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Chapter 7 Fractional powers with exponents of negative real part. Imaginary powers of operators
Chapter 7
Fractional Powers with Exponents of Negative Real Part. Imaginary Powers of Operators In this chapter we define and study the basic properties of the complex powers when the exponent has non-positive real part. We pay special attention to the imaginary powers because of the relationship to the existence, uniqueness and regularity of solutions of certain problems of abstract parabolic differential equations.
7.1
Definitions and Basic Properties
Throughout this section, we assume that A is an injective non-negative operator (not necessarily of dense domain or range) on a Banach space X. Remember that for non-negative operators, density of range implies injectivity. D e f i n i t i o n 7.1.1 Given c~ E C+ we define A - " = (A~) -1 .
(7.1)
Note t h a t since A is injective, by Corollary 5.2.4, A n is also injective and (A~) -1 - ( A - l ) a . Moreover, A n is closed and D ( A -'~) D R ( A p) if p is a positive integer such that p > Re a. Applying (5.20) to the operator ( A - l ) ~ yields A - ~ = (A + 1) p A - P A p+'~ (A + 1) -p . (7.2) D e f i n i t i o n 7.1.2 Given "r E IR we define A i~ = (A + 1) 2 A - 1 A 1+i~" (A + 1) -2 . 171
(7.3)
Note that A ~r is closed since A ~+ir (A 4- 1) -2 is bounded and (A + 1) 2 A -1 is closed. Moreover, D (A it) D D ( A ) N R ( A ) . Thus, if A E s and 0 e p(A), then Air is bounded and
Aide = - i sinh rT" 7r
[
Ai~ (A + A) -1
A A2 + 1
]
rT CdA+c~162
(r
(7.4) by (3.2). Moreover, an easy calculation shows that
ilA ll <_
(75)
where C only depends on the base operator A. It is not difficult to prove that A ir = (A + 1) q A-VA p+ir (A + 1) -q
(p, q e N, q > p),
(7.6)
and that in (7.2) and in (7.6) the terms (A + 1) • can be replaced by (A + A) • , with A > 0 (or A = 0 if A is positive). If T is a bounded operator that commutes with (A + 1) -1 , then A a T is an extension of T A ~ for all a E C. Finally, if p and q are positive integers such that q > p and q > p + Re a > 0, then for all r e X A p ( A + I ) - q C e D ( A ~)
and
A~A p ( A + I ) - q r
~+p(A+I) -qr
for all a E C. T h e o r e m 7.1.1 ( A d d i t i v i t y ) Let a, fl ft. C. Then
(i)
A"A ~ c A ~+~.
(ii) If r e D (A ~+~) N D (AZ) , then
A~r E D (A s)
and
A~Aar = A~+ar
In particular, if D (A ~+~) C D (An), AC'A ~ = A ~+~.
(iii)
If D (A) and R (A) are dense, then A~+~ _ A~A ~.
P r o o f . Let r E D (A~A ~) and let p be a positive integer such that p > max {IRe c~[, [Refll}. By (7.7) and by additivity for exponents in C+, A 2p (A + 1) -4p A~A~r = A~+VA p (A + 1) -@ A~r
= A~+VA ~+p (A + 1) -@ r = A ~+~+2p (A + 1) -@ r 172
(7.7)
= A ~+zA 2v (A +
1)-4P (~.
Thus, from (5.20), (7.2) or (7.6) (depending on the sign of Re (a + fl)) it follows that r E D (A ~+~) and A~A~r = Aa+~r The same reasoning works for statement (ii). Consider now r e D (Aa+~). Taking p as in case (i) and A, # > 0, the element A 2v (A + #)-2p (A + A)-2v r belongs to D (AaA~). So, A 2p (A + #)-2p (A + A)-2p A~+~r = A ~ A ~ A 2p (A + #)-2v (A + A)-2p r and multiplying by A2v and letting A---+ Go and # ~ O, r E D (A'~A ~) and A~+~r = A'~A~r since D (A) and R ( A ) are dense, m
Corollary 7.1.2 Given T E ]R\ {0}, if A i~ is bounded, then D (An) C D (A ~+ir
for all a e R.
Conversely, if D (A n) C D (A ~+ir) for all a e IR\ {0}, then A ir is bounded. If A is positive, then it is suO%ient to prove the inclusion for all a > O.
Proof. If A ir is bounded, then, by Theorem 7.1.1, D (A n) - D (AirA ~') C D (A'~+ir).
Conversely, if D (An) C D (A ~+'r) for all a e JR\ {0}, then D (A)C D (A 1+it)
and
D (A -1) C D (A -1+it)
and so, again by Theorem 7.1.1, R (A) C D (A i~')
and
D (A)C D (Air).
As every element r E X can be written as r = (1 + A) -~ r + A (1 + A) -~ r we have X C D (Air). By the closed graph theorem, A ir is bounded. The last assertion is evident, m T h e o r e m 7.1.3 (Multiplicativity) Let A be a sectorial operator, with spectral angle WA, and a E ~ such that IC~OAI < 7r. Then A n is sectorial and (A~) ~ - A ':'~ ( f l E C ) . 173
P r o o f . In Theorem 5.4.1 we proved that if A is sectorial and 0 < a W A "( 71", then A s is sectorial, with spectral angle WA,, <_ SWA. Moreover, the property (AS) ~ = A s~ has been proved in case Re/~ > 0. From Definition 7.1.1 and Corollary 5.2.4 it follows that this property is also true for s < 0 and Re B < 0. So, it is sufficient to consider case ~ = i7". Let p > Is I be a positive integer. Given r E D [(As)i~] , as A p (1 + A) -2p is a bounded operator that commutes with the resolvent of A s and its range is included in D (A m) M R (AS), A p (1 + A) -2p (AS) i" r = (A~) ir A ~ A p - s (1 + A) -2p r = (AS) 1+i" Ap-~ (1 + A) -2p r = A~+Sir p-~ (1 + A) -2p r = A~i~A p (1 + A) -2p r
Hence r e D (A s'~) and A si~r = (A s)i~ r Conversely, if r E D ( A ~ r ) , then the same reasoning as before yields A p (1 + A) -2p A ~ i ' r = (A~) i~ A p (1 + A) -2p r and if we multiply this expression by A m (1 + A s)-2, then A ~ (1 + A") -2 A~'~r = (A~) '~ A ~ (1 + A~) -2 r Therefore r E D [(AS)i~ 1 and (A~) '~ r = Aairr I The following result is an immediate consequence of Theorems 7.1.1 and 7.1.3. k
d
C o r o l l a r y 7.1.4 A ir is one-to-one and (A'") -1 = A -'~" = (A-X) '" .
(7.8)
P r o o f . The first equality is a consequence of additivity (Theorem 7.1.1) and the second one follows from Theorem 7.1.3. I P r o p o s i t i o n 7.1.5 (Analyticity) Let so E C+. If r E D (A ~~ M n ( A s ~ then the function a ~ ASr is holomorphic in {s E C" - R e s 0 < R e a < Re s0}. If r E R (A a~ , then a H A~r is holomorphic in
{s E C ' - R e s 0
< R e s < 0}.
P r o o f . By additivity, r E D (A m) if a E C such that IRe s] < Re so, Re a ~ 0. Fix m E N such that •m < Rea0. Then r E D (A 1/m) M R (A 1/m) and as A x/m is non-negative,
r
1174
Therefore r E D (A s) for all c~ E C, with IRe a[ < Re a0. Let r = A~~ E D (A~~ R (A~~ By additivity, A~r = A~+~~ , IReaI < Rea0 and by Corollary 5.1.13 this function is analytic in
{a E C :0 < R e a + Rea0 < 2Rea0} which proves the first assertion. The second one follows from Definition 7.1.1 and Corollary 5.1.13. I
7.2
The Balakrishnan and Komatsu Operators
Similarly to the case of the Balakrishnan operator introduced in Chapter 3 for exponents with positive real part, the following definitions and properties are closely related to the concept of complex power of non-negative operators. Throughout this section we assume that A is a non-negative operator on a Banach space X. D e f i n i t i o n 7.2.1 Let ~ E C+ and let n be the minimum integer number such that n > Rec~. We define the operator J - a with domain R(A '~) and range included in R (A) J - ~ r = J~-%?
(r = A"~ E R(A'~)).
(7.9)
By (5.14) we obtain, for m >__n and r E R (Am) 9 J - - (1~r
---
F(m)
Am-l-~ (A + A)-m r dA.
~0~176
(7.10)
Note that in order to define J - ~ it is not necessary to assume that A is injective. It is not difficult to show that (7.9) does not depend on rI. If A is injective, then it is obvious that A m extends J - ~ . In fact, a simple change of variable in (7.10) shows that JA ~ - J~_~. D e f i n i t i o n 7.2.2 Let T E I~. We define the operator jir, with domain
D(J ~) = D(A) M R(A) (and with range contained in D(A) M R(A)) as (7.11)
for r = AT E D(A) MR(A). complements of Euler
If~" ~ O, then, by (3.~) and the formula of
7r l" (iT) i" (1 -- iT) = sin iTrr'
we obtain j i b e _ ~ s i n~T h ~'T
~0~
Ai~( A + A)_2ACdA.
175
As in Definition 7.2.1, it is not necessary to assume that A is injective to define j i r . If A is injective, then A ir extends j i r . The estimates that we prove in the following proposition show that J - ~ and ji~ are closable, even thought A is not injective. P r o p o s i t i o n 7.2.1 Let a E C+ and let n > Re a be a positive integer. r = A'~7 E R(A'~), then
Ilj- r
<
C(n -
n, M)IIr
' - R,-~F-117/11R,:~,.
If
(7.12)
If r E lt~ and r = A ' ~ E D(A m-'~) M R(A'~), with m > n positive integers, then IlJi'r
<_ C(n + it, m, M)117/111-#'' IlAm-nr
'~' 9
(7.13)
The constants C(n - a, n, M) and C(n + it, m, M) are given by (3.11). Consequently, the operators J - ~ and j i r are closable. Proof. The estimates (7.12) and (7.13) are an immediate consequence of (3.10). From these estimates it follows that for - # E p(A), the operators J - ~ [A(# + A)-I] n , n > R e a , and Ji~A(# + A) -1 (1 + A) -1 are bounded. The proof of the closable character of J - ~ and ji~- is similar to that of Theorem 3.1.8, taking into account that
R ( J -~) U R ( J i~) C R(A) and letting # ~ 0. I If A is injective, then the relationship between J - ~ and ji~ and the corresponding powers of A is given in the following result, which is similar to Corollary 5.1.12. P r o p o s i t i o n 7.2.2 Let a E C+ and T E JR. Assume that A is injective. Then
and
Consequently, if the range of A is dense, then A -a - j - d . If, in addition, the domain is also dense, then A ir = jir. P r o o f . The inclusion
is evident. Conversely, if r E D (A i~) M D (A) N R (A) and Aide E D (A) M R (A), 176
then A (A + A) -1 # ( # + A )
-1 A i r r = j i r A (A + A ) - I # (# + A)-I r
(~, ~ > 0) /-.--\
and taking limits as # tends to infinity and l to zero we get r 6 D ( j i ~ ) and j i r r = Airr The equality (7.15) is analogous, m If A is non-negative with dense domain and range, then its adjoint A* is also non-negative and one-to-one but, in general, has non-dense domain or range. Moreover, A ir has dense domain, so we can consider its adjoint (Air) * . It is natural to ask if (Air) * = (A*) ir . The answer is given in the following:
T h e o r e m 7.2.3 If A has dense domain and range, then (Air
- ( A * ) ir "
P r o o f . As A has dense domain and range,
(y;)"
=
by Proposition 7.2.2. Let us prove that ( j i r ) * is an extension of j~r. Fix r e D (A) N R (A) and v e D (A*) N R (A*). Then (J~.v, r
= sinh (rT) 7rT
/0
Air A* (A + A*) -2 v, r
(
)
dA
where the second equality is due to the fact that [A (A + A)-2] * = A* (A + A*) -2 which follows easily from the definition of the adjoint operator. In order to prove that (Air) * - ( j i r ) * C (A*) ir let us consider an element Taking into account that the commutativity between J iT and
vED[(Jir)*]. L
.,J
A (1 + A) -2 also holds if we take adjoints, A* (1 + A*) -2 ( J i r
= =
n - ( 1 + A') J~.A*(I+A*)-2v
and by Definition 7.1.2 we conclude that v E D [(A* )iT] and (A*)ir v = (Air) . i,,. It only remains to prove that ( j i r ) * is an extension of (A*) ir . Let v e D [(A*) ir] and r E D ( j i r ) .
Given ,k,/z > 0 and taking into account that 177
(ji,)* is an extension of J~., (u, A (A + A) -1A (# + A) -1 jibe)
- (~(~+z)-,z(.+z)-l~,
j"~)
-
(J~.A(A+A*)-IA*(#+A*)-lu,
=
(:~(~ + A') -1 A" ( .
=
((A*)'" u, ,~ (,~ + A ) - I A (tt + A) -1 r
r
+ A') -1 (A')".,.
~,)
Finally, taking limits as ,~ tends to infimty and # to zero we conclude that
B
7.3
Examples
E x a m p l e 7.3.1 On the space X - / ~ 1 7 6 composed of the bounded sequences of complex numbers, with the supremum norm, let us consider the multiplication operator A(r r = (r162162 with domain D(A) - {r E X : Ar E X}. From Subsection 1.3.4, this operator is positive and non-densely defined. Its imaginary power A i~" is given by Ai~(r162
= (r162
3iT r
which is a bounded operator. However, ji~ is unbounded since D ( J i~) C D(A). By considering its inverse we obtain a non-negative operator with non-dense range and with bounded imaginary powers. Finally, in the Banach space X x X, the operator A~162r = (Ar162 defined on its maximal domain is a non-negative operator, with non-dense domain and range, and with bounded imaginary powers. E x a m p l e 7.3.2 Let A be a normal and injective operator defined in a Hilbert space. Assume that a(A) C ~ (with w e]0, r[). The imaginary power A i" coincides with the operator (z it) (A) obtained by means of the Functional Calculus associated to normal operators described in Subsection 1.3.7. Then, A i" is bounded ,ha IIA"II < ~ ' J E x a m p l e 7.3.3 [H. Komatsu] Let us consider in the space co (C) of the sequences of complex numbers converging to zero the operator A = ( 1 - S) -1 , 178
where S is the shift operator (for more details see Subsection 1.3.6). The operator A is non-negative and densely defined. The imaginary power A i~ is given by iv(iT + 1)$2 iv(iv + 1)(i7 + 2)$3 + ..... A i~ = 1 + iTS + + 2 3! If this operator was bounded its norm would be given by
IIA~[I
=
1+
I~-I + I~-I (ITI ~ + 1) ~/~ 2
>_ l~l
1+~+5+
ITI (ITI ~ + 1)~/2(IrI 2 + 4) ~12 +
3
+""
.... - o o ,
which proves that A i~" is unbounded if T # 0. As A ir = (A - 1 ) - i v (see Corollary 7.1.4) and the operator S - 1 generates an equibounded C0-semigroup, we have an example of a bounded operator whose negative generates an equibounded C0semigroup and such that its imaginary powers are unbounded. Combining this example with Theorems 7.1.3 and A.7.6 we obtain an example of a bounded operator whose negative is the generator of an analytical semigroup and its imaginary powers are unbounded. The following example shows a positive operator whose imaginary powers are bounded or unbounded, depending on the exponent. E x a m p l e 7.3.4 [A. Venni] The space X =
x - (xl,x2, ...) E C N 9 s,~ =
xk has a finite limit in C k--1
endowed with the norm nEN
is a Banach space since the mapping ( ~ , ~2, ...) ~ ( ~ , ~2, ...)
is an isometry from X onto the subspace of/~ of the convergent sequences. As Ix~] = is,~- s~-ll i 2 llxll for all n > 1 and Ixll = 181] ~___ IIX[I, the projections p,.,(x) = x,~ are continuous. Given a, x E C N, we denote by ~.~ = (a~,
~2~,..-)
and by D,~= { x E X : a . x E X } .
Consider now the operator Tax=a'x
for 179
xEDa.
It is evident that Da is dense in X since for all x E X, lim (xx,x2,...,x,~,O,O,...)
x-
1"1,.--*O 0
and obviously (Xl,X2,...,Xn,O,O,...) E Da for all n E N. Moreover, T~ is closed since if {Xm}meN C Da converges to x and Tax m converges to y, then for all p~ we have: lim p,~(x m) = pn(x)
lim p,.,(Tax m) =
and
m - - - * O0
m---,oo
lim a,~p~(x ~) = a,~p,~(x). m--,
oo
On the other hand,
lim B~(To~ ~) = p,,(y)
rt't---,O0
and therefore p,~(y) = anp,~ (x) for all n e N, i.e. x e D~ and Tax = y. Given z - A + i# E C, it is evident that z + T~ is injective if and only if
-z r {~,-~
e N}.
Let us now prove that if a E (R+) N is an unbounded increasing sequence of positive numbers, then T~ is positive and a (T~) = {a,~'n E. N}. Fix - z {an "n E N} and denote by bn = (z + a n ) -x . The sequence {Imb~},~e N is monotone and {Re b,~}n>n ~ (no being the first index such that A + ano >_ I#l) is monotone decreasing. So, if no < k < n, then n
n
n--1
E bpxp -- E bp (Sp -p=k p=k
8p_l)
--"
bnsn-bk.sk_l+ E (bp -
bp+l)sp
for all x E X
p=k
which implies that
• bpxp
n-1
max <_ Ib,~s,~l + Ibksk-~l + k<_p
p--k
p'-k
<- Ilxil ( 2sup lbpl + lRe(bk - b'~)i + lIm (bk < o Ilzll sup lb,! --, o p_>k
as k --, oo.
oo
Hence the series ~p=l bvxv is convergent and therefore the operator z + Ta is a bijection from Da onto X whose inverse is Tb. So, - z E p(Ta). In the case that # - 0 and A _> 0, then _b,~ls,~l+bl
t ere ore
§
II
max
X
Isvl<__2()~+ax)-lllxl I
II
Iir §
II-
II
II
for
0,
for a l l n e N
imp es
positive (here we do not need to assume that a has bounded projections). 180
Denote by a ir = (aT, a~~, ...). Let us prove that if T E IR\ {0}, (T~) ir If x E D~ and n E N, then, by the continuity of the projections,
[(To)
z
-
.x"'p~
(.x + To)-"
---" T a i T
o
2
7rT
=
I
)it 7rr
Jo
-1
(~ + aT,)
]
2
ir a,, d,~ - a,~ .
Since (T=) i~ and T=,~ are closed operators and both commute with (A + T a ) - 1 , (T=) ir = T=,~, due to the density of Da. Finally, let us consider an element a E (IR+) N , whose projections form an unbounded increasing sequence of positive numbers, such that for k - 1, 2, ... 2kTri 9 9 the imaginary power (Ta) coincides with the identity operator, and such that (Ta) (2k+1)~i is unbounded. Let 9 E X gl~ N such that ~,~--1 ]~,,[-- oo and let s E ]RN such that 1 if ~,~___0 sn_ if ~ , ~ < 0 " It is evident that 9 ~ D8 and therefore T8 is unbounded. non-decreasing sequence { f (n) },~--1 such that
f(n)
Let now a,., -
ef("),n
to an even number to an odd number
= -
if if
xn>_O
x~ < 0
1,2, .... It is evident that for k -
coincides with the identity operator. unbounded.
Consider now a
1,2,..., (Ta) 2kTri
However, (Ta) (2k+1)'i = Ts, which is
R e m a r k 7.3.1 The preceding example can be easily generalized to any complex Banach space which has a basis (in particular, to any separable Hilbert space). This is the way in which this example appears in [193]. R e m a r k 7.3.2 In Example 7 . 3 . 4 , A 2"i unbounded and therefore A ' ~ i A '~i ~ A 2~i.
I, which is bounded, but A ~i is
R e m a r k 7.3.3 If A is a bounded and positive operator, then a (Ar162= {z 'r
z E a(A)}
(7.16)
since in this case the imaginary power A ir is the operator associated to the function z i~ by means of the Dunford Functional Calculus. In general, under weaker conditions, we have no spectral mapping theorem. For instance, in Example 7.3.4, a (Ta) - {al, a2,...}. However, T~" is closed and unbounded and (T~ ~) 2, which is unbounded, satisfies (T~')
2
_
181
TiTr2 -a ,
by additivity (Theorem 7.1.1). As T_~'r2 = I, (Ta~) 2 # (T~) 2 and hence (T~) 2 is not closed. Therefore, by Lemma 5.3.6,
a (T~') = C without connection with { z i ~ ' z E a(Ta)}. R e m a r k 7.3.4 For the applications to partial differential equations (see [161, Theorem 6.3.1] and [185, Theorem 1.18]) it is interesting to know if
U
D(A'~) = z .
aEC+
In order to answer this question let us prove that if A is positive, then
U D (A '~) c
for
aEC+
Given a e C+ and r E D (A~ as D (A -'~+'~) = X, A-'~+'~A'~r = A'~r by additivity. Moreover, as R (A -~+'~) = D (A ~-'~) C D (A), A'~r e D (A). Proposition 7.2.2, part (iii), now yields r E D (~Ar) . We can now answer the above question related to the domains of the fractional powers. The operators studied in Examples 7.3.3 and 7.3.4 are positive, sectorial and densely defined. Its imaginary powers are unbounded. Therefore U
D (A m) c D (~TA~) r X.
(7.17)
aEC+
Examples of operators whose negative is the infinitesimal generator of an analytic semigroup and which satisfies (7.17) can easily be obtained by applying Theorems 7.1.3 and A.7.6 to the above examples.
7.4
Limit Operators Related to the Imaginary Power
Throughout this section, A will be a non-negative operator on a Banach space X. This section is devoted to the study of the operators s - lim~_.0+ (A + s ) - ~ , s - lima__.0+ A - ~ + ~ (when A is positive) and s - lime__.0+ (A + s ) ~ . In particular, we study the relationship between the last two operators and the imaginary power A i~. We also include a perturbation theorem for operators which have bounded imaginary powers. P r o p o s i t i o n 7.4.1 Let a E C+. Then J - ~ = s - lim (A + c) -~ . e---*O+
182
(7.18)
Proof. Denote by T = s - lime__.0+ (A + ~)-~. Given r e D (T) it is evident that lim ~ (A + ~)-~ r - 0 e---*0
and therefore r E R (A), by Theorem 6.1.1. Moreover, lim A ~ (A + e) -~ r = r
e---*0
(7.19)
which implies, since A ~ is closed, that Tr E D (A ~) and A~'Tr = r On the other hand, as r e R (A), (A + e)-~' r e R (A) for all e > 0, and therefore Tr E R (A). Given r E D (A a) M R (A), lim (A + e) -~ A~r = r
e---*0
and hence A~r E D ( T ) a n d TA~r = r So, T is the inverse of the part of A ~ in the closed subspace R (A). Therefore T is closed. Let n > Re a be a positive integer and let r E D (Y-~). By the dominated convergence theorem, lim
~--*0
/0
,V~-1-~ (A + ,~ + e) -'~ r d,~ =
and so 4) E D (T) and Tr = J-'~r A (A + #)-1
r
/0
,V~-1-'~ (A + ,~)-'~ r d,~
Thus, by multiplying T and J - ~ by
with # > 0, (taking r E D (T) and r e D ( J - ~ ) , respec-
tively) and letting/~ ---, 0 we get (7.18). I From now on in this section we assume that ,4 is injective. If A is positive, then A -~+i', o~ > 0, is bounded and its limit (as c~ ~ 0) has a strong connection with A i~ and j i r as we will see later. Definition 7.4.1 Let A be a positive operator. We denote by Ai~ the strong limit s - lim A -~+~ cr
defined on the domain D(Ai,)={r
~--.0+limA - ~ + i ' r ex/sts}.
R e m a r k 7.4.1 As the range of A -~+i~ is included in D (A), the range of Ai~ is also included in D (A). Therefore, if A has non-dense domain, then Ai0 C I. In the following results related to Air we assume that 0 E p (A) and that the domain of A is not necessarily dense. L e m m a 7.4.2 If r C D (A) M D ( j 1 + i 4 ) , then
Ar c D (Ai~)
and
Ai~Ar = j l + i ~ = AAi~r 183
Proof. As 0 E p (A), D (A) C D (Air). Hence, if r E D (A) N D (jl+i~) , then, by Theorem 7.1.1, Aide e D (A)
and
AAi~r = jl+i~r
Let 0 < a < 1. By additivity A-~+i~Ar = Al-~+i~r = AI-~Ai~r = j l - ~ A i ~ r where the last equality is due to Corollary 5.1.12, since R (A -~+~) C D (A). If we take limits as a ~ 0 in this identity, then, by Theorem 3.1.6, part (i), we conclude that Ar E D (Air)
and
Ai~Ar = AAi~r = jl+i~r
m
R e m a r k 7.4.2 By Proposition 7.1.5, D (A) C D ( A ~ ) and A ~ r = J ~ r T h e o r e m 7.4.3 Air is closed. Moreover, J~: c A~ c A ~. Proof. Let {r
C D (Air) such that lira r
=r
and
lim AirCn = r
n---*O0
n-.-400
As (1 + A) -1 and A (1 + A) -1 are bounded, lim (1 + A) -1 r
= (1 + A) -1 r
n - - - * OO
and lim A (1 + A) -a Ai~r
= A (1 + A) -x r
n---~OO
Moreover, as (1 + A) -x commutes with Air, (1 + A) -x Ai~r
= ji~ (1 + A) -1Cn E D (A)
by Remark 7.4.2. Thus, A (1 + A) -x A~r
= A J ~ (1 + A) -1 r
E D (A),
since A (1 + A) -1Ai~r
= Ai~r
- ( 1 + A) - 1 A ~ r
Hence +
and by additivity (1 + A) -1 r
E D (jl+ir) 184
e D (A).
and A (1 + A) -t Airr
= AJ ir (1 + A) - t Cn = jl+ir (1 + A) - t Cn.
Letting n --~ oo
(I + A)-I c E D ( j 1+it)
and
j i + i r (1 + A) -1 r = A (1 + A) -1 r
From the preceding lemma it is deduced that A (1 + A) -1 r E D (Air), which yields r E D (Air) and Ai,r = ~. The inclusion jir c Air follows easily from the fact that Air is closed and from Remark 7.4.2. Finally, the inclusion Air C A ir is due to the relation
ji~
(I +
A) -1 r = Ai,
(I +
A) -1 r =
(I +
A) - t Aide
which is valid for all r E D (Air). m C o r o l l a r y 7.4.4 If D (Air) C D (A), then
Air = jir. In particular, this happens if A is densely defined or if JA ir is continuous. P r o o f . By Proposition 7.2.2 and taking into account that 0 E p (A),
Moreover, by the preceding theorem and by Remark 7.4.1 it follows that the set
{ r e D (Ai~) is included in D ( 7 ) . ~
n
D (A)" Aide e D (A) } = D (Air)
Hence ji~: is an extension of Air, which implies that
f
Air = jir. Obviously, this happens if D (A) = X. On the other hand, if j - i t is continuous, then D (j-i~') _- D (A) and as R(Air) C D (A), by additivity and by the preceding theorem, J-irAirr = r for all r E D (Air). Hence D (Air) C R ( j ' i §
C D (A).
I The identity Air = yir also holds if we remove the hypothesis of density of D (A) and even if j - i t is not continuous. To see this it is sufficient to consider in l ~176 (C) • co (C) the operator A (r r
- (At r A2r 185
where A1r = (r162
3r
...) in l ~ (C)
and A2r = (r
Cx + r
r
+ ~2 + r
".-) in co (C).
The domain of A is not dense since D (A1) is not dense, and JA ir is t i n u o u s since j - J r is not continuous. However, A2 (A1)ir
--
j iA1 r
and
1-tot c o n -
(A2)ir - j iA2 r
since JA~ r is continuous and A2 has dense domain. So, Air = j~r. R e m a r k 7.4.3 Let us consider the operator A (r
r
r
...) = (r162162
...) in l ~ (C).
Since j - i t is continuous, Air = Jji§ Therefore A it, which is bounded, is a proper extension of Air. In the following results we deal with the strong limit of (A + 6) ir as 6 ~ 0. P r o p o s i t i o n 7.4.5 The domain of (A + 6)ir does not depend on 6 > O. If 0 E p (A), then this domain coincides with D (Air). P r o o f . Fix 6, 6' positive and # > max {6, 6'}. By (7.4), J~+~ (~ + A) -1 r i sin.h (lr'r) 71"
Jy+~, (# + A) -1 r
[
]
Air (A + 6 + A) -1 - (A + 6' + A) -1 (# + A) -1 r dA
for all r E X. Taking into account the resolvent formula and the fact that (6 + A) -1 and (e' + A) -1 are bounded, ~ (# + A) -1 r JA+~
J~+~, (# + A) -1 r e D (A)
If 0 E p(A), then the same reasoning is valid for 6' = 0, which yields
In"
L"
/
J
R e m a r k 7.4.4 I f 0 ~ p ( A ) , then it is not true that D (A it) - D [(A q-6)ir] . To see this it is sufficient to consider in co (C) the operator 1 - S, where S is the shift operator. The power (1 - S) ir is unbounded. ~However, (1 - S + 6), 6 > 0, is bounded and 0 E p (1 - S + 6). Consequently, (1 - S + 6) ir is bounded. 186
Definition 7.4.2 We denote by Sir the strong limit s-
lim (A + e) ir e--.~O+
with domain
T h e o r e m 7.4.6 Let A be an injective non-negative operator. Then: (i) Sir is closable and
3~
C
Sir
C
Ai~.
In particular, if A has dense domain and range, then Sir = A it.
(ii) If 0 E p (A) , then Sir is closed and Sir = A it. Proof. Since A ir is closed and j~r is closable, to prove (i) it is sufficient to show that j i r C Sir c A ir . Fix r = AT E D (A) M R ( A ) . By (7.4) and by writing f o = f : + f6~1766 = e 1/2, we have
II(
(2 (M + 1)Ilvll + M2 I1r fll
II
%.|!
I!
a/~
(7.20)
where M - s u p ~ I I A ( A + A ) - l l ] ' A > 0~. Letting e --, 0, r E D(Sir) and
&~r
= j~r
Let us prove that A ir is an extension of Sir. Given r E D (S/r), A(I+A) -2Sire
=
lim A ( I + A ) - 2 ( A + C ) i r e ~--.0+
=
lim j i r e~0+
=
A(I+A) -2r
A+e~x
JirA (1 + A) -2 r
Hence r E D (A it) and Airr = Sire. Finally, statement (ii) follows easily from the fact that (7.20) remains valid for 0 E p (A) and r e D (Air). m In Theorem 7.4.6, part (ii), we have proved that if 0 E p (A), then s-
lira (~ + A) ~ = A i~.
e---*0
This result also holds if it is only assumed that R (A) is dense, but we have to assume that A ir is bounded. P r o p o s i t i o n 7.4.7 Let A be non-negative with dense range. Suppose that A ir is bounded. Then lim (~ + A) ir r = Airr e---,O+
187
for all C E X.
Proof. As A i~" is bounded, by Corollary 7.1.2, we know that +
Then, (A + e) i~" is bounded, since by Theorem 7.1.1, (A + ~)i~- = (A + e) 89
(A + e) -'~ .
Moreover, given r E X, (A + e)i~ r
(A + e)89
_
A89
(A + ~)-89r 1
By Proposition 5.1.14, there is a positive constant C1 such that <_
~ and, as the range of A is dense, e--.olimr189(A + r189 r = 0 and
e--.olim[A (A + e)-'] 89r = r
by Theorem 6.1.1. m
7.5
Negative and Imaginary Powers on Locally Convex Spaces
All the results (with exception the Proposition 7.1.5) in Sections 7.1 and 7.2 continue to be for sequentially complete 1.c.s. However, we have to make the following remarks: 9 The inequality (7.5) will take the following form: If { [[lip- p E P } is a directed family of seminorms which describes the topology of X, given p E :P, there is an index s E P such that
where C only depends on the base operator A. 9 In Corollary 7.1.2 we have to have A i~" as a continuous operator defined on X. 188
9 Estimates 7.12 and 7.13 will take the following form: If a E C+ and n > Rec~ is a positive integer and r = A'~U E R(A'~), given p E :P, there is an index s E P such that I[J-~r
_ C(n - c~, n, M)I1r
~eo
Re
II lls
if ~- E II~ and r - A~D E D(A m-'~) MR(A~), with m > n positive integers, then r Ji'_,,p _< C(n + iT, m,M)1177111-~s IlAm-~r '~'~ . [I The index s E :P only depends on the operator A and on the index p. 9 In Theorem 7.2.3 we have to add the hypothesis that the space X ~endowed with the strong topology will be sequentially complete.
7.6
Notes on Chapter 7
The first papers about the imaginary powers of operators are due to Kober [107], Kalish [100], M. J. Fisher [70] and E. R. Love [130] who studied the pure imaginary powers of the Riemann-LiouviUe derivative operator on L p with 1 < p < c~ and proved that they were bounded operators. It is known that the part in L p (1 < p < c~) of certain linear differential operators have bounded imaginary powers ([8], [60], [61], [62], [75, Th. 1], [76, Wh. A], [77, Appendix A] and [165]). It is not known if the part in L p (1 < p < c~) of every sectorial differential operator admits bounded imaginary powers. M. Hieber in [90] constructs, given any 0 e [0, ~[, an example of a positive sectorial pseudo-differential operator A, with spectral angle 8 and such that does not have bounded imaginary powers. The imaginary powers of non-negative operators can be defined from the McIntosh functional calculus for sectorial operators (developed by A. McIntosh for Hilbert spaces in [150] and by M. Cowling, I. Doust, A. McIntosh and A. Yagi for Banach spaces in [39, Section 2]). However, this could not be done in locally convex spaces where there are non-negative operators which are not sectorial. Example 7.3.3 was given by H. Komatsu in [109, Section 14, Examples 6 and 8] and Example 7.3.4 is due to A. Venni and is found in [193]. Proposition 7.4.7 has been proved by J. P r o s and H. Sohr in [164, Th. 3], but with the additional conditions of the operator A being densely defined and of the class B I P ( X , OA). In [10, Th. A] J. P. Baillon and P. Clement give an example of a sectorial operator in a Hilbert space with spectral angle 0 and such that A is are not bounded operators for all s E IR\ {0}. In [152, Th. 4] A. McIntosh and A. Yagi give a general method for constructing positive operators on a Hilbert space which do not have bounded imaginary powers, by using the fact that this property is equivalent, by Theorem 10.2.6, to the non-existence of an 7-/~bounded functional calculus for the operator A. 189
Fractional Powers with Exponents of Negative Real Part. Imaginary Powers of Operators In this chapter we define and study the basic properties of the complex powers when the exponent has non-positive real part. We pay special attention to the imaginary powers because of the relationship to the existence, uniqueness and regularity of solutions of certain problems of abstract parabolic differential equations.
7.1
Definitions and Basic Properties
Throughout this section, we assume that A is an injective non-negative operator (not necessarily of dense domain or range) on a Banach space X. Remember that for non-negative operators, density of range implies injectivity. D e f i n i t i o n 7.1.1 Given c~ E C+ we define A - " = (A~) -1 .
(7.1)
Note t h a t since A is injective, by Corollary 5.2.4, A n is also injective and (A~) -1 - ( A - l ) a . Moreover, A n is closed and D ( A -'~) D R ( A p) if p is a positive integer such that p > Re a. Applying (5.20) to the operator ( A - l ) ~ yields A - ~ = (A + 1) p A - P A p+'~ (A + 1) -p . (7.2) D e f i n i t i o n 7.1.2 Given "r E IR we define A i~ = (A + 1) 2 A - 1 A 1+i~" (A + 1) -2 . 171
(7.3)
Note that A ~r is closed since A ~+ir (A 4- 1) -2 is bounded and (A + 1) 2 A -1 is closed. Moreover, D (A it) D D ( A ) N R ( A ) . Thus, if A E s and 0 e p(A), then Air is bounded and
Aide = - i sinh rT" 7r
[
Ai~ (A + A) -1
A A2 + 1
]
rT CdA+c~162
(r
(7.4) by (3.2). Moreover, an easy calculation shows that
ilA ll <_
(75)
where C only depends on the base operator A. It is not difficult to prove that A ir = (A + 1) q A-VA p+ir (A + 1) -q
(p, q e N, q > p),
(7.6)
and that in (7.2) and in (7.6) the terms (A + 1) • can be replaced by (A + A) • , with A > 0 (or A = 0 if A is positive). If T is a bounded operator that commutes with (A + 1) -1 , then A a T is an extension of T A ~ for all a E C. Finally, if p and q are positive integers such that q > p and q > p + Re a > 0, then for all r e X A p ( A + I ) - q C e D ( A ~)
and
A~A p ( A + I ) - q r
~+p(A+I) -qr
for all a E C. T h e o r e m 7.1.1 ( A d d i t i v i t y ) Let a, fl ft. C. Then
(i)
A"A ~ c A ~+~.
(ii) If r e D (A ~+~) N D (AZ) , then
A~r E D (A s)
and
A~Aar = A~+ar
In particular, if D (A ~+~) C D (An), AC'A ~ = A ~+~.
(iii)
If D (A) and R (A) are dense, then A~+~ _ A~A ~.
P r o o f . Let r E D (A~A ~) and let p be a positive integer such that p > max {IRe c~[, [Refll}. By (7.7) and by additivity for exponents in C+, A 2p (A + 1) -4p A~A~r = A~+VA p (A + 1) -@ A~r
= A~+VA ~+p (A + 1) -@ r = A ~+~+2p (A + 1) -@ r 172
(7.7)
= A ~+zA 2v (A +
1)-4P (~.
Thus, from (5.20), (7.2) or (7.6) (depending on the sign of Re (a + fl)) it follows that r E D (A ~+~) and A~A~r = Aa+~r The same reasoning works for statement (ii). Consider now r e D (Aa+~). Taking p as in case (i) and A, # > 0, the element A 2v (A + #)-2p (A + A)-2v r belongs to D (AaA~). So, A 2p (A + #)-2p (A + A)-2p A~+~r = A ~ A ~ A 2p (A + #)-2v (A + A)-2p r and multiplying by A2v and letting A---+ Go and # ~ O, r E D (A'~A ~) and A~+~r = A'~A~r since D (A) and R ( A ) are dense, m
Corollary 7.1.2 Given T E ]R\ {0}, if A i~ is bounded, then D (An) C D (A ~+ir
for all a e R.
Conversely, if D (A n) C D (A ~+ir) for all a e IR\ {0}, then A ir is bounded. If A is positive, then it is suO%ient to prove the inclusion for all a > O.
Proof. If A ir is bounded, then, by Theorem 7.1.1, D (A n) - D (AirA ~') C D (A'~+ir).
Conversely, if D (An) C D (A ~+'r) for all a e JR\ {0}, then D (A)C D (A 1+it)
and
D (A -1) C D (A -1+it)
and so, again by Theorem 7.1.1, R (A) C D (A i~')
and
D (A)C D (Air).
As every element r E X can be written as r = (1 + A) -~ r + A (1 + A) -~ r we have X C D (Air). By the closed graph theorem, A ir is bounded. The last assertion is evident, m T h e o r e m 7.1.3 (Multiplicativity) Let A be a sectorial operator, with spectral angle WA, and a E ~ such that IC~OAI < 7r. Then A n is sectorial and (A~) ~ - A ':'~ ( f l E C ) . 173
P r o o f . In Theorem 5.4.1 we proved that if A is sectorial and 0 < a W A "( 71", then A s is sectorial, with spectral angle WA,, <_ SWA. Moreover, the property (AS) ~ = A s~ has been proved in case Re/~ > 0. From Definition 7.1.1 and Corollary 5.2.4 it follows that this property is also true for s < 0 and Re B < 0. So, it is sufficient to consider case ~ = i7". Let p > Is I be a positive integer. Given r E D [(As)i~] , as A p (1 + A) -2p is a bounded operator that commutes with the resolvent of A s and its range is included in D (A m) M R (AS), A p (1 + A) -2p (AS) i" r = (A~) ir A ~ A p - s (1 + A) -2p r = (AS) 1+i" Ap-~ (1 + A) -2p r = A~+Sir p-~ (1 + A) -2p r = A~i~A p (1 + A) -2p r
Hence r e D (A s'~) and A si~r = (A s)i~ r Conversely, if r E D ( A ~ r ) , then the same reasoning as before yields A p (1 + A) -2p A ~ i ' r = (A~) i~ A p (1 + A) -2p r and if we multiply this expression by A m (1 + A s)-2, then A ~ (1 + A") -2 A~'~r = (A~) '~ A ~ (1 + A~) -2 r Therefore r E D [(AS)i~ 1 and (A~) '~ r = Aairr I The following result is an immediate consequence of Theorems 7.1.1 and 7.1.3. k
d
C o r o l l a r y 7.1.4 A ir is one-to-one and (A'") -1 = A -'~" = (A-X) '" .
(7.8)
P r o o f . The first equality is a consequence of additivity (Theorem 7.1.1) and the second one follows from Theorem 7.1.3. I P r o p o s i t i o n 7.1.5 (Analyticity) Let so E C+. If r E D (A ~~ M n ( A s ~ then the function a ~ ASr is holomorphic in {s E C" - R e s 0 < R e a < Re s0}. If r E R (A a~ , then a H A~r is holomorphic in
{s E C ' - R e s 0
< R e s < 0}.
P r o o f . By additivity, r E D (A m) if a E C such that IRe s] < Re so, Re a ~ 0. Fix m E N such that •m < Rea0. Then r E D (A 1/m) M R (A 1/m) and as A x/m is non-negative,
r
1174
Therefore r E D (A s) for all c~ E C, with IRe a[ < Re a0. Let r = A~~ E D (A~~ R (A~~ By additivity, A~r = A~+~~ , IReaI < Rea0 and by Corollary 5.1.13 this function is analytic in
{a E C :0 < R e a + Rea0 < 2Rea0} which proves the first assertion. The second one follows from Definition 7.1.1 and Corollary 5.1.13. I
7.2
The Balakrishnan and Komatsu Operators
Similarly to the case of the Balakrishnan operator introduced in Chapter 3 for exponents with positive real part, the following definitions and properties are closely related to the concept of complex power of non-negative operators. Throughout this section we assume that A is a non-negative operator on a Banach space X. D e f i n i t i o n 7.2.1 Let ~ E C+ and let n be the minimum integer number such that n > Rec~. We define the operator J - a with domain R(A '~) and range included in R (A) J - ~ r = J~-%?
(r = A"~ E R(A'~)).
(7.9)
By (5.14) we obtain, for m >__n and r E R (Am) 9 J - - (1~r
---
F(m)
Am-l-~ (A + A)-m r dA.
~0~176
(7.10)
Note that in order to define J - ~ it is not necessary to assume that A is injective. It is not difficult to show that (7.9) does not depend on rI. If A is injective, then it is obvious that A m extends J - ~ . In fact, a simple change of variable in (7.10) shows that JA ~ - J~_~. D e f i n i t i o n 7.2.2 Let T E I~. We define the operator jir, with domain
D(J ~) = D(A) M R(A) (and with range contained in D(A) M R(A)) as (7.11)
for r = AT E D(A) MR(A). complements of Euler
If~" ~ O, then, by (3.~) and the formula of
7r l" (iT) i" (1 -- iT) = sin iTrr'
we obtain j i b e _ ~ s i n~T h ~'T
~0~
Ai~( A + A)_2ACdA.
175
As in Definition 7.2.1, it is not necessary to assume that A is injective to define j i r . If A is injective, then A ir extends j i r . The estimates that we prove in the following proposition show that J - ~ and ji~ are closable, even thought A is not injective. P r o p o s i t i o n 7.2.1 Let a E C+ and let n > Re a be a positive integer. r = A'~7 E R(A'~), then
Ilj- r
<
C(n -
n, M)IIr
' - R,-~F-117/11R,:~,.
If
(7.12)
If r E lt~ and r = A ' ~ E D(A m-'~) M R(A'~), with m > n positive integers, then IlJi'r
<_ C(n + it, m, M)117/111-#'' IlAm-nr
'~' 9
(7.13)
The constants C(n - a, n, M) and C(n + it, m, M) are given by (3.11). Consequently, the operators J - ~ and j i r are closable. Proof. The estimates (7.12) and (7.13) are an immediate consequence of (3.10). From these estimates it follows that for - # E p(A), the operators J - ~ [A(# + A)-I] n , n > R e a , and Ji~A(# + A) -1 (1 + A) -1 are bounded. The proof of the closable character of J - ~ and ji~- is similar to that of Theorem 3.1.8, taking into account that
R ( J -~) U R ( J i~) C R(A) and letting # ~ 0. I If A is injective, then the relationship between J - ~ and ji~ and the corresponding powers of A is given in the following result, which is similar to Corollary 5.1.12. P r o p o s i t i o n 7.2.2 Let a E C+ and T E JR. Assume that A is injective. Then
and
Consequently, if the range of A is dense, then A -a - j - d . If, in addition, the domain is also dense, then A ir = jir. P r o o f . The inclusion
is evident. Conversely, if r E D (A i~) M D (A) N R (A) and Aide E D (A) M R (A), 176
then A (A + A) -1 # ( # + A )
-1 A i r r = j i r A (A + A ) - I # (# + A)-I r
(~, ~ > 0) /-.--\
and taking limits as # tends to infinity and l to zero we get r 6 D ( j i ~ ) and j i r r = Airr The equality (7.15) is analogous, m If A is non-negative with dense domain and range, then its adjoint A* is also non-negative and one-to-one but, in general, has non-dense domain or range. Moreover, A ir has dense domain, so we can consider its adjoint (Air) * . It is natural to ask if (Air) * = (A*) ir . The answer is given in the following:
T h e o r e m 7.2.3 If A has dense domain and range, then (Air
- ( A * ) ir "
P r o o f . As A has dense domain and range,
(y;)"
=
by Proposition 7.2.2. Let us prove that ( j i r ) * is an extension of j~r. Fix r e D (A) N R (A) and v e D (A*) N R (A*). Then (J~.v, r
= sinh (rT) 7rT
/0
Air A* (A + A*) -2 v, r
(
)
dA
where the second equality is due to the fact that [A (A + A)-2] * = A* (A + A*) -2 which follows easily from the definition of the adjoint operator. In order to prove that (Air) * - ( j i r ) * C (A*) ir let us consider an element Taking into account that the commutativity between J iT and
vED[(Jir)*]. L
.,J
A (1 + A) -2 also holds if we take adjoints, A* (1 + A*) -2 ( J i r
= =
n - ( 1 + A') J~.A*(I+A*)-2v
and by Definition 7.1.2 we conclude that v E D [(A* )iT] and (A*)ir v = (Air) . i,,. It only remains to prove that ( j i r ) * is an extension of (A*) ir . Let v e D [(A*) ir] and r E D ( j i r ) .
Given ,k,/z > 0 and taking into account that 177
(ji,)* is an extension of J~., (u, A (A + A) -1A (# + A) -1 jibe)
- (~(~+z)-,z(.+z)-l~,
j"~)
-
(J~.A(A+A*)-IA*(#+A*)-lu,
=
(:~(~ + A') -1 A" ( .
=
((A*)'" u, ,~ (,~ + A ) - I A (tt + A) -1 r
r
+ A') -1 (A')".,.
~,)
Finally, taking limits as ,~ tends to infimty and # to zero we conclude that
B
7.3
Examples
E x a m p l e 7.3.1 On the space X - / ~ 1 7 6 composed of the bounded sequences of complex numbers, with the supremum norm, let us consider the multiplication operator A(r r = (r162162 with domain D(A) - {r E X : Ar E X}. From Subsection 1.3.4, this operator is positive and non-densely defined. Its imaginary power A i~" is given by Ai~(r162
= (r162
3iT r
which is a bounded operator. However, ji~ is unbounded since D ( J i~) C D(A). By considering its inverse we obtain a non-negative operator with non-dense range and with bounded imaginary powers. Finally, in the Banach space X x X, the operator A~162r = (Ar162 defined on its maximal domain is a non-negative operator, with non-dense domain and range, and with bounded imaginary powers. E x a m p l e 7.3.2 Let A be a normal and injective operator defined in a Hilbert space. Assume that a(A) C ~ (with w e]0, r[). The imaginary power A i" coincides with the operator (z it) (A) obtained by means of the Functional Calculus associated to normal operators described in Subsection 1.3.7. Then, A i" is bounded ,ha IIA"II < ~ ' J E x a m p l e 7.3.3 [H. Komatsu] Let us consider in the space co (C) of the sequences of complex numbers converging to zero the operator A = ( 1 - S) -1 , 178
where S is the shift operator (for more details see Subsection 1.3.6). The operator A is non-negative and densely defined. The imaginary power A i~ is given by iv(iT + 1)$2 iv(iv + 1)(i7 + 2)$3 + ..... A i~ = 1 + iTS + + 2 3! If this operator was bounded its norm would be given by
IIA~[I
=
1+
I~-I + I~-I (ITI ~ + 1) ~/~ 2
>_ l~l
1+~+5+
ITI (ITI ~ + 1)~/2(IrI 2 + 4) ~12 +
3
+""
.... - o o ,
which proves that A i~" is unbounded if T # 0. As A ir = (A - 1 ) - i v (see Corollary 7.1.4) and the operator S - 1 generates an equibounded C0-semigroup, we have an example of a bounded operator whose negative generates an equibounded C0semigroup and such that its imaginary powers are unbounded. Combining this example with Theorems 7.1.3 and A.7.6 we obtain an example of a bounded operator whose negative is the generator of an analytical semigroup and its imaginary powers are unbounded. The following example shows a positive operator whose imaginary powers are bounded or unbounded, depending on the exponent. E x a m p l e 7.3.4 [A. Venni] The space X =
x - (xl,x2, ...) E C N 9 s,~ =
xk has a finite limit in C k--1
endowed with the norm nEN
is a Banach space since the mapping ( ~ , ~2, ...) ~ ( ~ , ~2, ...)
is an isometry from X onto the subspace of/~ of the convergent sequences. As Ix~] = is,~- s~-ll i 2 llxll for all n > 1 and Ixll = 181] ~___ IIX[I, the projections p,.,(x) = x,~ are continuous. Given a, x E C N, we denote by ~.~ = (a~,
~2~,..-)
and by D,~= { x E X : a . x E X } .
Consider now the operator Tax=a'x
for 179
xEDa.
It is evident that Da is dense in X since for all x E X, lim (xx,x2,...,x,~,O,O,...)
x-
1"1,.--*O 0
and obviously (Xl,X2,...,Xn,O,O,...) E Da for all n E N. Moreover, T~ is closed since if {Xm}meN C Da converges to x and Tax m converges to y, then for all p~ we have: lim p,~(x m) = pn(x)
lim p,.,(Tax m) =
and
m - - - * O0
m---,oo
lim a,~p~(x ~) = a,~p,~(x). m--,
oo
On the other hand,
lim B~(To~ ~) = p,,(y)
rt't---,O0
and therefore p,~(y) = anp,~ (x) for all n e N, i.e. x e D~ and Tax = y. Given z - A + i# E C, it is evident that z + T~ is injective if and only if
-z r {~,-~
e N}.
Let us now prove that if a E (R+) N is an unbounded increasing sequence of positive numbers, then T~ is positive and a (T~) = {a,~'n E. N}. Fix - z {an "n E N} and denote by bn = (z + a n ) -x . The sequence {Imb~},~e N is monotone and {Re b,~}n>n ~ (no being the first index such that A + ano >_ I#l) is monotone decreasing. So, if no < k < n, then n
n
n--1
E bpxp -- E bp (Sp -p=k p=k
8p_l)
--"
bnsn-bk.sk_l+ E (bp -
bp+l)sp
for all x E X
p=k
which implies that
• bpxp
n-1
max <_ Ib,~s,~l + Ibksk-~l + k<_p
p--k
p'-k
<- Ilxil ( 2sup lbpl + lRe(bk - b'~)i + lIm (bk < o Ilzll sup lb,! --, o p_>k
as k --, oo.
oo
Hence the series ~p=l bvxv is convergent and therefore the operator z + Ta is a bijection from Da onto X whose inverse is Tb. So, - z E p(Ta). In the case that # - 0 and A _> 0, then _b,~ls,~l+bl
t ere ore
§
II
max
X
Isvl<__2()~+ax)-lllxl I
II
Iir §
II-
II
II
for
0,
for a l l n e N
imp es
positive (here we do not need to assume that a has bounded projections). 180
Denote by a ir = (aT, a~~, ...). Let us prove that if T E IR\ {0}, (T~) ir If x E D~ and n E N, then, by the continuity of the projections,
[(To)
z
-
.x"'p~
(.x + To)-"
---" T a i T
o
2
7rT
=
I
)it 7rr
Jo
-1
(~ + aT,)
]
2
ir a,, d,~ - a,~ .
Since (T=) i~ and T=,~ are closed operators and both commute with (A + T a ) - 1 , (T=) ir = T=,~, due to the density of Da. Finally, let us consider an element a E (IR+) N , whose projections form an unbounded increasing sequence of positive numbers, such that for k - 1, 2, ... 2kTri 9 9 the imaginary power (Ta) coincides with the identity operator, and such that (Ta) (2k+1)~i is unbounded. Let 9 E X gl~ N such that ~,~--1 ]~,,[-- oo and let s E ]RN such that 1 if ~,~___0 sn_ if ~ , ~ < 0 " It is evident that 9 ~ D8 and therefore T8 is unbounded. non-decreasing sequence { f (n) },~--1 such that
f(n)
Let now a,., -
ef("),n
to an even number to an odd number
= -
if if
xn>_O
x~ < 0
1,2, .... It is evident that for k -
coincides with the identity operator. unbounded.
Consider now a
1,2,..., (Ta) 2kTri
However, (Ta) (2k+1)'i = Ts, which is
R e m a r k 7.3.1 The preceding example can be easily generalized to any complex Banach space which has a basis (in particular, to any separable Hilbert space). This is the way in which this example appears in [193]. R e m a r k 7.3.2 In Example 7 . 3 . 4 , A 2"i unbounded and therefore A ' ~ i A '~i ~ A 2~i.
I, which is bounded, but A ~i is
R e m a r k 7.3.3 If A is a bounded and positive operator, then a (Ar162= {z 'r
z E a(A)}
(7.16)
since in this case the imaginary power A ir is the operator associated to the function z i~ by means of the Dunford Functional Calculus. In general, under weaker conditions, we have no spectral mapping theorem. For instance, in Example 7.3.4, a (Ta) - {al, a2,...}. However, T~" is closed and unbounded and (T~ ~) 2, which is unbounded, satisfies (T~')
2
_
181
TiTr2 -a ,
by additivity (Theorem 7.1.1). As T_~'r2 = I, (Ta~) 2 # (T~) 2 and hence (T~) 2 is not closed. Therefore, by Lemma 5.3.6,
a (T~') = C without connection with { z i ~ ' z E a(Ta)}. R e m a r k 7.3.4 For the applications to partial differential equations (see [161, Theorem 6.3.1] and [185, Theorem 1.18]) it is interesting to know if
U
D(A'~) = z .
aEC+
In order to answer this question let us prove that if A is positive, then
U D (A '~) c
for
aEC+
Given a e C+ and r E D (A~ as D (A -'~+'~) = X, A-'~+'~A'~r = A'~r by additivity. Moreover, as R (A -~+'~) = D (A ~-'~) C D (A), A'~r e D (A). Proposition 7.2.2, part (iii), now yields r E D (~Ar) . We can now answer the above question related to the domains of the fractional powers. The operators studied in Examples 7.3.3 and 7.3.4 are positive, sectorial and densely defined. Its imaginary powers are unbounded. Therefore U
D (A m) c D (~TA~) r X.
(7.17)
aEC+
Examples of operators whose negative is the infinitesimal generator of an analytic semigroup and which satisfies (7.17) can easily be obtained by applying Theorems 7.1.3 and A.7.6 to the above examples.
7.4
Limit Operators Related to the Imaginary Power
Throughout this section, A will be a non-negative operator on a Banach space X. This section is devoted to the study of the operators s - lim~_.0+ (A + s ) - ~ , s - lima__.0+ A - ~ + ~ (when A is positive) and s - lime__.0+ (A + s ) ~ . In particular, we study the relationship between the last two operators and the imaginary power A i~. We also include a perturbation theorem for operators which have bounded imaginary powers. P r o p o s i t i o n 7.4.1 Let a E C+. Then J - ~ = s - lim (A + c) -~ . e---*O+
182
(7.18)
Proof. Denote by T = s - lime__.0+ (A + ~)-~. Given r e D (T) it is evident that lim ~ (A + ~)-~ r - 0 e---*0
and therefore r E R (A), by Theorem 6.1.1. Moreover, lim A ~ (A + e) -~ r = r
e---*0
(7.19)
which implies, since A ~ is closed, that Tr E D (A ~) and A~'Tr = r On the other hand, as r e R (A), (A + e)-~' r e R (A) for all e > 0, and therefore Tr E R (A). Given r E D (A a) M R (A), lim (A + e) -~ A~r = r
e---*0
and hence A~r E D ( T ) a n d TA~r = r So, T is the inverse of the part of A ~ in the closed subspace R (A). Therefore T is closed. Let n > Re a be a positive integer and let r E D (Y-~). By the dominated convergence theorem, lim
~--*0
/0
,V~-1-~ (A + ,~ + e) -'~ r d,~ =
and so 4) E D (T) and Tr = J-'~r A (A + #)-1
r
/0
,V~-1-'~ (A + ,~)-'~ r d,~
Thus, by multiplying T and J - ~ by
with # > 0, (taking r E D (T) and r e D ( J - ~ ) , respec-
tively) and letting/~ ---, 0 we get (7.18). I From now on in this section we assume that ,4 is injective. If A is positive, then A -~+i', o~ > 0, is bounded and its limit (as c~ ~ 0) has a strong connection with A i~ and j i r as we will see later. Definition 7.4.1 Let A be a positive operator. We denote by Ai~ the strong limit s - lim A -~+~ cr
defined on the domain D(Ai,)={r
~--.0+limA - ~ + i ' r ex/sts}.
R e m a r k 7.4.1 As the range of A -~+i~ is included in D (A), the range of Ai~ is also included in D (A). Therefore, if A has non-dense domain, then Ai0 C I. In the following results related to Air we assume that 0 E p (A) and that the domain of A is not necessarily dense. L e m m a 7.4.2 If r C D (A) M D ( j 1 + i 4 ) , then
Ar c D (Ai~)
and
Ai~Ar = j l + i ~ = AAi~r 183
Proof. As 0 E p (A), D (A) C D (Air). Hence, if r E D (A) N D (jl+i~) , then, by Theorem 7.1.1, Aide e D (A)
and
AAi~r = jl+i~r
Let 0 < a < 1. By additivity A-~+i~Ar = Al-~+i~r = AI-~Ai~r = j l - ~ A i ~ r where the last equality is due to Corollary 5.1.12, since R (A -~+~) C D (A). If we take limits as a ~ 0 in this identity, then, by Theorem 3.1.6, part (i), we conclude that Ar E D (Air)
and
Ai~Ar = AAi~r = jl+i~r
m
R e m a r k 7.4.2 By Proposition 7.1.5, D (A) C D ( A ~ ) and A ~ r = J ~ r T h e o r e m 7.4.3 Air is closed. Moreover, J~: c A~ c A ~. Proof. Let {r
C D (Air) such that lira r
=r
and
lim AirCn = r
n---*O0
n-.-400
As (1 + A) -1 and A (1 + A) -1 are bounded, lim (1 + A) -1 r
= (1 + A) -1 r
n - - - * OO
and lim A (1 + A) -a Ai~r
= A (1 + A) -x r
n---~OO
Moreover, as (1 + A) -x commutes with Air, (1 + A) -x Ai~r
= ji~ (1 + A) -1Cn E D (A)
by Remark 7.4.2. Thus, A (1 + A) -x A~r
= A J ~ (1 + A) -1 r
E D (A),
since A (1 + A) -1Ai~r
= Ai~r
- ( 1 + A) - 1 A ~ r
Hence +
and by additivity (1 + A) -1 r
E D (jl+ir) 184
e D (A).
and A (1 + A) -t Airr
= AJ ir (1 + A) - t Cn = jl+ir (1 + A) - t Cn.
Letting n --~ oo
(I + A)-I c E D ( j 1+it)
and
j i + i r (1 + A) -1 r = A (1 + A) -1 r
From the preceding lemma it is deduced that A (1 + A) -1 r E D (Air), which yields r E D (Air) and Ai,r = ~. The inclusion jir c Air follows easily from the fact that Air is closed and from Remark 7.4.2. Finally, the inclusion Air C A ir is due to the relation
ji~
(I +
A) -1 r = Ai,
(I +
A) -1 r =
(I +
A) - t Aide
which is valid for all r E D (Air). m C o r o l l a r y 7.4.4 If D (Air) C D (A), then
Air = jir. In particular, this happens if A is densely defined or if JA ir is continuous. P r o o f . By Proposition 7.2.2 and taking into account that 0 E p (A),
Moreover, by the preceding theorem and by Remark 7.4.1 it follows that the set
{ r e D (Ai~) is included in D ( 7 ) . ~
n
D (A)" Aide e D (A) } = D (Air)
Hence ji~: is an extension of Air, which implies that
f
Air = jir. Obviously, this happens if D (A) = X. On the other hand, if j - i t is continuous, then D (j-i~') _- D (A) and as R(Air) C D (A), by additivity and by the preceding theorem, J-irAirr = r for all r E D (Air). Hence D (Air) C R ( j ' i §
C D (A).
I The identity Air = yir also holds if we remove the hypothesis of density of D (A) and even if j - i t is not continuous. To see this it is sufficient to consider in l ~176 (C) • co (C) the operator A (r r
- (At r A2r 185
where A1r = (r162
3r
...) in l ~ (C)
and A2r = (r
Cx + r
r
+ ~2 + r
".-) in co (C).
The domain of A is not dense since D (A1) is not dense, and JA ir is t i n u o u s since j - J r is not continuous. However, A2 (A1)ir
--
j iA1 r
and
1-tot c o n -
(A2)ir - j iA2 r
since JA~ r is continuous and A2 has dense domain. So, Air = j~r. R e m a r k 7.4.3 Let us consider the operator A (r
r
r
...) = (r162162
...) in l ~ (C).
Since j - i t is continuous, Air = Jji§ Therefore A it, which is bounded, is a proper extension of Air. In the following results we deal with the strong limit of (A + 6) ir as 6 ~ 0. P r o p o s i t i o n 7.4.5 The domain of (A + 6)ir does not depend on 6 > O. If 0 E p (A), then this domain coincides with D (Air). P r o o f . Fix 6, 6' positive and # > max {6, 6'}. By (7.4), J~+~ (~ + A) -1 r i sin.h (lr'r) 71"
Jy+~, (# + A) -1 r
[
]
Air (A + 6 + A) -1 - (A + 6' + A) -1 (# + A) -1 r dA
for all r E X. Taking into account the resolvent formula and the fact that (6 + A) -1 and (e' + A) -1 are bounded, ~ (# + A) -1 r JA+~
J~+~, (# + A) -1 r e D (A)
If 0 E p(A), then the same reasoning is valid for 6' = 0, which yields
In"
L"
/
J
R e m a r k 7.4.4 I f 0 ~ p ( A ) , then it is not true that D (A it) - D [(A q-6)ir] . To see this it is sufficient to consider in co (C) the operator 1 - S, where S is the shift operator. The power (1 - S) ir is unbounded. ~However, (1 - S + 6), 6 > 0, is bounded and 0 E p (1 - S + 6). Consequently, (1 - S + 6) ir is bounded. 186
Definition 7.4.2 We denote by Sir the strong limit s-
lim (A + e) ir e--.~O+
with domain
T h e o r e m 7.4.6 Let A be an injective non-negative operator. Then: (i) Sir is closable and
3~
C
Sir
C
Ai~.
In particular, if A has dense domain and range, then Sir = A it.
(ii) If 0 E p (A) , then Sir is closed and Sir = A it. Proof. Since A ir is closed and j~r is closable, to prove (i) it is sufficient to show that j i r C Sir c A ir . Fix r = AT E D (A) M R ( A ) . By (7.4) and by writing f o = f : + f6~1766 = e 1/2, we have
II(
(2 (M + 1)Ilvll + M2 I1r fll
II
%.|!
I!
a/~
(7.20)
where M - s u p ~ I I A ( A + A ) - l l ] ' A > 0~. Letting e --, 0, r E D(Sir) and
&~r
= j~r
Let us prove that A ir is an extension of Sir. Given r E D (S/r), A(I+A) -2Sire
=
lim A ( I + A ) - 2 ( A + C ) i r e ~--.0+
=
lim j i r e~0+
=
A(I+A) -2r
A+e~x
JirA (1 + A) -2 r
Hence r E D (A it) and Airr = Sire. Finally, statement (ii) follows easily from the fact that (7.20) remains valid for 0 E p (A) and r e D (Air). m In Theorem 7.4.6, part (ii), we have proved that if 0 E p (A), then s-
lira (~ + A) ~ = A i~.
e---*0
This result also holds if it is only assumed that R (A) is dense, but we have to assume that A ir is bounded. P r o p o s i t i o n 7.4.7 Let A be non-negative with dense range. Suppose that A ir is bounded. Then lim (~ + A) ir r = Airr e---,O+
187
for all C E X.
Proof. As A i~" is bounded, by Corollary 7.1.2, we know that +
Then, (A + e) i~" is bounded, since by Theorem 7.1.1, (A + ~)i~- = (A + e) 89
(A + e) -'~ .
Moreover, given r E X, (A + e)i~ r
(A + e)89
_
A89
(A + ~)-89r 1
By Proposition 5.1.14, there is a positive constant C1 such that <_
~ and, as the range of A is dense, e--.olimr189(A + r189 r = 0 and
e--.olim[A (A + e)-'] 89r = r
by Theorem 6.1.1. m
7.5
Negative and Imaginary Powers on Locally Convex Spaces
All the results (with exception the Proposition 7.1.5) in Sections 7.1 and 7.2 continue to be for sequentially complete 1.c.s. However, we have to make the following remarks: 9 The inequality (7.5) will take the following form: If { [[lip- p E P } is a directed family of seminorms which describes the topology of X, given p E :P, there is an index s E P such that
where C only depends on the base operator A. 9 In Corollary 7.1.2 we have to have A i~" as a continuous operator defined on X. 188
9 Estimates 7.12 and 7.13 will take the following form: If a E C+ and n > Rec~ is a positive integer and r = A'~U E R(A'~), given p E :P, there is an index s E P such that I[J-~r
_ C(n - c~, n, M)I1r
~eo
Re
II lls
if ~- E II~ and r - A~D E D(A m-'~) MR(A~), with m > n positive integers, then r Ji'_,,p _< C(n + iT, m,M)1177111-~s IlAm-~r '~'~ . [I The index s E :P only depends on the operator A and on the index p. 9 In Theorem 7.2.3 we have to add the hypothesis that the space X ~endowed with the strong topology will be sequentially complete.
7.6
Notes on Chapter 7
The first papers about the imaginary powers of operators are due to Kober [107], Kalish [100], M. J. Fisher [70] and E. R. Love [130] who studied the pure imaginary powers of the Riemann-LiouviUe derivative operator on L p with 1 < p < c~ and proved that they were bounded operators. It is known that the part in L p (1 < p < c~) of certain linear differential operators have bounded imaginary powers ([8], [60], [61], [62], [75, Th. 1], [76, Wh. A], [77, Appendix A] and [165]). It is not known if the part in L p (1 < p < c~) of every sectorial differential operator admits bounded imaginary powers. M. Hieber in [90] constructs, given any 0 e [0, ~[, an example of a positive sectorial pseudo-differential operator A, with spectral angle 8 and such that does not have bounded imaginary powers. The imaginary powers of non-negative operators can be defined from the McIntosh functional calculus for sectorial operators (developed by A. McIntosh for Hilbert spaces in [150] and by M. Cowling, I. Doust, A. McIntosh and A. Yagi for Banach spaces in [39, Section 2]). However, this could not be done in locally convex spaces where there are non-negative operators which are not sectorial. Example 7.3.3 was given by H. Komatsu in [109, Section 14, Examples 6 and 8] and Example 7.3.4 is due to A. Venni and is found in [193]. Proposition 7.4.7 has been proved by J. P r o s and H. Sohr in [164, Th. 3], but with the additional conditions of the operator A being densely defined and of the class B I P ( X , OA). In [10, Th. A] J. P. Baillon and P. Clement give an example of a sectorial operator in a Hilbert space with spectral angle 0 and such that A is are not bounded operators for all s E IR\ {0}. In [152, Th. 4] A. McIntosh and A. Yagi give a general method for constructing positive operators on a Hilbert space which do not have bounded imaginary powers, by using the fact that this property is equivalent, by Theorem 10.2.6, to the non-existence of an 7-/~bounded functional calculus for the operator A. 189