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Chapter 3 The balakrishnan operator
Chapter 3
The Balakrishnan Operator The operator J~ defined and studied by A. V. Balakrishnan in 1960 is today a classic reference which, directly or indirectly, has a bearing on all the known ways of constructing the complex powers of non-negative operators. In this chapter we set out its elemental properties.
3.1
Definition of Balakrishnan and Basic P r o p e r t i e s
This section is devoted to the study of the basic properties of the operators J~, a E C+ = {z E C : Re z > 0}, introduced by A. V. Balakrishnan in [12]. Throughout this section, A will be a non-negative operator on a Banach space X and M will be its non-negativity constant. Definition 3.1.1 Let ~ ~. C+. Then: (i) If 0 < Re a < 1, D (J~) = D (A) and oo
j ~ r = sin a~r
A~-1 (A + A) -1 Ar dA
(r e D (A)).
(3.1)
71"
(ii) If R e a = 1, D (J~) = D (A 2) and j ~ r = sina~r f0~176 A~_~ [(A + A)_ ~ 7r
~7r
A2+l
Ar dA + sin - ~ A r
jo~ r ~ D (A~) (iii) If n < R e a < n + l, n e N, D (J ~) = D (A ~+1) and
j-+ = j--~A~ 57
(~ e D (A~§
(3.2)
(iv) I f R e a - n + 1, n E N, D (J~) - D (A ''+2) and (r E D (An+2)).
J~r = J~-nAnr
W h e n it is necessary to specie' the base of the power we write J~. The power A~-1 is understood with respect to the principal determination of the argument and the above integrals are considered in the Bochner sense (see Definition A.6.4). As the integrand is continuous, then the above integrals also can be considered are in the sense of improper Riemann integrals of a continuous function defined in [0. oo[ and taking values in the Banach space X. The existence of the integral in the case 0 < Re a < 1 is due to the fact that
[~--1 (~ _~.d)-I Ar ~ (M -~-1)[]r ,~Rea-1 (r
D (d))
and to the fact that
I[~a-1 (~ ~' A) -1Ar __
(,,)k-'[--A) -1
Aa - 1
A
Ar
A2+l
< ( M + 1)IlCll Al:bea-1 -~" IIACil A2 + 1
and to the fact that
:
§
(,§
~Rea--I _< ((M + 1)II~II+ M [IA=r
~= + 1"
F r o m Definition 3.1.1 it follows easily that if A is bounded, then j n r = Anr T h e continuity of the function c~ 9 > J~ is established in the following proposition.
P r o p o s i t i o n 3.1.1 Let a E C+ and n > R e a , n E N. I f r E O ( A n ) , then lim J ~ r ~--,a
= J=r
P r o o f . By the way in which J~ is defined, it is sufficient to consider the case Re c~ = 1 and r e D (A2). Moreover, as (3.2) is continuous with respect to a 0 < Re a < 2, we must only prove that
=
[
sin ~Tr r
A~-1
(A q- A) -1
sin~
A~-1 (A + A) - 1 A r dA
58
]
A Ar dA + sin ~ A2 + 1 -~- Ar
for 0 < Re/5 < 1 and r E D (A2), and that sin37r 7r
/o
,X~-i
sin ( / 3 - 1)7r
[
(,~ + A) - i
"~ ~2+I
}
Ar dA + sin
Ar
A~-2 (A + A) -1 A2r d)~
71"
for 1 < Re/5 < 2 and r E D (A2). The key to prove both equalities is the formula oo Ao., d A = 7r k2 + 1 2cos (lrw/2) ( - 1 < Rew < 1) (3.3)
L
which can be proved by integrating the function z ~ ' / ( z 2 + 1) along the path with positive orientation: F2 = {pe iO : -Tr + e < 0 < 7 r - e}
Fi = {re i~ : -Tr + e <_ 0 <__r r - e} Fa = Ae i('r-e) : p <_ A <_ r}
=
- p <
<
with e, p, r > 0, and taking limits as e and p go to zero, and r to infinity. I P r o p o s i t i o n 3.1.2 Let c~ E C+. Then:
(i) D (J~) = D (A). (ii) R (J~) C D (A) M R (A). (iii) J~ commutes with (# + A) -x , - # E p (A) . In general, J~ commutes with any bounded operator that commutes with the resolvent. (iv) If r is an eigenvector of A, corresponding to the eigenvalue # E C \ JR_, then r is also an eigenvector of Ja and #c, (calculated with respect to the principal determination of the argument) is its corresponding eigenvalue. If 0 is an eigenvalue of A, then it is also an eigenvalue of j a and the eigenvectors corresponding to both eigenvalues are equal. P r o o f . Statement (i) follows easily from part (iii) of Proposition 1.1.3. Assertions (ii) and (iii) are a straightforward consequence of Definition 3.1.1. Statement (iv) is evident if # = 0. If # E C \ IR_, then (A + A) -1 Ar = (A + # ) - I #r and therefore, from Defirfition 3.1.1 it follows that J ~ r = #~r by (3.3). I The operators J~ can be defined by a unique unified formula, as we see in the following proposition. This formula will be very useful in Section 6.1 to describe the domains of the fractional powers. P r o p o s i t i o n 3.1.3 Let a E C+ and n > Re a, n E N. If r E D (A n ) and if m >_ n is a positive integer, then r(m)
J ~ r = F (a) F (m - a)
f0
),~-I
59
[
A (,k + A)-
i ''~
]
r dA.
(3.4)
Proof. Let us first suppose that n - 1 < Re a < n. As d (A + A) -p = - p (A + A) -p-1 d-'A
(p e N)
and as lim A'~-n+v (A + A) -v A'~r
A--,0 lim A'~-'~+1 [A (A + A)-I] p-1 ( 1 - A (A + A) -1) A n - I r ,k---,o and lim A~-'~+v (A + A) -v A'~r = 0
~---*oo
for 1 < p < n - 1, by integrating by parts we get j~r
--
sin (a - n + 1)7r
71"
A~-n (A + A) - 1 A n r dA
~0~176
F(n) ~~176 F (a) F (n - a)
] r A (A + A) -~ '~
where we have used the f o r m u l a of c o m p l e m e n t s of E u l e r
71" F ( a ) F(1
-
(0 < R e a < 1)
a)=sin~.a
which is obtained from the well-known relation F (a) F (1 - a) = B (a, 1 - a) =
j~01 x=-1(1
-
x)-=dx
and the value of the last integral follows from (3.3). We now proceed by induction on m. Suppose that (3.4) holds for a fixed m. Then
/0
Act-1 A (A + A) -1
--
A ( A + A ) -1
o~
[
r
I
0
r dA
+~
(A+A)
m+l q~dA
since
converges to zero both as A goes to zero and as A goes to infinity. Moreover, since the integrand of (3.5) can be written as A~Am [(A + A ) - l l m + l r Aa-lAm
+
+ -
60
A)-Ir
(3.4) follows easily for m + 1. Finally, if Re a is a positive integer, (3.4) is a consequence of the continuity of jc~ (Proposition 3.1.3) and of the continuity with respect to a of the second member of (3.4). I R e m a r k 3.1.1 If z E C \ N_ and rn > Reck, m E N, then the formula F (m)
z ~ r = r (~) r (m - ~)
/o
,k~-1 [z (,~ + z)] - m r d,k
(3.6)
is a particular case of (3.4). C o r o l l a r y 3.1.4 Let 0 < R e a < Re/3 < m, rn E N, and r E D ( A m ) . Then
r(/~)
~_1
J ~ r = F (a) F (~ - a)
JA(X+A) -~ r d)~.
(3.7)
P r o o f . Note first that as A (A + A) -1 is non-negative, the integrand in (3.7) makes sense. Writing J~A(~+A)_t r by means of formula (3.4) and applying the Tonelli theorem we obtain that the integral in (3.7) is convergent in norm and that F (a) F (/~ - a) =
F(a) F(m-a)
/o /0
JA(~,+A)-' r dA A ( A + A ) -1 m
where we have used the well-known identity fo ~ t ~-~-1(1 + t ) a - m d t = B (~ - ~, m - ~) =
r (Z - ~) r (m - Z) r(~n-~)
I T h e o r e m 3.1.5 ( A n a l y t i c i t y ) Let n E N and r E D (A'~). The function
is analytic in {a E C : 0 < R e a < n}.
P r o o f . Taking into account that the term log,k that appears after derivating the integrand in ( 3 . 4 ) i s o (,~8/ as ~ ---, oo, and o ""()~-8) as )~---, 0 , 6 > 0, the \ / \ / result is a straightforward consequence of the theorem of derivation of integrals depending on parameters. I In relation to the dependence of J~ with respect to a, another interesting question we should investigate is if lim
jar
Ar
and
lim J a r 1 6 2
c~---, 1
c~---,0
O
R.e r~ > 0
61
(3.8)
hold for r e D (A). As R (J~) c D (A) M R (A), the first limit in (3.8) fails for r e D (A) such that Ar ~ D (A). For the same reason, the second limit in (3.8) could only hold for r E R (A). If we assume these hypotheses, then (3.8) holds in a certain sense, as the following theorem shows. T h e o r e m 3.1.6 (i) Let S1 be a fixed sector about 1 and contained in {a E C" 0 < Re a < 1}. Then lim J ~ r = Ar ct--* 1 ctES 1
for r e D (A) such that A r
D (A).
(ii) Let So be a fixed sector about 0 and contained in {a E C" 0 < Re a < 1 }. Then lim J~r = r ct---*0 ES 0
for r e D (A) M R (A). P r o o f . Note first that J~r
Aa-1 [()~ b A ) - I Ar = sin7ra'Tr f0~176
-
1---k--]ACdA
(3.9)
~+1
since r
~O
,~a-- 1
71"
A+I dA=sina~.
as we can easily see from (3.3). Decomposing now (3.9) as f o L > 0, we deduce that
[IJ~r162
<
sin a~r
[(
[(M + 1)[Ir
= fo
L= a
+ IIAr
7r
+ suPll(A(A+A)-I-1)Ar A>L
+JT,
L~-I a + M [IAr
Lar-2/ 2- a
where a = Re a. On the other hand, S1 can be written as ~1 --" ( O~--
a+ir"
0 < a < 1 and
ITI
_K}
for s o m e K > 0 ,
and so Isin a~r I = Isin (a - 1)7r I < sin (~ (1 - a))cosh KTr + sinh (K (1 - c~)7r). 62
Moreover, (1 - (7) -1 sin ((1 - a) r ) and (1 - a) -1 sinh (K (1 - a) zr) are bounded for 0 < a < 1. Taking into account these considerations as well as the fact that the term
II
*>L
+
converges to zero as L ~ oo, since A r D (A), one deduces (i). The proof of (ii) is similar to that of (i) and we omit the details. First we write j ~ r _ r = s!nrc~Tr oo ,Xa-1 (,X + A) - 1 A r - ,X+ 1
/0
[
and the proof runs as before, taking into account Proposition 1.1.3. m Our next goal is to prove that the operators J a are closable. We need a preliminary lemma. L e m m a 3.1.7 ( M o m e n t I n e q u a l i t y ) Let a E C+ and n > Reck, n E N. Then there is a positive function C (c~, n, M) (which is increasing with respect to the third variable) such that
IIJ':'r
< c (~, n, M)llA"r
Re'~/'~ I1r ('~-m''~)/"
(r e D (A")).
(3.10)
P r o o f . Fix r E D (A~). Decomposing f o = f : + f6~176 in (3.4) it can be deduced that
IIJ~r
F (n) IC(a) V ( n - a ) l
1
n -- Rec~
(
1 $p,.ea ( M + 1) '~ I1r Rea
5Re'~-nMr~
IIA"r
and minimizing the second term with respect to 5 > 0 we obtain (3.10). The function C (a, n, M) that appears is equal to
F ( n + 1) MRe~ (M + 1) '~-P~ ~ C ( a , n , M ) = I r ( ~ ) r ( ~ - ~)1 R e a ( n - Reck) "
(3.11)
m R e m a r k 3.1.2 In particular, if 0 < Re c~ < 1, then C (c~, 1, M) _< ]sin aTr] M Rea (M + 1) 1-Rea lr R e a (1 - R e a ) < e,rlima[ M R e a (M + 1) 1 - R e a
-
R e a (1 - R e a )
"
It is easy to show that if 0 < a < 1, then
IIJ~r ~ 2(M
+
1)max(llr 63
ilAr
(3.12)
If A is bounded, Lemma 3.1.7 shows that J~' is a bounded operator. If A is compact, then J~ also is. T h e o r e m 3.1.8 Ja is closable. P r o o f . Let n E N such that n - 1 _< Rec~ < n and let (era)meN C D (A '~) be a sequence such that lim e r a = 0
and
m---,(x:~
lim J ~ r 1 6 2 m---,oo
[
We must prove that r = 0. From (3.10) it follows that J~ (# + A) -1
in is
bounded. Hence, as (# + A) -1 ~ > O, commutes with J~
[(, + A/-']~
=
,~-~lim[(# + A ) - l ] n J ~ r
=
J~ [(# + A)-l] '~ lim Cm = 0. m---~(X)
Therefore r = 0, since (# + A) -1 is a one-to-one operator. I D e f i n i t i o n 3.1.2 Given ~ E C+, Balakr'ishnan defines the power with base A and exponent ~ as the operator Ja. m
m
From the definition of J~ it can be deduced that if A is bounded, then J~ also is. As a consequence of (3.10) we deduce that if A is compact, then J'-~ also is. On the other hand, from Defimtion 3.1.1 it can be deduced that Proposition 3.1.2 its also valid for J~. If D (A) is non-dense, p ( ~ is empty and therefore it makes no sense to study a relationship between the spectrum of J a and the spectrum of A (spectral __.._
mapping theorem) or to study the identity Jj~-z-- J ~ . Moreover, J-"ffdoes not coincide with A '~ (n E N), by part (iv) of Corollary 1.1.4. To solve these deficiencies, in Chapter 5 we will introduce a new concept of fractional power for non-densely defined operators, which coincides with the one of Balakrishnan in the case that D (A) is dense. P r o p o s i t i o n 3.1.9 Let c~ E C+. Then r E D ( ~ n E N and - # E p (A) such that
if and only if there exist
[A(#+A)-X]~r e D (~. P r o o f . The condition is necessary since j"-5 commutes with (# + A) -1 . To prove the converse, by induction, it is sufficient to consider only the case n = 1. If A (# + A) -1 r E D ( ~ , taking n E N such that n > Reck, the relation
l<_p
64
implies that r E D (J--g). I Similarly, r E D(AP), for some p E N, (or r E D ( A ) ) if and only if r[A (# + A) -1] '~ r E D (A p) (D (A)). The following result gives us a criterion to know if an element r E X belongs to D ( ~ . T h e o r e m 3.1.10 Let a E C+ and r E X. If the limit lim
N---,oo
Jo
[
~c~--I A (,~ + A) -1
r d)~ = r
(3.13)
exists for some positive integer n > Re c~, then
r E D (~
and
r(~)
J s r = ~ (~) F (n - c~) r
Proof. Let us first prove that if (3.13) holds, then r E D (A). For ,~ > 0 we have =
[)~()~+A) -1 + A ( ) ~ + A ) -1 ]~r
-- [A(~-~-A)-I]nr
- ~
(p) [A(~ -~-A)-lln-P [~(~-~_A)-I] p
o
l_
Multiplying this expression by ) a - - I and integrating, a simple calculation shows that %
N---g
=
A (3, + A) -1
r d,~ r dA
l <_p<_n
and as, by hypothesis, the first integral is a bounded function (with respect to N > 0 ) we get r = lim
c~
~
(;)/0"
N--.oo- ~ 1<_p<_n
1i
1
A~-1 3,(,k+A) -1 p A (,~ + A) -1 '~-Pr d,~.
Therefore r E D (A), since the integrands belong to D (A). On the other hand, by Proposition 3.1.3, [k (k + A) -1 ] n r = .r (a)F(n)r (n - a) j~ [k (k + A)_I]jn r and as, by Proposition 1.1.3, lim
k---,cx~
k (k + A) -~
r - r
and 65
lim k (k + A) -~
k--,c~
r - ~
since, by the comment that follows Proposition 3.1.9, r C D (A). So,
-and J~r
Ce D ( ~
r(n) r (a) V(n _ a) r
NI R e m a r k 3.1.3 In the preceding proposition we only need to assume that the weak limit (3.13) exists for some unbounded sequence (Nj > 0)jeN. That is because the concepts of closed operator and weakly closed operator are equivalent, by the Theorem of Mazur.
3.2
Expressions of the Balakrishnan Operator w h e n - A is t h e I n f i n i t e s i m a l G e n e r a t o r o f an Equibounded C0-Semigroup
Throughout this section A E G b (X), that is, - A is the generator of an equibounded C0-semigroup {Pt }t>0 on a Banach space X. In this case the Balakrishnan operators can be expressed by means of formulas referred to the semigroup, in which neither the resolvent nor the base operator A appear. P r o p o s i t i o n 3.2.1 Let a E C+\ N, n E N, n > R e a . If r E D (A'~), then j~r
_-
'[Jo ~t-~-I (P~-
r(-~) +
t - ~ - l P t r dt +
~
(-I)~A~ k!--
O
E
)
r dt
(-1)k k! (k - a) Ak
. (3.14)
O
In particular, if n - 1 < Re a < n, this formula can be reu~itten as
1/0 [
J ~ r _- F ( - a )
t-~-I
]
P t - o
(3.15)
P r o o f . By induction and taking into account (A.11) it is not hard to show that if r E D (An), then
_< M IIA"r
~.v'
0
where IIPtll < M (t ___0). The absolute convergence of the integrals of (3.14) is due to the preceding estimate (for t - 0) and the fact that the semigroup is equibounded (for t - cxD). 66
Suppose first that n - 1 < Re c~ < n. It is well-known that the resolvent can be expressed as the Laplace transform of the semigroup, i.e.
e-XtPtdt
(A+A) -1=
(A>0)
and thus
e -xt (1 - Pt) dt.
A (A + A) -1 = A
(3.16)
Substituting (3.16) into the definition of J~ and applying the Tonelli theorem we get j~,r = n - a - 1 t_,~+n_ 2 An_Xr F (n - c~) (Pt - 1) dt. From this, repeated integrations by parts enables us to write (3.15). Suppose now that 0 < Re c~ < n, c~ it N. The second term in (3.14) is an analytic function of a, and the value of this second term equals the second term in (3.15) when n - 1 < Re c~ < n, as we can see by a simple calculation. Therefore, by Theorem 3.1.5, (3.14) holds. Is In the following proposition we give a formula that expresses the operator J~ only in terms of the semigroup. Despite the importance of this formula from a theoretical point of view, in practice this formula is not applicable in many cases, since the constant K~,m can take the value zero. T h e o r e m 3.2.2 Let c~ E C+, m, n E N, m > n > R e a . If r E D (A'~), then fo ~ t - a - 1 (1
-
-
P t ) m r de = Ka,mJar
(3.17)
where Ka,m = f o t - a - 1 ( 1 - e-t) m d t . Moreover, a sufficient condition to know if r E D (-Ya-) is that for some positive integer m > Re a such that K~,m ~ O, the limit lim 6.--.0
t - ~ - 1 (1 - Pt) m r dt = r
exists. In this case J'--gr = (K~,m)- x r P r o o f . The convergence of the integral that appears in (3.17) is due to the fact that (1 - Pt)mr = O(t m) for all t close to 0, and that ( 1 - Pt) m is bounded. Suppose that n - 1 < Re c~ < n. Integrating by parts,
f0
t - ' - 1 (1 - p t ) m r dt = _1
1 c~ (c~ - 1)-.-(c~ - n + 2)
/o 67
fo
t - ~ d (1 - P t ) m r dt
t-~+,~_2 d ~-1 )m dtn_ 1 (1 - Pt r dt (3.19)
where we have used the fact that the functions dq
t-~+q~ttq (1- Pt)mr
(O < q < n - 1 )
vanish when t goes to zero, since r E D (A'~), and also vanish at infinity. On the other hand,
dt'~-~ (1 - p~)m r
=
dt'~-~ ~
(-1)k
Pk~r
k=0
=
E
(-1)k+n-'
kn-lPktAn-lr
(3.20)
k=l
(where the sum includes k = 0 if n = 1). Moreover, m
1
1:
dt'~-~ I~=0 (1 - _ ~ ) m = 0.
k--1
Substracting the last expression from (3.20) and substituting the result into (3.19) we get
f0
Pt)mr dt
~176 - a - 1 (1 -
~-~.k,~__l(_l)k+n (,~)k,~ = =
a(a-
foo 1)--. ( a - n + 2) Jo t -~+'~-2 (1 -
E(--1)k
Pt)A'~-~r dt
k=F(-c~) J ~ r
k=l
the last equality being a particular case of (3.15) applied to J=r = J=-n+lAn-lr This proves (3.17), since
K~,m =
/0
t -~-~ (1 - _ ~ ) m dt
as we can see by substituting the operator A by the identity operator into the above calculations. By analyticity, (3.17) holds for 0 < R e a < n. The proof of the second assertion runs as in the proof of Theorem 3.1.10. II R e m a r k 3.2.1 The condition (3.18) can be replaced by the weaker condition of existence of the weak limit w - lim j---,0
t -~-1 (1 - Pt)mr
for some sequence (~Y)j~N convergent to zero. 68
dt
R e m a r k 3.2.2 Formula (3.17) is useful whenever K~,m ~- 0, since in this case we can calculate the operator J~. This always happens if c~ > 0. Although some authors write (3.17) as 1
t -c~-1 (1 - P t ) m r dt
j~0~176
and therefore they suppose that K~,m # 0, it is evident that Kc,,2 vanishes at ce - 1 + i2pTr/ log 2 (p = 1,2,.--). Moreover, {Rec~ - Kc,,3 = 0} = [0,1] tO [2, 3] and K~,3 -if- 0 if Re c~ = 1 or if Re c~ = 2. In general, if IIm c~I < rr/log m, then K~,m # 0, m = 4, 5,---, and given c~ E C, Re ce > 0, there is at least one positive integer m such that Ka,m # 0 (see [135]). We conclude this chapter by calculating the Balakrishnan operator associated to some non-negative operators that we have considered in Chapter 1.
3.3
Examples
E x a m p l e 3.3.1 Let (l'l, ,4,/1) be a measurable space, where the open sets have non-zero measure, and let f 9 ~ --+ C be a continuous function such that R a n f C S~, (0 < w < lr). In LP(f2) (1 < p < oo) the operator M / considered in Subsection 1.3.4 is sectorial, with spectral angle less than or equal to w. Let 0 < Re c~ < m, m e N, and r e D [(M/)m]. As the convergence in L p implies the pointwise convergence of some subsequence, (3.4) implies that
(J ~ r ) (x) -
r (ce) r (m - ce)
._,
j~0~176
[f(x) (A + f(x))]
r
(a.e. x e f2)and, by (3.6), = M
or
Consequently, MI,, is an extension of J~MI" Taking into account that Ms,~ commutes with the resolvent operator, it is not hard to show that if D (Ms-) is dense (p r oo), then MI,, - J~I" If D (MI) is non-dense (p = oo), taking an element r e D [(M/) m] such that (M/)m r r D (M/) we have M I , , M / ..... r = (M/) m r which implies that the range of M/~ is bigger than D (M/) and therefore M/~ extends properly Ja. E x a m p l e 3.3.2 Let A H. We will see that jc~ function z ~ by means of described in Subsection
be a normal non-negative operator in a Hilbert space coincides with the operator (z ~) (A) associated to the the Functional Calculus associated to normal operators 1.3.7. Since D (A) is dense and (z ~) (A) is closed and 69
commutes with the resolvent, it is sufficient to prove that (z ~) (A) is an extension of Ja. Let 0 < R e a < m, m E N, and r E D(Am). Then
(~) \ z +
L(So
:
,~a--1
Z
(A)
=
~)
z§ J
dA
d#• (z)
r (~)Fr (m) (m ~) L z~ d#m (z) .... (A) -
r(.)r(m-.) =
F(m)
(r (z~) (A) r
the third equality being a consequence of the Tonelli theorem, since
,~o~--1 (
where H = sup A>O
{ ARe~_l(l+H)m Z )m z+~ -< Iz[" ~ . - m - i H . ,
and where we have used the fact that f-(A)]zl
d#, (z)
&Ea(A)
is finite, since r e D (Am). A consequence of the identity jc/ = (z~)(.4) is that the operator ~/~ is normal and its domain D (~
- {r E H" z ~ E L 2 ( a ( A ) , ~ r
only depends on Re c~. Moreover, if A is self-adjoint and if ~ > 0, then J~ is self-adjoint. We will prove in Corollary 5.1.12 that if Re c~ > 0, then j"-5 j"~ _ j~+~. If A is normal, this additivity follows from the property (iii) of the Functional Calculus described in Subsection 1.3.7 and from the inclusion D [(z ~+z) (A)] C D [(z ~) (A)] which is a simple consequence of the fact that ~r is finite. E x a m p l e 3.3.3 Consider the operator defined in Subsection 1.3.6
A (r
r
r
") -" (r
r -~ r
r -~" r "~"r
")
in co. Let S be the shift operator. If 0 < Re c~ < n and if r E D (An), then
r(n)r(n- ~) (J~O) _
m=o
re+n-1 m
(n-l)! 70
~ 1 ( + )~+nSmr
da
where we have used (1.15) and (1.17). As the k - t h projection is a linear and continuous operator, it commutes with the integral and so, F (a) F (n
-
_ \m=0 =
Pk
a)(Pk (J~r m + n - 1 (n - 1)! m (i
1)m+np k (Smr
dA
+
(m+n-1)-..(m+l)
( A + I ) m+'~ dA sine
.
\m=O
Thus
pk (J~r
- pk
k-1 (m+a-1)(m+a-2)---a m!
r + E
sine
)
m=l since
~0 ~176Arn-ba-1
(A + 1) m+" dA = B(n - a, m + a) - F (n -F (ma)F+(mn)+ a)
This proves that the operator T~ defined on the domain D~ = {r 6 co" r
a (a + l) s2r + " " 6 c~
aSr
as
= r + aSr +
a (a + 1) r 2
+...
is an extension of J~. Note that pk (Tar is a finite stun for any sequence r Moreover, Ta is closed, since the operators pkTa, k 6 N, are bounded. Considering also the projections, it can be deduced that Ta commutes with the resolvent, since (# + A) -1 , # > 0, can be written as a power series with base the operator S (see Subsection 1.3.6). This, together with the fact that D (A) is dense, implies that Ta = ~'~.
3.4
Notes
on Chapter 3
The bibliography on fractional powers of operators is extensive. The semigroup of the fractional powers of a bounded operator had already been studied in 1939 by E. Hille in [91]. S. Bochner [19], R. S. Phimps [162], K. Yosida [203] and A. V. Balakrishnan [11] worked on the powers of operators of class Gb(X ). In 1960 A. V. Balakrishnan in [12] defined the class of non-negative operators and constructed a theory of fractional powers for this kind of operators. More or less at the same time, M.A. Krasnosel'skii and P. E. Sobolevskii [116] and T. Kato [102] proposed different defmitions of fractional power but only for particular cases and equivalent to that given by Balakrishnan. Other authors 71
who worked on the same subject were H. W. Hovel and U. Westphal [98], T. gato [102]-[104], H. Komatsu [109]-[114], V. Nollau [159], J. W'atanabe [196], U. Westphal [197], etc. One can find several surveys on this subject in books on differential equations and functional analysis (see, for instance, {7, Section III.4.6], [66, Sections 6.3-6.5], [73, Section 2.14], [87, Sections 1.4-1.5], [117, Section 1.5], [161, Section 2.6], [185, Section 1.2], {186, Section 2.3] and [204, Section IX.11]. In some of these mentioned sources only positive operators were considered and in all of them a complete study was only made for when the base operator was densely defined. There is an extensive and widely discussed bibliography in [66, 6.6], [96], [117, p. 362] and [198] The formula (3.4) was obtained by H. Komatsu in [110, Th. 2.10]. If - A generates an equibounded C0-semigroup {Pt}t>o, Balakrishnan in [11, Wh. 6.3] had suggested the formulas (3.15) and (3.17) to define the fractional power A s, by means of the semigroup Pt. The formula (3.17)) was also proved in general by S. Komatsu in [110, Wh. 4.4] and by J. L. Lions and J. Peetre in {129] (there is also a proof of this representation integral in [15]). For this reason, this formula is known as the B a l a k r i s h n a n - K o m a t s u - L i o n s - P e e t r e algorithm.
72
The Balakrishnan Operator The operator J~ defined and studied by A. V. Balakrishnan in 1960 is today a classic reference which, directly or indirectly, has a bearing on all the known ways of constructing the complex powers of non-negative operators. In this chapter we set out its elemental properties.
3.1
Definition of Balakrishnan and Basic P r o p e r t i e s
This section is devoted to the study of the basic properties of the operators J~, a E C+ = {z E C : Re z > 0}, introduced by A. V. Balakrishnan in [12]. Throughout this section, A will be a non-negative operator on a Banach space X and M will be its non-negativity constant. Definition 3.1.1 Let ~ ~. C+. Then: (i) If 0 < Re a < 1, D (J~) = D (A) and oo
j ~ r = sin a~r
A~-1 (A + A) -1 Ar dA
(r e D (A)).
(3.1)
71"
(ii) If R e a = 1, D (J~) = D (A 2) and j ~ r = sina~r f0~176 A~_~ [(A + A)_ ~ 7r
~7r
A2+l
Ar dA + sin - ~ A r
jo~ r ~ D (A~) (iii) If n < R e a < n + l, n e N, D (J ~) = D (A ~+1) and
j-+ = j--~A~ 57
(~ e D (A~§
(3.2)
(iv) I f R e a - n + 1, n E N, D (J~) - D (A ''+2) and (r E D (An+2)).
J~r = J~-nAnr
W h e n it is necessary to specie' the base of the power we write J~. The power A~-1 is understood with respect to the principal determination of the argument and the above integrals are considered in the Bochner sense (see Definition A.6.4). As the integrand is continuous, then the above integrals also can be considered are in the sense of improper Riemann integrals of a continuous function defined in [0. oo[ and taking values in the Banach space X. The existence of the integral in the case 0 < Re a < 1 is due to the fact that
[~--1 (~ _~.d)-I Ar ~ (M -~-1)[]r ,~Rea-1 (r
D (d))
and to the fact that
I[~a-1 (~ ~' A) -1Ar __
(,,)k-'[--A) -1
Aa - 1
A
Ar
A2+l
< ( M + 1)IlCll Al:bea-1 -~" IIACil A2 + 1
and to the fact that
:
§
(,§
~Rea--I _< ((M + 1)II~II+ M [IA=r
~= + 1"
F r o m Definition 3.1.1 it follows easily that if A is bounded, then j n r = Anr T h e continuity of the function c~ 9 > J~ is established in the following proposition.
P r o p o s i t i o n 3.1.1 Let a E C+ and n > R e a , n E N. I f r E O ( A n ) , then lim J ~ r ~--,a
= J=r
P r o o f . By the way in which J~ is defined, it is sufficient to consider the case Re c~ = 1 and r e D (A2). Moreover, as (3.2) is continuous with respect to a 0 < Re a < 2, we must only prove that
=
[
sin ~Tr r
A~-1
(A q- A) -1
sin~
A~-1 (A + A) - 1 A r dA
58
]
A Ar dA + sin ~ A2 + 1 -~- Ar
for 0 < Re/5 < 1 and r E D (A2), and that sin37r 7r
/o
,X~-i
sin ( / 3 - 1)7r
[
(,~ + A) - i
"~ ~2+I
}
Ar dA + sin
Ar
A~-2 (A + A) -1 A2r d)~
71"
for 1 < Re/5 < 2 and r E D (A2). The key to prove both equalities is the formula oo Ao., d A = 7r k2 + 1 2cos (lrw/2) ( - 1 < Rew < 1) (3.3)
L
which can be proved by integrating the function z ~ ' / ( z 2 + 1) along the path with positive orientation: F2 = {pe iO : -Tr + e < 0 < 7 r - e}
Fi = {re i~ : -Tr + e <_ 0 <__r r - e} Fa = Ae i('r-e) : p <_ A <_ r}
=
- p <
<
with e, p, r > 0, and taking limits as e and p go to zero, and r to infinity. I P r o p o s i t i o n 3.1.2 Let c~ E C+. Then:
(i) D (J~) = D (A). (ii) R (J~) C D (A) M R (A). (iii) J~ commutes with (# + A) -x , - # E p (A) . In general, J~ commutes with any bounded operator that commutes with the resolvent. (iv) If r is an eigenvector of A, corresponding to the eigenvalue # E C \ JR_, then r is also an eigenvector of Ja and #c, (calculated with respect to the principal determination of the argument) is its corresponding eigenvalue. If 0 is an eigenvalue of A, then it is also an eigenvalue of j a and the eigenvectors corresponding to both eigenvalues are equal. P r o o f . Statement (i) follows easily from part (iii) of Proposition 1.1.3. Assertions (ii) and (iii) are a straightforward consequence of Definition 3.1.1. Statement (iv) is evident if # = 0. If # E C \ IR_, then (A + A) -1 Ar = (A + # ) - I #r and therefore, from Defirfition 3.1.1 it follows that J ~ r = #~r by (3.3). I The operators J~ can be defined by a unique unified formula, as we see in the following proposition. This formula will be very useful in Section 6.1 to describe the domains of the fractional powers. P r o p o s i t i o n 3.1.3 Let a E C+ and n > Re a, n E N. If r E D (A n ) and if m >_ n is a positive integer, then r(m)
J ~ r = F (a) F (m - a)
f0
),~-I
59
[
A (,k + A)-
i ''~
]
r dA.
(3.4)
Proof. Let us first suppose that n - 1 < Re a < n. As d (A + A) -p = - p (A + A) -p-1 d-'A
(p e N)
and as lim A'~-n+v (A + A) -v A'~r
A--,0 lim A'~-'~+1 [A (A + A)-I] p-1 ( 1 - A (A + A) -1) A n - I r ,k---,o and lim A~-'~+v (A + A) -v A'~r = 0
~---*oo
for 1 < p < n - 1, by integrating by parts we get j~r
--
sin (a - n + 1)7r
71"
A~-n (A + A) - 1 A n r dA
~0~176
F(n) ~~176 F (a) F (n - a)
] r A (A + A) -~ '~
where we have used the f o r m u l a of c o m p l e m e n t s of E u l e r
71" F ( a ) F(1
-
(0 < R e a < 1)
a)=sin~.a
which is obtained from the well-known relation F (a) F (1 - a) = B (a, 1 - a) =
j~01 x=-1(1
-
x)-=dx
and the value of the last integral follows from (3.3). We now proceed by induction on m. Suppose that (3.4) holds for a fixed m. Then
/0
Act-1 A (A + A) -1
--
A ( A + A ) -1
o~
[
r
I
0
r dA
+~
(A+A)
m+l q~dA
since
converges to zero both as A goes to zero and as A goes to infinity. Moreover, since the integrand of (3.5) can be written as A~Am [(A + A ) - l l m + l r Aa-lAm
+
+ -
60
A)-Ir
(3.4) follows easily for m + 1. Finally, if Re a is a positive integer, (3.4) is a consequence of the continuity of jc~ (Proposition 3.1.3) and of the continuity with respect to a of the second member of (3.4). I R e m a r k 3.1.1 If z E C \ N_ and rn > Reck, m E N, then the formula F (m)
z ~ r = r (~) r (m - ~)
/o
,k~-1 [z (,~ + z)] - m r d,k
(3.6)
is a particular case of (3.4). C o r o l l a r y 3.1.4 Let 0 < R e a < Re/3 < m, rn E N, and r E D ( A m ) . Then
r(/~)
~_1
J ~ r = F (a) F (~ - a)
JA(X+A) -~ r d)~.
(3.7)
P r o o f . Note first that as A (A + A) -1 is non-negative, the integrand in (3.7) makes sense. Writing J~A(~+A)_t r by means of formula (3.4) and applying the Tonelli theorem we obtain that the integral in (3.7) is convergent in norm and that F (a) F (/~ - a) =
F(a) F(m-a)
/o /0
JA(~,+A)-' r dA A ( A + A ) -1 m
where we have used the well-known identity fo ~ t ~-~-1(1 + t ) a - m d t = B (~ - ~, m - ~) =
r (Z - ~) r (m - Z) r(~n-~)
I T h e o r e m 3.1.5 ( A n a l y t i c i t y ) Let n E N and r E D (A'~). The function
is analytic in {a E C : 0 < R e a < n}.
P r o o f . Taking into account that the term log,k that appears after derivating the integrand in ( 3 . 4 ) i s o (,~8/ as ~ ---, oo, and o ""()~-8) as )~---, 0 , 6 > 0, the \ / \ / result is a straightforward consequence of the theorem of derivation of integrals depending on parameters. I In relation to the dependence of J~ with respect to a, another interesting question we should investigate is if lim
jar
Ar
and
lim J a r 1 6 2
c~---, 1
c~---,0
O
R.e r~ > 0
61
(3.8)
hold for r e D (A). As R (J~) c D (A) M R (A), the first limit in (3.8) fails for r e D (A) such that Ar ~ D (A). For the same reason, the second limit in (3.8) could only hold for r E R (A). If we assume these hypotheses, then (3.8) holds in a certain sense, as the following theorem shows. T h e o r e m 3.1.6 (i) Let S1 be a fixed sector about 1 and contained in {a E C" 0 < Re a < 1}. Then lim J ~ r = Ar ct--* 1 ctES 1
for r e D (A) such that A r
D (A).
(ii) Let So be a fixed sector about 0 and contained in {a E C" 0 < Re a < 1 }. Then lim J~r = r ct---*0 ES 0
for r e D (A) M R (A). P r o o f . Note first that J~r
Aa-1 [()~ b A ) - I Ar = sin7ra'Tr f0~176
-
1---k--]ACdA
(3.9)
~+1
since r
~O
,~a-- 1
71"
A+I dA=sina~.
as we can easily see from (3.3). Decomposing now (3.9) as f o L > 0, we deduce that
[IJ~r162
<
sin a~r
[(
[(M + 1)[Ir
= fo
L= a
+ IIAr
7r
+ suPll(A(A+A)-I-1)Ar A>L
+JT,
L~-I a + M [IAr
Lar-2/ 2- a
where a = Re a. On the other hand, S1 can be written as ~1 --" ( O~--
a+ir"
0 < a < 1 and
ITI
_K}
for s o m e K > 0 ,
and so Isin a~r I = Isin (a - 1)7r I < sin (~ (1 - a))cosh KTr + sinh (K (1 - c~)7r). 62
Moreover, (1 - (7) -1 sin ((1 - a) r ) and (1 - a) -1 sinh (K (1 - a) zr) are bounded for 0 < a < 1. Taking into account these considerations as well as the fact that the term
II
*>L
+
converges to zero as L ~ oo, since A r D (A), one deduces (i). The proof of (ii) is similar to that of (i) and we omit the details. First we write j ~ r _ r = s!nrc~Tr oo ,Xa-1 (,X + A) - 1 A r - ,X+ 1
/0
[
and the proof runs as before, taking into account Proposition 1.1.3. m Our next goal is to prove that the operators J a are closable. We need a preliminary lemma. L e m m a 3.1.7 ( M o m e n t I n e q u a l i t y ) Let a E C+ and n > Reck, n E N. Then there is a positive function C (c~, n, M) (which is increasing with respect to the third variable) such that
IIJ':'r
< c (~, n, M)llA"r
Re'~/'~ I1r ('~-m''~)/"
(r e D (A")).
(3.10)
P r o o f . Fix r E D (A~). Decomposing f o = f : + f6~176 in (3.4) it can be deduced that
IIJ~r
F (n) IC(a) V ( n - a ) l
1
n -- Rec~
(
1 $p,.ea ( M + 1) '~ I1r Rea
5Re'~-nMr~
IIA"r
and minimizing the second term with respect to 5 > 0 we obtain (3.10). The function C (a, n, M) that appears is equal to
F ( n + 1) MRe~ (M + 1) '~-P~ ~ C ( a , n , M ) = I r ( ~ ) r ( ~ - ~)1 R e a ( n - Reck) "
(3.11)
m R e m a r k 3.1.2 In particular, if 0 < Re c~ < 1, then C (c~, 1, M) _< ]sin aTr] M Rea (M + 1) 1-Rea lr R e a (1 - R e a ) < e,rlima[ M R e a (M + 1) 1 - R e a
-
R e a (1 - R e a )
"
It is easy to show that if 0 < a < 1, then
IIJ~r ~ 2(M
+
1)max(llr 63
ilAr
(3.12)
If A is bounded, Lemma 3.1.7 shows that J~' is a bounded operator. If A is compact, then J~ also is. T h e o r e m 3.1.8 Ja is closable. P r o o f . Let n E N such that n - 1 _< Rec~ < n and let (era)meN C D (A '~) be a sequence such that lim e r a = 0
and
m---,(x:~
lim J ~ r 1 6 2 m---,oo
[
We must prove that r = 0. From (3.10) it follows that J~ (# + A) -1
in is
bounded. Hence, as (# + A) -1 ~ > O, commutes with J~
[(, + A/-']~
=
,~-~lim[(# + A ) - l ] n J ~ r
=
J~ [(# + A)-l] '~ lim Cm = 0. m---~(X)
Therefore r = 0, since (# + A) -1 is a one-to-one operator. I D e f i n i t i o n 3.1.2 Given ~ E C+, Balakr'ishnan defines the power with base A and exponent ~ as the operator Ja. m
m
From the definition of J~ it can be deduced that if A is bounded, then J~ also is. As a consequence of (3.10) we deduce that if A is compact, then J'-~ also is. On the other hand, from Defimtion 3.1.1 it can be deduced that Proposition 3.1.2 its also valid for J~. If D (A) is non-dense, p ( ~ is empty and therefore it makes no sense to study a relationship between the spectrum of J a and the spectrum of A (spectral __.._
mapping theorem) or to study the identity Jj~-z-- J ~ . Moreover, J-"ffdoes not coincide with A '~ (n E N), by part (iv) of Corollary 1.1.4. To solve these deficiencies, in Chapter 5 we will introduce a new concept of fractional power for non-densely defined operators, which coincides with the one of Balakrishnan in the case that D (A) is dense. P r o p o s i t i o n 3.1.9 Let c~ E C+. Then r E D ( ~ n E N and - # E p (A) such that
if and only if there exist
[A(#+A)-X]~r e D (~. P r o o f . The condition is necessary since j"-5 commutes with (# + A) -1 . To prove the converse, by induction, it is sufficient to consider only the case n = 1. If A (# + A) -1 r E D ( ~ , taking n E N such that n > Reck, the relation
l<_p
64
implies that r E D (J--g). I Similarly, r E D(AP), for some p E N, (or r E D ( A ) ) if and only if r[A (# + A) -1] '~ r E D (A p) (D (A)). The following result gives us a criterion to know if an element r E X belongs to D ( ~ . T h e o r e m 3.1.10 Let a E C+ and r E X. If the limit lim
N---,oo
Jo
[
~c~--I A (,~ + A) -1
r d)~ = r
(3.13)
exists for some positive integer n > Re c~, then
r E D (~
and
r(~)
J s r = ~ (~) F (n - c~) r
Proof. Let us first prove that if (3.13) holds, then r E D (A). For ,~ > 0 we have =
[)~()~+A) -1 + A ( ) ~ + A ) -1 ]~r
-- [A(~-~-A)-I]nr
- ~
(p) [A(~ -~-A)-lln-P [~(~-~_A)-I] p
o
l_
Multiplying this expression by ) a - - I and integrating, a simple calculation shows that %
N---g
=
A (3, + A) -1
r d,~ r dA
l <_p<_n
and as, by hypothesis, the first integral is a bounded function (with respect to N > 0 ) we get r = lim
c~
~
(;)/0"
N--.oo- ~ 1<_p<_n
1i
1
A~-1 3,(,k+A) -1 p A (,~ + A) -1 '~-Pr d,~.
Therefore r E D (A), since the integrands belong to D (A). On the other hand, by Proposition 3.1.3, [k (k + A) -1 ] n r = .r (a)F(n)r (n - a) j~ [k (k + A)_I]jn r and as, by Proposition 1.1.3, lim
k---,cx~
k (k + A) -~
r - r
and 65
lim k (k + A) -~
k--,c~
r - ~
since, by the comment that follows Proposition 3.1.9, r C D (A). So,
-and J~r
Ce D ( ~
r(n) r (a) V(n _ a) r
NI R e m a r k 3.1.3 In the preceding proposition we only need to assume that the weak limit (3.13) exists for some unbounded sequence (Nj > 0)jeN. That is because the concepts of closed operator and weakly closed operator are equivalent, by the Theorem of Mazur.
3.2
Expressions of the Balakrishnan Operator w h e n - A is t h e I n f i n i t e s i m a l G e n e r a t o r o f an Equibounded C0-Semigroup
Throughout this section A E G b (X), that is, - A is the generator of an equibounded C0-semigroup {Pt }t>0 on a Banach space X. In this case the Balakrishnan operators can be expressed by means of formulas referred to the semigroup, in which neither the resolvent nor the base operator A appear. P r o p o s i t i o n 3.2.1 Let a E C+\ N, n E N, n > R e a . If r E D (A'~), then j~r
_-
'[Jo ~t-~-I (P~-
r(-~) +
t - ~ - l P t r dt +
~
(-I)~A~ k!--
O
E
)
r dt
(-1)k k! (k - a) Ak
. (3.14)
O
In particular, if n - 1 < Re a < n, this formula can be reu~itten as
1/0 [
J ~ r _- F ( - a )
t-~-I
]
P t - o
(3.15)
P r o o f . By induction and taking into account (A.11) it is not hard to show that if r E D (An), then
_< M IIA"r
~.v'
0
where IIPtll < M (t ___0). The absolute convergence of the integrals of (3.14) is due to the preceding estimate (for t - 0) and the fact that the semigroup is equibounded (for t - cxD). 66
Suppose first that n - 1 < Re c~ < n. It is well-known that the resolvent can be expressed as the Laplace transform of the semigroup, i.e.
e-XtPtdt
(A+A) -1=
(A>0)
and thus
e -xt (1 - Pt) dt.
A (A + A) -1 = A
(3.16)
Substituting (3.16) into the definition of J~ and applying the Tonelli theorem we get j~,r = n - a - 1 t_,~+n_ 2 An_Xr F (n - c~) (Pt - 1) dt. From this, repeated integrations by parts enables us to write (3.15). Suppose now that 0 < Re c~ < n, c~ it N. The second term in (3.14) is an analytic function of a, and the value of this second term equals the second term in (3.15) when n - 1 < Re c~ < n, as we can see by a simple calculation. Therefore, by Theorem 3.1.5, (3.14) holds. Is In the following proposition we give a formula that expresses the operator J~ only in terms of the semigroup. Despite the importance of this formula from a theoretical point of view, in practice this formula is not applicable in many cases, since the constant K~,m can take the value zero. T h e o r e m 3.2.2 Let c~ E C+, m, n E N, m > n > R e a . If r E D (A'~), then fo ~ t - a - 1 (1
-
-
P t ) m r de = Ka,mJar
(3.17)
where Ka,m = f o t - a - 1 ( 1 - e-t) m d t . Moreover, a sufficient condition to know if r E D (-Ya-) is that for some positive integer m > Re a such that K~,m ~ O, the limit lim 6.--.0
t - ~ - 1 (1 - Pt) m r dt = r
exists. In this case J'--gr = (K~,m)- x r P r o o f . The convergence of the integral that appears in (3.17) is due to the fact that (1 - Pt)mr = O(t m) for all t close to 0, and that ( 1 - Pt) m is bounded. Suppose that n - 1 < Re c~ < n. Integrating by parts,
f0
t - ' - 1 (1 - p t ) m r dt = _1
1 c~ (c~ - 1)-.-(c~ - n + 2)
/o 67
fo
t - ~ d (1 - P t ) m r dt
t-~+,~_2 d ~-1 )m dtn_ 1 (1 - Pt r dt (3.19)
where we have used the fact that the functions dq
t-~+q~ttq (1- Pt)mr
(O < q < n - 1 )
vanish when t goes to zero, since r E D (A'~), and also vanish at infinity. On the other hand,
dt'~-~ (1 - p~)m r
=
dt'~-~ ~
(-1)k
Pk~r
k=0
=
E
(-1)k+n-'
kn-lPktAn-lr
(3.20)
k=l
(where the sum includes k = 0 if n = 1). Moreover, m
1
1:
dt'~-~ I~=0 (1 - _ ~ ) m = 0.
k--1
Substracting the last expression from (3.20) and substituting the result into (3.19) we get
f0
Pt)mr dt
~176 - a - 1 (1 -
~-~.k,~__l(_l)k+n (,~)k,~ = =
a(a-
foo 1)--. ( a - n + 2) Jo t -~+'~-2 (1 -
E(--1)k
Pt)A'~-~r dt
k=F(-c~) J ~ r
k=l
the last equality being a particular case of (3.15) applied to J=r = J=-n+lAn-lr This proves (3.17), since
K~,m =
/0
t -~-~ (1 - _ ~ ) m dt
as we can see by substituting the operator A by the identity operator into the above calculations. By analyticity, (3.17) holds for 0 < R e a < n. The proof of the second assertion runs as in the proof of Theorem 3.1.10. II R e m a r k 3.2.1 The condition (3.18) can be replaced by the weaker condition of existence of the weak limit w - lim j---,0
t -~-1 (1 - Pt)mr
for some sequence (~Y)j~N convergent to zero. 68
dt
R e m a r k 3.2.2 Formula (3.17) is useful whenever K~,m ~- 0, since in this case we can calculate the operator J~. This always happens if c~ > 0. Although some authors write (3.17) as 1
t -c~-1 (1 - P t ) m r dt
j~0~176
and therefore they suppose that K~,m # 0, it is evident that Kc,,2 vanishes at ce - 1 + i2pTr/ log 2 (p = 1,2,.--). Moreover, {Rec~ - Kc,,3 = 0} = [0,1] tO [2, 3] and K~,3 -if- 0 if Re c~ = 1 or if Re c~ = 2. In general, if IIm c~I < rr/log m, then K~,m # 0, m = 4, 5,---, and given c~ E C, Re ce > 0, there is at least one positive integer m such that Ka,m # 0 (see [135]). We conclude this chapter by calculating the Balakrishnan operator associated to some non-negative operators that we have considered in Chapter 1.
3.3
Examples
E x a m p l e 3.3.1 Let (l'l, ,4,/1) be a measurable space, where the open sets have non-zero measure, and let f 9 ~ --+ C be a continuous function such that R a n f C S~, (0 < w < lr). In LP(f2) (1 < p < oo) the operator M / considered in Subsection 1.3.4 is sectorial, with spectral angle less than or equal to w. Let 0 < Re c~ < m, m e N, and r e D [(M/)m]. As the convergence in L p implies the pointwise convergence of some subsequence, (3.4) implies that
(J ~ r ) (x) -
r (ce) r (m - ce)
._,
j~0~176
[f(x) (A + f(x))]
r
(a.e. x e f2)and, by (3.6), = M
or
Consequently, MI,, is an extension of J~MI" Taking into account that Ms,~ commutes with the resolvent operator, it is not hard to show that if D (Ms-) is dense (p r oo), then MI,, - J~I" If D (MI) is non-dense (p = oo), taking an element r e D [(M/) m] such that (M/)m r r D (M/) we have M I , , M / ..... r = (M/) m r which implies that the range of M/~ is bigger than D (M/) and therefore M/~ extends properly Ja. E x a m p l e 3.3.2 Let A H. We will see that jc~ function z ~ by means of described in Subsection
be a normal non-negative operator in a Hilbert space coincides with the operator (z ~) (A) associated to the the Functional Calculus associated to normal operators 1.3.7. Since D (A) is dense and (z ~) (A) is closed and 69
commutes with the resolvent, it is sufficient to prove that (z ~) (A) is an extension of Ja. Let 0 < R e a < m, m E N, and r E D(Am). Then
(~) \ z +
L(So
:
,~a--1
Z
(A)
=
~)
z§ J
dA
d#• (z)
r (~)Fr (m) (m ~) L z~ d#m (z) .... (A) -
r(.)r(m-.) =
F(m)
(r (z~) (A) r
the third equality being a consequence of the Tonelli theorem, since
,~o~--1 (
where H = sup A>O
{ ARe~_l(l+H)m Z )m z+~ -< Iz[" ~ . - m - i H . ,
and where we have used the fact that f-(A)]zl
d#, (z)
&Ea(A)
is finite, since r e D (Am). A consequence of the identity jc/ = (z~)(.4) is that the operator ~/~ is normal and its domain D (~
- {r E H" z ~ E L 2 ( a ( A ) , ~ r
only depends on Re c~. Moreover, if A is self-adjoint and if ~ > 0, then J~ is self-adjoint. We will prove in Corollary 5.1.12 that if Re c~ > 0, then j"-5 j"~ _ j~+~. If A is normal, this additivity follows from the property (iii) of the Functional Calculus described in Subsection 1.3.7 and from the inclusion D [(z ~+z) (A)] C D [(z ~) (A)] which is a simple consequence of the fact that ~r is finite. E x a m p l e 3.3.3 Consider the operator defined in Subsection 1.3.6
A (r
r
r
") -" (r
r -~ r
r -~" r "~"r
")
in co. Let S be the shift operator. If 0 < Re c~ < n and if r E D (An), then
r(n)r(n- ~) (J~O) _
m=o
re+n-1 m
(n-l)! 70
~ 1 ( + )~+nSmr
da
where we have used (1.15) and (1.17). As the k - t h projection is a linear and continuous operator, it commutes with the integral and so, F (a) F (n
-
_ \m=0 =
Pk
a)(Pk (J~r m + n - 1 (n - 1)! m (i
1)m+np k (Smr
dA
+
(m+n-1)-..(m+l)
( A + I ) m+'~ dA sine
.
\m=O
Thus
pk (J~r
- pk
k-1 (m+a-1)(m+a-2)---a m!
r + E
sine
)
m=l since
~0 ~176Arn-ba-1
(A + 1) m+" dA = B(n - a, m + a) - F (n -F (ma)F+(mn)+ a)
This proves that the operator T~ defined on the domain D~ = {r 6 co" r
a (a + l) s2r + " " 6 c~
aSr
as
= r + aSr +
a (a + 1) r 2
+...
is an extension of J~. Note that pk (Tar is a finite stun for any sequence r Moreover, Ta is closed, since the operators pkTa, k 6 N, are bounded. Considering also the projections, it can be deduced that Ta commutes with the resolvent, since (# + A) -1 , # > 0, can be written as a power series with base the operator S (see Subsection 1.3.6). This, together with the fact that D (A) is dense, implies that Ta = ~'~.
3.4
Notes
on Chapter 3
The bibliography on fractional powers of operators is extensive. The semigroup of the fractional powers of a bounded operator had already been studied in 1939 by E. Hille in [91]. S. Bochner [19], R. S. Phimps [162], K. Yosida [203] and A. V. Balakrishnan [11] worked on the powers of operators of class Gb(X ). In 1960 A. V. Balakrishnan in [12] defined the class of non-negative operators and constructed a theory of fractional powers for this kind of operators. More or less at the same time, M.A. Krasnosel'skii and P. E. Sobolevskii [116] and T. Kato [102] proposed different defmitions of fractional power but only for particular cases and equivalent to that given by Balakrishnan. Other authors 71
who worked on the same subject were H. W. Hovel and U. Westphal [98], T. gato [102]-[104], H. Komatsu [109]-[114], V. Nollau [159], J. W'atanabe [196], U. Westphal [197], etc. One can find several surveys on this subject in books on differential equations and functional analysis (see, for instance, {7, Section III.4.6], [66, Sections 6.3-6.5], [73, Section 2.14], [87, Sections 1.4-1.5], [117, Section 1.5], [161, Section 2.6], [185, Section 1.2], {186, Section 2.3] and [204, Section IX.11]. In some of these mentioned sources only positive operators were considered and in all of them a complete study was only made for when the base operator was densely defined. There is an extensive and widely discussed bibliography in [66, 6.6], [96], [117, p. 362] and [198] The formula (3.4) was obtained by H. Komatsu in [110, Th. 2.10]. If - A generates an equibounded C0-semigroup {Pt}t>o, Balakrishnan in [11, Wh. 6.3] had suggested the formulas (3.15) and (3.17) to define the fractional power A s, by means of the semigroup Pt. The formula (3.17)) was also proved in general by S. Komatsu in [110, Wh. 4.4] and by J. L. Lions and J. Peetre in {129] (there is also a proof of this representation integral in [15]). For this reason, this formula is known as the B a l a k r i s h n a n - K o m a t s u - L i o n s - P e e t r e algorithm.
72