Functions of Linear Operators

Appendix A

Functions of Linear Operators Here we collect some facts of operator theory which are in use in the course of our main considerations. As therein, we let :~ be a complex Banach space and accept all the attendant notation introduced in the book.

A.1

A n a l y t i c continuation of the resolvent

In this section, given a linear operator s we denote by p(s the resolvent set of ~. It is well known that if s is a linear operator and if ~ E p(s then p(s contains as well some open disk centered at A (cf. Hille & Phillips [84, Theorem 5.8.2]), which can be shown by analytic continuation. As a consequence of this fact, we obtain that p(~) is, in fact, an open set. For our purposes in the main parts of the book, we need, however, to have at our disposal some quantative estimates accompanying the procedure of analytic continuation. T h e o r e m A.1. Let s be a linear operator and let # C p(2~), with

II(,~- ~)--llt <_ M. Then, for any )~ C C such that ! ~ - tt I = r < M, there exists an operatot | )~) E B(2~) which is permutable with the resolvent of s obeys the equality ( ~ I - ~)-1 = ( ~ I - ~ ) - 1 ~ ( ~ , ~), and satisfies the estimate 1

ll~(~ ~)11 < ' - 1-rM" 255

256

A P P E N D I X A. F U N C T I O N S OF L I N E A R O P E R A T O R S

As a consequence, it holds

)-111

ll( Xx -

M <

-1-rM

Proof. The claim follows when inserting dn

d# n

(/.tI- ~)-1 = ( - 1 ) n n ! ( # I - e)-(n+ 1)

into the Taylor expansion of the resolvent

(~,]'_ ~)-1

oo

n=o

dttn ( # I - ,~)-1

n!

with the aid of a standard argument (cf. Hille & Phillips [84, Section 5.8]). D Now the last result is used for analytic continuation of the resolvent of a sectoriM operator.

1 T h e o r e m A.2. Let s E 8 ( M ; qa, 0) for some qa E (0, 7r/2) and M > sinqa and let 991 be an arbitrary number such that qa- arcsin ~ < qal < qa. Then for each A E E~o \ Int E~I there exists an operator | E B(2~) which is permutable with the resolvent of,g, obeys the equality

(AI - ~)-1 _ (cos(~o - ~1)I - s and satisfies the estimate

IlO( )ll <_ 1 -

M sin(qa - ~1)"

As a consequence, it holds, for all )~ ~ Int

I](AI- s

Eqol,

M ___ i - M s i n ( ~ - ~1)

Proof. Let A E C be arbitrary with arg A = (ill. Then, for # - cos(v-v~)e~, we have [ # - A [ - [#] s i n ( ~ - ~1). Using this result and noting that

[[(#Z- s

< M[#[-1,

A.1.

A N A L Y T I C C O N T I N U A T I O N OF T H E R E S O L V E N T

257

since 1~ E S ( M ; qo, 0) and arg # = qo, by Theorem A.1 we therefore find, with some ~(A) E B(Js for all A with arg A = 991, that

( ~ I - ,~)-1 __ ( # I - ~)-1~(~), where

1 II~(A)I I <

1 - M sin(~o - ~Ol)"

Now, in view of the inequality It*{ > IAI, the claim follows at least for all with arg A = g)l and, by symmetry, for all A with arg A = -t-~Ol. It is however clear, using the above reasons, that the same assertion remains valid for all ~ Int E~I as well. El Now we consider some applications of the above results. L e m m a A.1. Suppose that s E ,5(~o, a) with some ~o E (0, 7r/2) and a E IR. Then there exist a qOl E (0, qO) and an R > 0 such that for all )~ ~ Int(E~l U

Z)(R)), II(xI - ~)-1 {i <__ c{~1-1. In the case a >_ O, (A. 1) holds with R -

(A.1)

O.

Proof. By Theorem A.2 there exists a ~o2 E (0, ~o) such that

)-111 <_ via -

1-1,

if X - a ~ Int E~2. Selecting further any fixed ~01 E (~02, q0), we can take R >__0 sufficiently large to be confident that (A.1) holds for ,~ ~ Int ( E ~ U 79(R)). The more precise wording in the case a > 0 is obvious. [-1 L e m m a A.2. Let s E 8(qo, a) and let a seminorm {. l form a ({lm{qo, a)concordant pair with P~, for some qO E (0, r / 2 ) , a E IR, ~ > 0, and m E N such that ~ < m. Then there exist a r E (0, qo) and an R >_ 0 such that for all A ~ Int(E~l tO 2)(R)) and u E ~,

{(;XI- s

I <__CI)~l~-ml{ul{.

In the case a > 0 the above holds with R -

O.

Proof. The claim is established in a manner similar to that used in the proof of L e m m a A.1. In fact, it suffices to mention that it follows from Theorem A.2 that with some 992 E (0, qo), for a r g ( A - a) -- q02, a r g ( # - a) -- qo, and (AI- s

= ( > I - s174

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A P P E N D I X A. FUNCTIONS OF L I N E A R O P E R A T O R S

where

ILe(a)ll < c. El

A.2

Functions of bounded

and sectorial operators

We now discuss some means to define functions of linear operators; using such functions is an essential ingredient of our techniques in the present book. For our purposes we employ a simplified notion of contour which differs slightly from what is usually accepted in the theory of functions of a complex variable. More precisely, by a contour we mean a simple piecewise smooth curve in the complex plane, which either is of finite length and closed or passes through oc. It is accepted in the book that a contour is always oriented counter-clockwise or downwards if it is finite or infinite, respectively, unless it is specified differently. Instead of contour sometimes we say on equal terms integration path. Let first s E B(E) and let ~ be a finite contour such that the open set A, of which S is the boundary, contains the spectrum of s Let further f(A) be a complex function that is holomorphic in a neighbourhood 1 of A. Then the operator f ( ~ ) given by f(s

1 L f(s

- 2~ri

( ) ~ I - J~)-ld/~

(A.2)

is defined and bounded on X (see, e.g., Hille & Phillips [84, Chapter V]). Furthermore, the following implications hold, for a l , 0/2 E C, f(A) - 1 ~ f ( ~ ) :

I,

f(A) = A =~ f ( ~ ) :

~,

g(/~) : O~lfl (/~) -!- o~2f2()~) ==~ g(~) : oLlfl (,~) -t- 012f2(~),

g(A) :/i(A)/2(A)

(A.3) (A.4) (A.5) (A.6)

Apart from these properties, there exist some other premises allowing us to think of the operator f ( s given by (A.2) as a function of s The above representation (A.2) is often called the Dunford operator calculus formula. Next we consider possibilities of defining functions of closed unbounded operators. Actually, for our purposes it suffices to deal with sectorial operators. 1By a neighbourhood of a set g C C we mean any open set containing CI G.

A.2. BOUNDED A N D S E C T O R I A L O P E R A T O R S

259

Let now s E $(qa, a) for some ~ E (0, 7r/2) and a E R. Suppose further that a complex function f(A) satisfies the following conditions, for any fixed C (0, ~), with some (r > 0 depending possibly on r f(A) is holomorphic in a neighbourhood of Er + a,

(A.7)

there exists a finite limit f(oc) = lim Ixl-~, f(A),

(A.8)

f(

)l < c ( 1 +

e

+

o.

(A.9)

By Lemma A.1 there exist a ~1 E (0, ~) and an R >__0 such that (A.1) holds. Then, letting, for any 5 > 0, ~ be the contour given by ~ = 0 ( E ~ U7:)(5)), owing to (A.1) and to the accepted conditions on f(A) the expression

f (2.) - f (oc)I + ~ 1 ~= R+, (f(A) - f ( o o ) ) ( ~ I - ~)-ldA

(A.10)

makes sense and defines a linear bounded operator on ~ for any fixed e k 0 subject to R + e > 0. It is easy to show, using Cauchy's Theorem (see, e.g., Uille & Phillips [84, Theorem 3.11.1]), that the definition of f ( s does not depend on ~. Moreover, by Cauchy's Theorem there exist other deformations of the integration path in (A.10), which are tolerable as far as it is regulated by the accepted conditions on s and f(A). Further, by standard means (cf. Taylor [160] or Hille & Phillips [84, Chapter V]) it is established that the properties (A.3), (A.5), and (A.6) are still in force while (A.4) is now absurd for the reason that formula (A.10) involves bounded functions only. In principle, following Taylor [160] one could show that formula (A.10) possesses all the features that are intrinsic in an operator calculus, dealing with the class of functions f(A) with the above stated properties ( A . 7 ) - - ( A . 9 ) . Fortunately, there is no necessity in showing this fact because, for the aims of this book, it suffices to treat the algebra of functions which is constituted by e -A and all suitable rational functions. At the same time, if f(A) is a rational function, using Cauchy's Theorem yields that formula (A.10) defines f(.!1) just the same as Taylor's operator calculus formula (see [84, Formula (5.11.2)]), while in the case f(A) = e -A, applying (A.10) defines the operator e -n in conformity with semigroup theory (see, e.g., Reed & Simon [133, Section X.8]). Nevertheless, we remark, following [160], that if f(A) satisfies ( A . 7 ) - - ( i . 9 ) and if with some m E N, there exists a finite limit g(oc) -

lim (f(A)Am),

(A.11)

I~1-,cr ),El~r

for any fixed r E (0, ~), it then holds, with g(A) - f(,k)A m, for u e D o m s m, g(s

= f(s163

(A.12)

APPENDIX A. FUNCTIONS OF LINEAR OPERATORS

260

where f ( ~ ) and g(s are defined by (A.10). The above reasoning can be extended to the case where/~ E 8 ( M ; ~, a), M _ 1, ~ E (0,~r/2), a E ]R (see Remark 6.4). Since, according to our assumptions, the constant M varies considerably afterwards, we cannot now apply Lemma A.1 in order to obtain (A.10). At the same time, if we assume in addition to the conditions ( A . 7 ) - - ( A . 9 ) that s E S ( M ; ~ I , a ) with some ~1 E (0,~) and with the same ~ E (0,7r/2) and a E IR as above, then formula (A.10) remains valid with R > m a x { - a , 0} even if the size of M varies considerably; however, the influence of M clearly appears in any estimates involving the norms of functions of the operator ~. Note also that under the new assumption on/~ the additional condition (A.11) implies, as above, (A. 12). In the next section we consider also other functions of sectorial operators.

A.3

Fractional powers of operators

Fractional powers are usually defined for sectorial operators, however, if the operator in question is unbounded, one then meets certain technical difficulties (see, e.g., Komatsu [93]). At the same time, for the aims of this book, it suffices to treat the positive fractional powers of bounded sectorial operators and certain negative fractional powers of sectorial operators possessing a bounded inverse, which facilitates our consideration. Let first s E S(~, 0)M B(:~) for some ~ E (0, 7r/2). In principle, there are several ways equivalent to each other that define the fractional powers /~e, ~ > 0. For the sake of being definite, we take the definition suggested by Salakrisnan [45] (cf. also g o m a t s u [93]), avoiding the difficulties that are proper for unbounded operators. More precisely, given ~ E ( m - 1, m) for m E N, the operator s is introduced by

s = 2nil ~= )~e_ms

I_s

A,

(A. 13)

where .=. = 0E~, while s is defined in the standard way if ~ is an integer. It then follows from [45, 93] that 2 s163 = s

for ~, 77 > 0.

(A.14)

It is significant that since s is bounded, by Cauchy's Theorem the contour -= in (A.13) can be replaced by a suitable closed (that is, finite) contour, 2We take into account that s E B(:~).

A.3. F R A C T I O N A L P O W E R S OF OPERATORS

261

for instance, F. can be replaced by ~L -- O(E~nV(L)) with L > 0 sufficiently large. Now let s E $(~, 0) for some ~ E (0, 7r/2) and let s E B(:~). Then, given ~ E (0, 1), the operator s is defined as follows (cf. Krasnosel'skii & Sobolevskii [95], Krein [96, Chapter I], or Komatsu [93]) s 1 6 2= 2~ri1f~ A_r

_ j~)_ldA,

(A.15)

where the contour F~ is defined the same as above. Also, it is proved in a standard way (see, e.g., Krein [96, Chapter I, Theorem 5.1] 3) that s163

= s162

for ~, r / > 0 such that ~ + 77 < 1.

(A.16)

In principle, s is defined for any ~ >_ 0 and the property (A.16) is then valid for any ~, r/_> 0, but we do not deal with such an extension. It is clear that the integral representations (A.13) and (A.15) are related to the operator calculus formulas (A.2) and (A.10). 4 Moreover, it appears that fractional powers are included in the algebra of functions of a bounded sectorial operator, as stated in the following result. T h e o r e m A.3. Let 2. E S(~,O)M B(X) for ~ E (0,1r/2), let L > 0 be sufficiently large so that the spectrum of 2, belongs to the set AL -- Int (E~ M 7)(L)) U {0}, and let f(A) be holomorphic in a neighbourhood of AL. Then it holds, with ~ L --- O A L , for any fixed ~ > 0, 5

s163

27ri

L

A~f(A) ( A I - s

(A.17)

In fact, this is established with the aid of the representations (A.2) and (A.13) by using a standard argument (see, e.g., Krein [96, Chapter I, Section

5]) We state also the so-called convexity inequality. 3It is assumed in [96] that the operator in question has dense domain. This assumption is not really needed for showing (A.16). 4The difference is that the integrand in (A.13) and (A.15) is unbounded (at 0), as a function of A. It is easily seen that the corresponding integrals nevertheless converge under the accepted conditions. 5In principle, "~'L may be replaced by another closed contour, among those which are obtained by a deformation of EL within the bounds of possibility, as regulated by Cauchy's Theorem.

A P P E N D I X A. FUNCTIONS OF L I N E A R O P E R A T O R S

262

T h e o r e m A.4. Let 2. e S(~, O)N B(X) for some ~ e (0, 7r/2). Then, for any fixed ~ and 77 such that 0 < ~ < ~7,

I1s162 < cII,C'ullr

1-r

.for

all u e X.

(A.18)

sl ~nstead ol s ~ #~(~) it is assumed that s ~ #3(:~), the last estimate holds with 2.-~ and .?..-'1 in place of ~ and 2.~, respectively, where ~ + ~] <_ 1. For a proof, see, e.g., Krein [96, Chapter I, Theorem 5.2].

A.4

F u n c t i o n s of o r d e r e d n o n - c o m m u t a t i v e ators

oper-

To meet our needs it suffices to discuss the subject for bounded operators only. Let therefore s163 E B(:~) and let =-1 and .=.2 be closed contours surrounding the spectrum of s and that of s respectively. Let further f(A2, ~1) be a holomorphic function on a connected set in C 2 containing E2 x ~1 in its interior. By the Dunford operator calculus formula (A.2), one can define the operator f(~2, A) =

1 L f(A2, A) ( ~ 2 I - ~2)-1dA2. 21ri 2

It is readily ascertained that f(~2, A) is a (B(E))-valued holomorphic function of ~ on a connected set containing ~1 in its interior. This fact further yields that the expression

[[f(~2, ~1)]]1 ._ 27ril~ 1 f(~2' )~)(/~I- ,~l)-ld/~

(A.19)

makes sense and defines a linear bounded operator on ~.

2 1

D e f i n i t i o n A.1. The operator [[f(~2,s tion of ordered operators.

given by (A.19) is called a func-

In fact, the above notation arises from Maslov [111]" the indices over operators show the order in which they act while autonomous brackets [-~ are employed to restrict the region where this symbolism is valid. It is easily checked that if, with 7:)j C C, j - 1, 2, some open disks centered at 0 and containing Ej, j - 1, 2, respectively, f(A2, ~1) is holomorphic in the bidisk 7:)2 x 7:)1 so that OO

f(~2,~1)--

E ajl,J~-~{l~J22' j~ ,j2=0

A.4. FUNCTIONS OF ORDERED OPERATORS

263

then the expression (X)

ajl,j2 s E jl ,j2 =0

s 1

defines a linear bounded operator on 2C which coincides with that given by (A.19).