let us put *«* f «={ o otherwise, where ξ and /^ are any positive number. By (4. 4), we have
176 I- MIYADERA ξ+h ) = -jr f <P (O dt. OO Since T(Φt,n)\\^M J\ φ iyh (t) \dt^m, we get Thus Kf)ll = lim 4- f <p(t)dt\\^m for almost all ξ >, so that <p(t) is an element in #([, ) X). If we now define the new operator T" on Li(, ) into X by T'() = Γ φ(t)ψ(t)dt, where (/) e Li(, ), then T' is a bounded linear operator Li(, ) into X The setz> = U {( ( e L 2 (, ω; and ( = for / > ω] is dense in Li(, ) and T() ^fφ(t) Thus we get <P(t) dt = T'() for any e D, so that Γ - T'. (4. 5) T(φ) = fψ(t)φ(t)dt i for each ( ewo, ). Let ( = e~ st. Then, by (4. 1) and (4. 5), for each x* <= X* and for each 5 > **(/(*)) = lim x*( f e- st L ki>t [f]dt) fc{ >OO * J ' so that Ίci > = x*(f e- st <p(f)dt),) /(s) = fe- st <P(t) dt (s> ). Thus the theorem is proved. Since the above method is quite general, Rney's result [4; Theorem 9. 2] is also true for a reflexive Banach space.
ON THE REPRESENTATION THEOREM 177 Theorem 4 shows that Hille's condition [2; Theorem 1. 3.5, (1. 3.15)] is also sufficient to represent the function /(A) as a Laplace transform when X is reflexive ( = locally weakly compact). We shall show by an example that the reflexivility is essential for the sufficiency of Theorems 3 and 4. EXSAMPLE. Let C [, ] be the family of all real-valued continuous functions x(u) of u on the closed interval [, ]. The norm of x(u) is defined as the maximum of its absolute value in [, ]. C[, ] is obviously a Banach space. Let X be the family of all bounded linear transformations on C[, ] into itself. It is well known that X is a Banach algebra. We define (4. 6) T(t) [*(ιθ] = e-*x(u + /), a ^ ). Then {TOO /1> } is a semi-group of operators satisfying the following conditions: (C) T(t) e X and T(t + s) = T(OT(s) (t, s;> ), CC 2 ) IITCOII^e- 1 tf^o), (CO lim T(t + h)x - T(t)x = (f ^, x e C[, ]). Λ-» Furthermore we have (CO T«+A) -TCOl^e-t «+A>, /> and AφO). In fact, we can always find the element of C[, ] such that x(t) = l, x(t + A) = 1 and max ΛΓ(/) = 1 for any given t and A, where t >, / + h > and AφO. For such an element x we have \\T(t + A)ΛΓ - TCO^II^^", so that (C 4 ) holds. We denote the infinitesimal generator of T(t) by A and the resolvent of A by R(s; A). From the theory of semi-group of operators, (4. 7) R(s;A)x= f e"*t(t)x dt for all s> and for all # e C[, ]. R(s; A) has derivatives (strong derivatives in the sense of X-norm) of all orders by the resolvent equation R(s A) - R(t A) = - (5 - O#(s il)«α A). By (4. 7) and so that -,.fc + l I*,»[i?( A)x\ = --j- (-J-) Ju*e-wT(u)xdu, ςfc + l ςfc + 1 J, (4. 8) -^j- R«Ks; A) \\ ^ -^y- J t*e"*dt - 1 (* =,1,2,...
178 I- MIYADERA and (4. 9) JII I*,.[/?C ; A)] " Λ ^ " Γ(«) f JM - 1/p ' (l<ί<;* = 1,2,...). Thus, the vector-valued function R(s A) on (, ) into X satisfies the sufficient conditions of Theorems 3 and 4. If there exists an element φ(t) of B p ([, ) X), 1 < p <:, such that then, by Theorem 1, R(s A) =Je- st <P CO Λ? > ), lim [ LftίRC A)] ^CO I = fc-> for almost all t >. On the other hand, we have from (4. 7), (C?) and Theorem 1 lim L fc,«[i?( A)x\ - T(t)x = λ for all ί > and for all x e C [, ]. Thus for almost all t >, so that TCO is a Bochner measurable function on the interval (, ) into X such that for < t, s < Then, by Hille's theorem [2, Theorem 8.3.1], lim Γ(* + h) - ΓCO I = for all / >. This is contrary to the condition (C 4 ). Thus Xis not reflexive and R(s A) can not be represented as Laplace transformations of function in B p ([, ) X) for each p, 1 < p <:. 5. Representation of vector-valued functions by Laplace transformations of functions in Bi([, ) X). THEOREM 5. Let X be a Banach space. A necessary and sufficient condition that f(s) can be expressed in the form dt (s>ΰ), where φ(t) e #i([, ) Ci") f(s) has strong derivatives of all orders in < s < and /C) =, (ii")
ON THE REPRESENTATION THEOREM 179 oc Ciii") lim f\\ U y,[/] - L 3ί,[/] dt =. PROOF OF SUFFICIENCY. By (iii"), there exists an element φ(t) of #i([, ) X) such that lim Γ I I*, * [/] - PCO I d/ =. Then there exists a positive integer k such that f\\l k, t [f]\\dt^l+j\\φ(t)\\dt for k ί> ko. Let #* be an arbitrary element in X*. By Widder's theorem [5; Chap. VII, Theorem 12a], there exists a function of bounded variation a x * (t) such that **[/(«)] = fe~ st da x *(t). On the other hand, we obtain from the inversion formula tfa*co-tf**c + ) = lim j #*(Lfc,u[/]) dw /() = implies α^co +) =, so that = f x*(<p(u)) du. Thus so that OO x*(f(sy) = f e- st x*(<p(t)) dt = x*(fe- st φ(t) dt), * /(s) = Γ e- si <P(f) dt (s > ). CO The necessity of the theorem may be proved similarly as in numericallyvalued case.
18 I. MIYADERA REFERENCES [1] S. BOCHNER AND A. E. TAYLOR, Linear f unctionals on certain spaces of abstractly-valued functions, Ann. of Math., 39(1938), 913-944. [2] E. HILLE, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publications, New York(1948). [3] B. J. PETTIS, Differentiation in Banach space, Duke Math. Journ., 5(1939), 254-269. [4] P. G, ROONEY, An inversion and representation theory for the Laplace integral of abstractly-valued functions, Canadian Journ. of Math., 5(1954), 19-29. [5] D. V. WiDDER, The Laplace transform, Princeton(l941). MATHEMATICAL INSTITUTE, TOKYO METROPOLITAN UNIVERSITY. ADDED IN PROOF. The formula (4.5) can also be obtained from the following theorem. // T is a bounded linear operator on L x (, ) into a reflexive Banach space X f then there exists an element φ(t) e 2?([, ) X) such that = f ψ(t)φ(t)dt, Φ(t) e XJCO, OO).