J. Math. Soc. Japan Vol. 40,E No. 2, 1988
Fractional powers of operators
By Celso MARTINEZ, Miguel SANZ and Luis MARCO
(Received Nov. 5, 1985) (Revised Dec. 5, 1986)
1. Introduction. The study of fractional powers of operators has an extensive history. The semigroup of fractional powers of a bounded operator was studied by Hille [5] (1939). Fractional powers for the negatives of the infinitesimal generators of bounded strongly continuous semigroups have been discussed by Bochner [3], Phillips [19], Yosida [22] and Balakrishnan [1], who afterwards (1960) [2] gives a new definition and extends this theory to closed linear operators A in a Banach space X such that ]-00, 0[ is contained in the resolvent set p(A), and the resolvent satisfies (1.1) IIp(p+A)-111<_ M < 00, o slna2r 1 l JaW_ For 0 (1.2) JwJaw = sma~ +00 a 1 (1~+A)1 -1+ ~ 2 ]Awd+sin(a)Aw. ;uZ ~ For n Ja Ji _ Ja+i 332 C. MARTINEZ,M. SANZ and L. MARCO Almost at the same time two different definitions appear, introduced by Krasnosel'skii-Sobolevskii [16] and T. Kato [9], limited to particular cases, and equivalent to the one given by Balakrishnan when A is densely defined as Nollau proves in [18] (1967). In his first work out of a series of six on fractional powers, Komatsu proves the additivity for the first time, namely : (1.3)«1~ = J«+~ with no restriction for A or for the exponent. Other ways to prove (1.3) can be found in [11], [18], [8], althoughall of them are long and difficult. When D(A)is dense in X, the resolvent of J" has been studied in several papers ([2], [9], [21], [11]) and consequentlythe spectralmapping theorem has been proved. Another property, deduced from the resolvent of J" was the multiplicativity(equality between the power of power and the power with base A and productexponent) also limited to the case in whichA is denselydefined. An extensiveand annotated bibliographycan be found in Fattorini [4], Hirsch [7] and Krein [17]. We have to point out that the f power has its domainand range contained in D(A), which makes the resolvent set of J" to be empty when A is not denselydefined. Consequently,with this notion of power, there is no sense in establishingthe two propertiesmentioned above (the spectral mappingtheorem and multiplicativity).Another obviousfault of the j' power is that J1~A when D(A)~ X. These disadvantagescan be avoidedwhen A-1 is boundedconsidering the definition well known in the literature (for examplesee Krein [17] and Krasnosel'skii- Sovolevskii[16]). Nevertheless,there is not yet a definitionwhich could be satisfactory in that sense when (1.4) D(A)~ X and OEQ(A) (wherec(A) is the spectrumof A). This is one of the purposesof this paper. The definitionwe shall take is originalonly when we are preciselyunder assumption(1.4) and it will allow us to develop a basic theory about fractional powerswith natural and direct proofs,in which the fact of D(A)being dense or not is irrelevant. Henceforth we shall keep the notationA" for this definition. On the other hand, we shall introducethe operator Fractional powers of operators 333 T(s, n) = A"- ~(-1)psp a C(A+~)'1]p(A+~)" p=o p (where(;)=a(a-1)...(a_P+1)) 1 about which there is nobibliographical background. At most, some authors (see [9] and [18]) have considered estimates and integral representations of the difference JA-JA+E (where the subindex indicates the base operator). This paper is divided as follows : In Section 2, we give the definition of the A" fractional power, we prove that the domain D[(A+s)a] does not depend on E>_O and the fundamental property CA(A-~~)-1]" = A"[(A+~)-1]" As a consequence A" turns out to be closed. The additivity (Aa+P=A"A~) and the connection between A" and J" are dis- cussed in Section 3. In particular we offer a very easy way to demonstrate property (1.3). Section 4 is devoted to the resolvent of A", the spectral mapping theorem and the multiplicativity. We establish two integral representations of the T(s, n) operator and we examine its range in Section 5. As an application we offer a counterexample for the separation of the domains D(Ac) and D(AB) even if Re a=RejS. Throughout this paper we assume that A is a non-negative operator in a Banach space X and a is a complex number with Re a>O. 2. Construction of fractional powers. t)EFINITION2.1. If A is bounded we define Aa=J". If A is unbounded and OE p(A), we define A"=[(A-1)a]-1 If A is unbounded and OE Q(A), we define A" by the formula (2.1) A"cp = lira (A+s)"cp 6-'0+ where the domain of A" is the set of all cpE X which the limit (2.1) exists. REMARK 2.1. It's obvious that Al=A. 2.2. If A is bounded and p(A) contains ]-oo, 0], A" is a bounded operator which can be obtained from functional calculus for bounded operators (see W. Rudin [20]). 2.3. If A is bounded and non-negative, it is easy to prove that A" is the 334 C. MARTINEZ,M. SANZ and L. MARCO limit, in the uniform operator topology of (A+E)" as E-+0+, which we shall denote henceforth by (2.2) A" = II II-- lim (A+~)". 2.4. If A is unbounded and OE p(A), the adopted definition can be found in [16], and it is known that A" is an extension of j'. 2.5. If A is unbounded and OE a(A), then for each E>0, A+E is non-negative and OE p(A+s), therefore applying the latter remark and making use the dominated convergence theorem, it follows that A" is an extension of J". THEOREM 2.1. Let Rea>0 and >0. Then we have that A(A+E)-1 is non- negative and (2.3) D[(A+] = D(A"), (2.4) [A(A+~)-1]a = Aa[(A+~)-1]a PROOF. The first statement is a consequence of the following identity (2.5) A(A+E)-1+~ = (+1)(A+_ rI +1 (valid for each i >0). In order to prove (2.3) and (2.4) we split the argument into four steps. I. If A is bounded and OE p(A), (2.4) is a consequence of functional calculus. II . Let A be bounded and OE Q(A). Since (A+s /(i +1)) satisfies the assumption of I, for each >0, it is enough to take powers of exponent a on both sides in (2.5) and then to apply (2.4) replacing A by A+e~/(~+1). Taking II II-lim~.,0+ we obtain (2.4). III. Let A be unbounded and OE p(A). Through the identity = and using that A-1 satisfies the assumption of II we have (2.6) [(A-f-~)-1]a_ -a(A-1)a[(A-1~~-1)-1]a Hence it follows that the ranges of (Aand [(A+e)-1]a are equal, there- fore (2.3) is valid. On the other hand, operating A" on both sides in (2.6) we obtain (2.4). IV. The general case when A is unbounded and OEa(A) can be proved in the same way as II. COROLLARY2.1. A" is a closedlinear operator. PROOF. In view of (2.4) and the commutativity of A" with [(A+~)-1]a, ~ Fractional powers of operators 335 for any >0, we obtain : (2.7) A" = (A+[A(A+~)-1]a and taking into account that [A(A+E)-1]a is bounded and (A+s)" is closed (since it is the inverse of a bounded operator) we conclude the closedness of A". REMARK2.6. If A is unbounded and OE p(A), identity (2.7) implies that (A+s)" = (EA-1+1)"A". Moreover, from the integral expressions (1.2) through a simple calculation we obtain 1= II Il--lim(~A-1+1)" E-+0+ and therefore (for each c E D(A")=D[(A+s)a]) A"cp = lim (A+E)"cp e-0+ thus the definition given in (2.1) for unbounded operators with OE Q(A) is valid for any unbounded operator. 3. Additivity. When A is bounded and OE p(A) the additivity is a consequence of func- tional calculus and if 0 o A) it follows immediately using (2.2). Thus, accord- ing to Definition 2.1, it is verified that A"A~ = AAA" = for all a, /3 such that Re a, Re /3>0 if A is bounded or if A is unbounded with OE p(A). Therefore there remains to prove the additivity when A is unbounded and OE Q(A). In order to prove it first we give the following : LEMMA3.1. Let Re 7> 0, Re 8> 0 and >0. Then we have co E D(A) i f and only if [A(A+~)-1]aco D(A7) and (3.1) (A+~)r[A(A+e)-1]acD = [A(A+~)-1]s(A+~)rcp. PROOF. Since [(A+E)-1+2]-1 and [A(A+~)-1+p]-1 commute for A, p>0, it is easily shown that the bounded operators [A(A+~)-1]a and [(A+1]E)-r com- mute. Hence, it follows that the condition is necessary and (3.1) holds. Conversely, if [A(A+~)-1]apED(Ar), taking an integer n such that n>Re8 and using the additivity of the powers of A(A+E)-1 and the first part of this 336 C. MARTINEZ,M. SANZand L. MARCO lemma, we conclude [A(A+E)-1]ncP E D(A). Writing b=[A(A+E)-1]7- o we have Therefore I-1 IT (l+2 °[(A+) J2°)((/'-s(A+s)~b) D(A) for m=1, 2, •••. That is : Hence taking m such that 2'n > Re r we have c D(AT) and by recurrence we obtain that CA(A-~-~)-1]7t-2~~.., , A(A+~)-lcp, ~o belong to D(AT). THEOREM3.1. Let A be unbounded and OEQ(A) and let Rea>"o, Re/3>O. Then = APA" = A"A~ . PROOF. Let cpcD(A"+P). From (2.3) we know that cp e D[(A+s)"+P] (for any s>O) and since we have, by (2.3) and (2.4), that coE D(A) and Hence, using Lemma 3.1, AacocD[(A+E)~] and Applying [A(A+E)-1]~ to both sides we obtain AAA"cp= A"+'9~o. Conversely, let SPE D(Aa) such that A"cpE D(AB). In view of (2.4) and the first part of this proof we have A~Aa[(A+s)-1]«+~W = A"+~[(A+E)-1]«+~~ (3.2) _ [A(A+eYh]a+1 c . On the left hand side of this equality the operator [(A+~)-1]can be placed on the left, since it commutes with the powers of A, as it is easily followed Fractional powers of operators 337 from Definition 2.1. Thus, from (2.4), we find that [A(A+~)-1]a+~p E D[(A+E)] and A~A«St,= (A+~)a+~[A(A+E)-1]a+Pco Hence, using Lemma 3.1 and Theorem 2.1 we conclude that coE D(A"+~) and A'9A"gyp= A"+~cp. In the following theorem we give a very simple proof of the well-known property (1.3) : J" JP - Ja+P (see [11], [8] and [18]). THEOREM3.2. Let Re a>O and Re jS>O. Then (i) E D(J") if and only i f ~OED(Aa) and Aaco D(A). (ii) J"=A" if and only if D(A)=X. (iii) Ja J!S=Ja+!S PROOF. A" is an extension of J" in view of Corollary 2.1 and Remark 2.5. The necessary condition of statement (i) is evident since RanJ"CD(A). Conversely, let ~oE D(A") such that A"co D(A) and let us take an integer n such that n > Re a. We know lim [A(A+A)-1]ncn= cp A +~ and lim J"[A(A A)-1].Th~c'= lim [A(A+AY 1]nAaco = A"~o since A is non-negative and A"~o belongs to D(A). Therefore ~ D(J) and the statement (i) is proved. If ~(A)=X, it is obvious by (i) that A"=J". Conversely, if D(A)~X and. n is an integer such that n> Re a, there is some cpED(AT) such that Ancp D(A). Consequently from A"An-a~o=Ancp we obtain that A" is a strict extension of J". Statement (iii) follows at once from (i) and Theorem 3.1. 4. Spectral theory. Multiplicativity. Balakrishnan on [2] proves the spectral mapping theorem for J", based on the integral representation of the resolvent of j' with a suitable choice of the exponent. In this section we shall give a similar theorem for the resolvent of A". 338 C. MARTINEZ,M. SANZand L. MARCO THEOREM4.1. Let 0< 131z f (,~) 2~ci1 + ,I~e-tiny1 _ 1 Then, for each E Up, we have that -p E p(AP) and (4.1) (p+A~)-1 _ fp(p, 2)(2+A)-1d2. PROOF. Let us denote the right hand side of (4.1) by R(oc). It is clear that the integral converges in the uniform operator topology and therefore R(4u) is a bounded operator. The proof will be given in four steps. I. If A is bounded and OEp(A), (4.1) follows easily from functional cal- culus for bounded operators (see [20]). II . Let A be bounded and OE Q(A). Then A+s satisfies the assumption of I for any s>0. Therefore --pE p[(A+s)~] and (4.1) holds for the operator A+s. An easy calculation gives 'u 'u Red 1--Re1 (C not depending on s). Hence we conclude that R(p)(p+A~) _ (a+N)R(;u) =1. III. Let A be unbounded and 0 E p(A). Given p E Up then ;u-1E U~ and taking into account that A-1 is an operator of the type considered in II we get ` fp(p-1, 2)(2+A-1)-1d2 0 replacing this expression in the identity and through a simple transformation in this integral we obtain (4.1). Iv. Let A be unbounded and 0E o (A). Taking into account that for all a>0 [~u+(A+s)~]-1 = 2)(2+A+s)'1d2 just like in II, that lim [IC+(A+s)']-1-R(p) ( = 0 F-o+ which yields that for all c E D(AP) _02 Fractional powers of operators 339 R(p)(p+A~)cp = cp. On the other hand, given cpE X and E>0, the element R(p)cp--[~C+(A+E)~]-1~ = f p(p, A)(A+A+E)-1(A+A)-1cpdA 0 belongs to D(A) and furthermore lim A[R(~C)~-(;u+(A+~)~)-l~p] = 0 which yields that lim [p+(A+s)~][R(4) o-[p+(A+s)~]-1~o] = 0 E-.o+ and consequently R(p)~pE D(AB) and (4a+A'5)R(4a)So= cP COROLLARY4.1. Let Rea>0. Then (4.2) o A") = z': zE U(A)}. PROOF. This result can be proved from Theorem 4.1 in a similar way to the corresponding well known statement for J" when D(A)=X, given by Bala- krishnan in [2]. THEOREM4.2 (Multiplicativity). Let 0 IIp(p+A")-11 M, o~p<+oo (where M=sups>olip(;~+A)-111). On the other hand, since (A")n=A"n for positive integers, by additivity, it is enough to show . (4.3) for 0 IIC( +)J ( )II- (4.4) (A")Pcp= A"Pcp for all cpE D(A"). Let cb D(A"~) and p>O and let us denote R(oc)=(p+A")'1. We have (4.5) (AR(p)cb = A"~R(p)cb = R(IC)A"~~b. Since A"=(A")P(A")1-P we obtain that A"R(p)cb E D[(A]. Hence we conclude that (bE D[(A")P] and commuting R(p) with (Ain (4.5) we get (AO = A"~cb. Conversely, let ~bED[(Aa)1S]. From (4.4) we have AaPR(~C)c~= (A«)!R(p)cb = R(p)(A")P~b. Since A"=AaPA"-"P we obtain that A"R(p)c1E D(A"~) which implies that cpED(A"~). 5. A Taylor rest. In this section we shall give two integral representations of the vector n (5.1) T(E, n)~o = A"gyp- a (-1)pEp(A+~)a-pSP p=o p where Rea>O, s>O, n is a positive integer, c D(A") and by (Ay -P we denote the operator [(A+E)-1]p(A+s)". The first one of them (Theorem 5.1) is a consequence of Taylor's formula and it is less interesting than the second one (Theorem 5.2) due to the fact that the integrand contains a fractional power, dependent on the variable of integration. As a consequence of these results we can study the range of the operator T(E, n). In order to prove the first integral representation we need the following Fractional powers of operators 341 lemma. LEMMA 5.1. Let Rea>0 and SPED(A"). Then the function E H (A+~)"cp defined from ]0, +oo[ into X is continuous. PROOF. When 0 II(A+~)«~p -(A+~')"~pll K(a 1-~, 1 Rea , M) Rea(1 --Rea) II~II easily obtained from the integral formula (1.2) (where K(a, M) is constant in relation to E and i'). When 0 (~1+~)«~-(A+~')"SP = (A-F-)"JZ(A+~)"~2~'--(A+~')"'2(A+~)"'ZSP +(A+s)a/2(A+s )a'Zcp-(A+s')"2(A+~')a'2cp and the latest estimate. Finally, case Re a >_2 comes down with no difficulty to the former two. THEOREM 5.1. Let n be positive integer and Re a>0, >0 and co D(A"). Then, the function tHtn(A+st)«-n-1p is integrable over [0, 1 ] and (5.2) T(e,n)cp = (--1)' 1~n+1(a-n) a 1)t'A+ ( st)a-n-lSDdt. n a PROOF. Let q> Re a with q an integer and ~bE D(A). Differentiating under the sign of integration in formulae (1.2) (refering to the operator A+), it is easy to obtain that the function ~ By induction on n, it follows that this function is infinitely differentiable and dsn (A+~)«~ = a(a-1).....(a-n+l)(A+s). Therefore using Taylor's formula we obtain r1 b p=o p ~ ~ (5.3) a 1 +(a-n) n (~~E)n+1 (1_t)nCA+~+t(~_e)]a-n-l~dt (for £>0 and 7>O). Applying (5.3) to vector cb=(A+p)-qp (where p E p(-A)) and commuting n 342 C. MARTINEZ,M. SANZ and L. MARCO (A+p)-1 with operators (A+s)'1, (A+E)" and (A+i )" we obtain _E)- (A+(A+2)a~o_ =0 a)(,P(A+e)aPSo] (5.4) p =(a-n) a (~!E)n+1 (1_t)n[A+E+t()]a-n-1(A+p)dt. n o Lemma 5.1 allows us to commute operator (A+,u)-q with the integral sign on the right hand side and hence we obtain (5.3) with cp replacing ~l' and passing to the limit as ri-- O+ we get 1 (5.5) T(E, n)~p = (-1)n+1~n+1(a_n) a lim (1_t)n[A+E+t(~_e)]«-n-1cpdt. n ,-+o+ We now show that (1-tY I[A+E+t(ii--E)]"-n-1cpll is majorized by a function of t (independent of r~) integrable. In order to prove it we shall consider three cases : If Re a--n-1 (5.6) T(~, n) cP _ _~n+1slna~c ~ +°°o (1+;)n+2 ~" A+ 1+2d2cp.E -l (A+~)a-n PROOF. First we shall prove the identity : [A(A+~)-1]a = (-1)p~p a (A+~)-p p=o (5.7) (+'0 o (l+;)n+2 2+l o1 Fractional powers of operators 343 Applying, then, both sides to (A+e)"cp and using (2.4) we shall get (5.6). Let's start considering 0 +sin a- A(A+e)-1 2 sina2r +°° 2a-1 -1 = (1 ±A)(1±2) ) [(1-A)A+e]A A+ e~ (A+e)-1d2 +sin a A(A+e)-1 2 after making some operations in the integrand. Using the identities [(1-A)A+e]AA+1~X)1 (1-A)A+ 22±1 +e2(A±21 (22±1)2 1+~ ( sina~r +°° 2«-1(1_2) it n o (1±2)(1+22) dA =1-sin a Z (the integrand is easily calculated by means of the residue theorem). We con- clude : ~ « - [A(A+e)-1]" = 1-ea(A+e)-1-e2 sma ~ ~r +o (1+2)s A.-}-1+~ e 1d,I (A+e)-1. Atthe same time, ifwe develop thelast integral inthe following way: 0 (1+2)3(A±~+1 )1d2 = [ 0 (1+/l)3(A+)(A±)1d2](A+1 we obtain, through easy calculations that e [A(AF)1J" =1-ea(A+e)-1+ 22I a(a_1)(A+e)-2 sina7r -1 (1+2) (A±) d2](A+2 after taking into account(by the residuetheorem) that sinair ) +°° 2" d2 a(a-1). ?C o (1+~)3 21 Iterating this process we obtain (5.7). Now let 0 < Re cx< n+1 with n>1, the identity (5.7) can be easily proved by induction on p such that p-<_Rea< p±1. COROLLARY5.1. Let n + 1 > Re a and a * 1, 2, , n. Let cpED(Aa) and £>O. Then, if OS h <1, the vector T(, n)cp belongs to D(A1) if and only if, coED(Aa+h). 2 344 C. MARTINEZ,M. SANZand L. MARCO PROOF. Since A is a non-negative operator, the integral +00 2 d~ Jo (1+A)n+2A "A+-j7) is convergent and (5.6) implies that T(, n)cpE D(A' 1). In the same way, from (5.6) it follows that A(A+E)T T (s, n)co= K(A+E)«cp+cb where K is a non-Zero complex number, depending on (E, a, n), and ~i5ED(A). As D(A)CD(Ah) we conclude that A(A+~Y T(E, n)cp belongs to D(Ah) if and only if (A+E)«cpE D(Ah), namely if and only if So D(A«+h). Application to the separation of domains of powers. From Corollary 5.1 it easily follows that if 0 (ptcp)(x) = cp(x+t) for each x >0, t>0 and cPE E. p : ]0, +oo[ --~ ]0, +oo[ will be a continuous, increasing, and bounded function and lima-.0+p(x)=0. YP = {WEE: pcp is continuous, bounded and such that lira p(x)cD(x) exists and is finite} x-o+ (understanding by pcp the usual product). IJco= Ilpcoll© (with 1k1'II= sup Icjb(x)I) for each cPEYp. x>0 L = {cpE Yp : p is differentiable and cp'E Y}. X p = L (closure with respect to (Ye, II II)). Fractional powers of operators 345 COUNTEREXAMPLE.We shall work in the space X,, with the norm induced by (Ye, II ID. The following statements can be easily verified : 1. (Ye, I II) is a Banach space (and therefore so is (Xe, II)). 2. Pt(Yp)CY, and Pt(L)CL. Moreover Pt is contrastive on Yp and strongly continuous on L. Consequently strongly continuous and contrastive on X,. 3. The semigroup Pt lx p has as generator an operator, A, that in its domain D(A) coincides with the derivative. 4. For all cpE D(A2) it is fulfilled (with respect to the operator - A as base of powers) (for each E>O). (5.8) (AFE ~ )(x) = sina2r~ o A«-1+°°e-2t(1_e-Et)W'l(x+t)dta d2. This is due to an integral expression of FET obtained from the integral formulae (1.2) and also to the fact that, being A an infinitesimal generator of a bounded strongly continuous semigroup, the resolvent is Laplace's transform of the semigroup. Let a * 1 with Re a=1 and let f E E such that f(x) - - (a+1)1a(a-1) x«+1 if xE]0, b[ (b>0) 0 if x E ]c, +oo[ (b c-x E t (.AFf) E (x) - sinarc~ I'(a) o t1-a(1-e t )f"(x+t)dtif 0 Through considerations of elementary calculation we notice that (AFEf )' can be written in the form (AFEf)'(x) = h(x)+K(a, ~)g(x) d~>0 where h is a function with finite limit when x tends to zero, K(a, E) is a non- 346 C. MARTINEZ, M. SANZ and L. MARCO zero constant, and b-x g(x) = 0tl-a(x+t)a-2dt if x E ]0, b[. Hence, we get xg'(x) = -(b-x)1-aba-1 obtaining afterwards with no difficulty that g(x) = go(x)-ln x if x E ]0, b[ go being a certain function with finite limit when x tends to zero. Consequently, once chosen function p so that lim p(x)lnx = -oo x-o we shall obtain that (AFEf )' Y,. For example, we can take the function p(x) - 1+ I In x 11/2 if 0 REMARK. This counterexample, at least with the same choice of function f, cannot be generalized to the space of measurable functions whose product by a suitable function p (with the imposed conditions here) is p-summable (1 <_ p<+oo). This is essentially due to the fact that ~lnx~gdx < +oo (8>0 and q€R). Bibliography [1] A.V. Balakrishnan, An operational calculus for infinitesimal generators of semi- groups, Trans. Amer. Math. Soc., 91 (1959), 330-353. (MR 21, X5904) [2] A.V. Balakrishnan, Fractional powers of closed operators and the semigroups gener- ated by them, Pacific J. Math., 10 (1960), 419437. (MR 22, #5899) [3] S. Bochner, Diffusion equation and stochastic processes, Proc. Nat. Acad. Sci. U.S.A., 35 (1949), 368-370. (MR 10, #720) [4] H.O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its appli- cations, 18, Addison-Wesley, Massachusetts, 1983. [5] E. Hille, Notes on linear transformations, II. Analyticity of semigroups, Ann. Math., 40 (1939), 1-47. [6] E. Hille and R.S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., 31, Providence, 1957. [7] F. Hirsch, Extension des proprietes des puissances fractionaires, Seminaire de Theorie du potentiel, Lecture notes in Mathematics, Springer, 563, pp. 100-120. [8] H. W. Hovel and U. Westphal, Fractional powers of closed operators, Studia Math., 42 (1972), 177-194. (MR 46, X2481) [9] T. Kato, Note on fractional powers of linear operators, Proc. Japan Acad, 36 s0 Fractional powers of operators 347 (1960), 94-96. (MR 22, #12400) [10] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13(1961), 246-274. (MR 25, #1453) [11] H. Komatsu, Fractional powers of operators, Pacific. J. Math., 19 (1966), 285-346. (MR 34, 41862) [12] H. Komatsu, Fractional powers of operators, II. Interpolation spaces, Pacific. J. Math., 21 (1967), 89-111. (MR 34, X6533) [13] H. Komatsu. Fractional powers of operators, III. Negative powers, J. Math. Soc. Japan, 21 (1969), 205-220. (MR 39, #3340) [14] H. Komatsu, Fractional powers of operators, 1V. Potential operators, J. Math. Soc. Japan, 21 (1969), 221-228. (MR 39, #3341) [15] H. Komatsu, Fractional powers of operators, V. Dual operators, J. Fac. Sci. Univ. Tokyo Sect. I, 17 (1970), 373-396. (MR 42, #6650) [16] M. A. Krasnosel'skii and P. E. Sobolevskii, Fractional powers of operators, acting in Banach spaces, Dokl. Akad. Nauk SSSR, 129 (1959), 499-502. (MR 21, X7447) [17] S. G. Krein, Linear differential equations in Banach spaces, (Russian), Izdat. "Nauka" Moscow, 1967. (English translation), Amer. Math. Soc., (1972). [18] W. Nollau, Uber potenzen von linearen operatoren in Banachschen. Raumen. Acta Sci. Math. Szeged, 28 (1967), 107-121. [19] R.S. Phillips, On the generation of semigroups of linear operators, Pacific J. Math., 2 (1952), 343-369. (MR 14, #383) [20] Wr. Rudin, Functional Analysis, McGraw-Hill, New York, 1973. [21] J. Watanabe, On some properties of fractional powers of linear operators, Proc. Japan Acad., 37 (1961), 273-275. [22] K. Yosida, Fractional powers of infinitesimal generators and the analyticity of the semigroups generated by them, Proc. Japan Acad., 36 (1960), 86-89. (MR 22, 112399) [23] K. Yosida, Functional analysis, 5th ed., Springer, 1978. Celso MARTINEZ Miguel SANZ Luis MARCO Departamento de Matematica Aplicada y Astronomia Facultad de Matematicas Bur jassot, Valencia Spain