Real sectorial operators Текст научной статьи по специальности «Математика»

MSC 47B44, 47A60, 47F05

DOI: 10.14529/mmp 170106

REAL SECTORIAL OPERATORS

A. Yagi, Department of Applied Physics, Osaka University, Suita, Osaka, Japan, [email protected]

Sectorial operators that act in complex Banach spaces and map real subspaces into themselves should be called real sectorial operators. These operators have already been used implicitly in the study of various diffusion equations. Meanwhile, in the Lojasiewicz -Simon theory which provides longtime convergence of solutions to stationary solutions, the real valued Lyapunov functions play an important role. In order to make general methods for studying longtime convergence problems on the basis of the Lojasiewicz -Simon theory, it may therefore be meaningful to give an explicit definition for these real sectorial operators and to show their basic properties that are inherited from those of complex sectorial operators.

Keywords: sectorial operators; fractional powers of operators; differential operators.

Dedicated, to Professor Angelo Favini on the occasion of his 70th birthday.

Introduction

When we want to study diffusion equations, there is an advantage of handling them in complex valued function spaces than in real valued function spaces, because the second order elliptic operators often generate an analytic (holomorphic) semigroup in a suitable complex Banach space. Using the techniques of functional analysis including such analytic semigroups, construction of local or global solutions can easily be carried out, see Krein [1], Tanabe [2], Favini, Yagi [3] and Yagi [4]. In the meantime, unknown functions of diffusion equations often denote densities or concentrations of some physical objects or chemical substances and they are real valued. Thereby, only real parts of unknown functions are meaningful in applications.

Fortunately, when diffusion equations are well posed, the solutions are always real if their initial functions are real. This fact means that one can construct desired real solutions in the framework of complex function spaces. Even more, as seen by [5, 6] for example, one can use the Lojasiewicz - Simon theory in order to prove longtime convergence of real solutions to stationary solutions. Furthermore, the arguments in [5, 6] suggest that one could make general methods for studying the longtime convergence problems in a unified way for various diffusion equations employing the Lojasiewicz - Simon theory. To the ends, however, we have to begin by constructing firm frameworks.

In the Lojasiewicz - Simon theory, the gradient inequality (see Chill [7] and Haraux, Jendoubi [8]) for the Frechet derivatives of real valued Lyapunov functions plays a crucial role. So, everything must be set in the sense of "real", namely, real Banach spaces, real sectorial operators, real analytic semigroups, interpolations of real Banach spaces by the complex methods, and so on. As a matter of fact, these things have already been used implicitly in the various complex settings. The objectives of this Note are then to introduce an explicit definition of real sectorial operators acting in the real subspaces and

to show their basic properties which are reasonably analogous to those of complex sectorial operators.

When a densely defined, closed linear operator acting in a complex Banach space has its spectrum contained in a sectorial complex domain and satisfies an optimal decay estimate of resolvent, the operator is called a sectorial operator (Brezis [9] and Yosida [10]). This notion has naturally been defined only for complex linear operators. Meanwhile, when an underlying complex Banach space X admits a conjugation f M f and is decomposed into XX = X + iX, where X is a real Banach subspace of XX, and when a sectorial operator A of XX maps D(A) fi X into X, we call its part A\X restricted in X a real sectorial operator induced from A. It is verified that A|X inherits basic properties from A.

1. Complex Banach Spaces with Conjugation

XX

complex Banach space with norm || • ||. Assume that XX is equipped with a correspondence f M f satisfying:

f + g = f + g for f, g e XX, (l)

af = af for a e C, f e X, (2)

f = f forf e X ^ (3)

_ for f e X. (4)

XX

and onto. In particular, 0 = 0. The vector f is called the conjugate vector of f. Such a

XX

For f e XX, we put

f+f f-J

Ref = U^ and Imf = (5)

Then, it is clear that f = Ref + ilmf. The vector Ref (resp. Imf) is called the real part (resp. imaginary part) of f e XX. The vectors satisfying Imf = 0 or equivalently f = f are called a real vector. By (1), (2) and (3), both Ref and Imf are a real vector. As noticed, 0 is also a real vector. On the other hand, the vectors satisfying Ref = 0 or equivalently f = —f are called a purely imaginary vector. Obviously, iImf is a purely imaginary vector. It holds that

Ref + iImf = Ref — iImf for f e X. (6)

We want to consider the space

X = {f e X; Imf = 0,i.e., J = f}.

By (1) and (2), X is a real vector space equipped with the norm || • ||. We see the following fact.

Theorem 1. X is a closed subset of XX and is a real Banach space.

Proof. By definition, X = Im_10. Meanwhile, f M Imf is continuous; therefore, X is a closed subset.

математическое моделирование

Since X is a real normed spac^ it suffices to verify its completeness. But, as X is a closed subset of a complete space X, X is naturally complete.

□

We call X the real Banach subspace induced from XX. Thereby, we have a decomposition of any f E XX into the form

f = Ref + iImf with Ref, Imf E X. (7)

Indeed, such a decomposition is unique.

Theorem 2. For any f E X, f can be written as f = fi + if2, fk E X (k = 1, 2) in a unique way. The correspondence f M fk is continuous from X onto X for k = 1, 2. In this sense, XX = X + iX.

Proof. Since (7) gives such a decomposition, it is sufficient to prove the uniqueness. Let f = fi + if2 = gi + ig2 with fk, gk E X for k = 1,2. Then, (fi — gi) + if — g2) = 0; at the same time, considering this conjugate, we have (fi — gi) — i(f2 — g2) = 0. Therefore,

fi = gi f2 = g2

By (4), we see that

\\f || < 11Ref|| + 11Imf\\< 2max{|Re f ||, ||Im f ||} < 2|f\\, f E X. (8)

This readily yields that f M Ref and f M Imf are continuous from XX onto X.

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Corollary 1. When XX is a complex Hilbert space with inner product (•, •), its real Banach X

Proof. Since

4(f, g) = \\f + g\\2 — \\f — g\\2 + i\\f + ig\\2 — i\\f — ig\\2,

it follows that

4(fg) = \\f + g\\2 — \\f — g\\2 — i\\f + ig\\2 + i\\f — ig\\2.

In view of (4), we observe that

fg) = (f, g) for f,g e xX. (9)

f, g E X (f, g) = (f, g) (• , • ) X

provides a Hilbert structure.

□

2. Real Sectorial Operators

XX

with norm \\ • \\. Assume that X is equipped with a conjugation f M f and let X be its real Banach subspace.

Let A : D(A) ^ XX be a linear operator of XX with domain D(A) C XX. Assume that A satisfies the conditions:

u G D(A) is equivalent to Reu, Imu G D(A), (10)

Re Au = A (Re u) and Im Au = A(Im u) for u G D(A). (11)

These conditions can be described in terms of conjugate.

Proposition 1. In order that a linear operator A : D(A) ^ XX satisfies (10) and (11), it

A

u G D(A) if and only if u G D(A), (12)

Au = Au for u G D(A). (13)

Proof. Let A satisfy (10) and (11). Then, u G D (A) if and only if Re u, Im u G D(A); and these are obviously equivalent to u G D(A). Moreover, by (6), it h olds for u G D(A) that

Au = ReAu + iImAu = Re Au — iImAu, Au = A(Re u — iIm u) = A(Re u) — iA(Im u).

Hence, (11) implies (13).

Conversely, let A satisfy (12) and (13). Then, u e D (A) implies u,u e D(A); then, (5) shows that Reu, Imu e D(A); hence, (10) is verified. Moreover, under (13),

Re Au = (Au + Au)/2 = (Au + Au)/2 = A (Re u), ImAu = (Au — Au)/2i = (Au — Au)/2i = A(Im u). Hence, (11) is verified.

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We thus observed that (10) and (11) imply that D(A) = [D(A) f X] + i[D(A) f X] and that A maps D(A) f X into X. ^n ^rase, we call A a real linear operator of XX.

AX

which is defined by

|D(A|X) = D(A) f X, 1 Ax u = Au.

By (11), we then have

Au = A\x (Re u) + iA\x (Im u) for u e D(A). Theorem 3. Let A : D(A) m X be a densely defined, closed linear operator of XX satisfying

A| X X

X

Proof. First, let us prove density of D(A|X) in X. For any f e X, there exists a sequence un e D(A) such that un converges to f in XX. Then, Re un e D(A|X^d Re un M Ref = f in XX and of course in X Hence, D(A|X) is ^rase in X.

Second, let us prove closedness of A^. sequences un e D(A|X^d fn = Aun

such that, as n M un M u and fn M f in X. By the closedness of A, u e D(A) and f = A^^s X is ^teed in XX (due to Theorem 1), u must be in X; thereby, u e D(A|X). Consequently, f = Au = A\Xu.

X □ XX

closed linear operator A is said to be a sectorial operator of XX if its spectr um a (A) is contained in a sectorial domain

a(A) c E = {A E C; | argA\ <u}, 0 < u < n, (15)

and its resolvent satisfies the estimate

M

\\(A — A) \\ < — forAE E, (16)

with some constant M > 1.

A XX A

XX

theorems.

A XX

A E p(A) if and only if A E p(A); (17)

(A — A)-if = (A — A)-lf for A E p(A), f E X. (18)

In particular, when A E p(A) is real, (A — A)-1 is a real operator and A belongs to the resolvent set p(A\X) of the part A\x.

Proof. From (13), the relation (A — A)u = f for u E D(A) and f E XX is equivalent to (A — A)u = f. This then shows that (17) holds true.

As seen, we have (A — A)u = f and (A — A)u = f for A, A E p(A). Thereby, u = (A — A)-if and u = (A — A)-if. Hence, (18) is also shown.

When A E p(A) is real, (18) means that (A — A)-1 satisfies (13). Hence, (A — A)-1 is a real operator.

~ 0 □

Theorem 5. Let A be a real sectorial operator of X. Let, for 0 < 9 < ro, A0 be its fractional powers. Then, for every exponent 9, A0 is also a real operator.

Proof. The spectrum condition (15) implicitly means that 0 E a(A). So, there exists 8 > 0 such that {A E C; |A| < 8} c p(A). We then introduce an integral contour r = r-ur0ur+ such that r±: A = re±Mi, 8 < r < m and r0 = 8ee%, —u < 9 < u. Its orientation is from to 8e^\ from 8eM to 8e-u\ and from 8e-M to me-Ui. By definition, A-d is given by the integral

A-' f = n L A-0(A — A)-ifdA for f E X. Taking the conjugate of each hand side, we obtain by (18) that

Af = — J A- (A — A)-i~fd\.

Here, A-0 = e-0(log'A'-iarg= (A)-0. And, as A varies on r in the positive sense, A varies on the same contour r in the negative sense. It therefore follows that

Af = 2ti L A-' (A — A)-lfdA = A-0f.

This means that A 0 satisfies (13). Thanks to Proposition 1, A 0 is a real operator (note

that, as A-0 is a bounded operator, (12) is automatically satisfied).

It is now easy to see that A0 is real. Indeed, if u E D(A0), then there exists f E XX for

which u = A-0f holds; therefore, u = A-0^d u E D(A0); furthermore, A0u = A0u. It

is clear that u E D(A0) conversely implies u = u E D(A0). Hence, Proposition 1 is again A0

□

Consider a real sectorial operator A of XX. As observed by Theorem 3, its part A\X in X is a densely defined, closed operator acting in X. Then, we can give a definition of fractional powers for A\X. In fact, noting that A0 is an operator from D(A0) if X into X, we set

[A|x ]0 = [A0 ] |x for any 0 <9 < to,

with the domain

D([A|x]0) = D(A0) f X. (19)

Then,

A0u = [A|x]0(Reu) + i[A\x]0(Imu) for u E D(A0).

Theorem 6. Let A be a real sectorial operator of X with angle u < 2. Then, for the analytic semigroup e-tA (0 < t < to) generated by —A e-tA a rea^ operator for any 0 <t < to.

Proof. Let r be a similar integral contour used in the proof of Theorem 5. As well known, for 0 < t < to the semigroup e-tA is given by

e-tAf = 2- lr e-tx(A — A)-1 fdX, f E X.

Taking the conjugate of each hand side, we obtain by the similar arguments as in the proof of Theorem 5 that

e^A =2^ L A-0 (A — A)-1fdA = e-tAf.

This means that e-tA satisfies (13) of Proposition 1 and is a real operator of XX. Consequently, e-tA is a real bounded operator acting on X.

□

Under the assumptions of Theorem 6, we define a semigroup on X generated by —A\X by the formula

e-tA i * = [e-tA]ix for 0 <t< to.

Then,

e-tAf = e-tAi* (Ref) + ie-tAi* (Imf) for f E X.

Moreover,

e-tAi* . e-sAi* = e-(t+s)Ai* on X.

3. Interpolation and Real Subspaces

Let Z and X b^two complex Banach spaces such that X C XX densely and continuously. We assume that XX has a conjugation f M f on it. In addition, we assume that this

zx

u E x if and only if u E x, (20)

\\u\\~ = \\u\\z for u E Z. (21)

For 0 < 9 < 1, let [XX, x]0 denote the complex interpolation space ( [11]). This space can also be decomposed into a sum of real part and imaginary part as in Theorem 2.

Theorem 7. For any 0 < 9 < 1,

u E [X, x]0 if and only if u E [X, x]0, (22)

MX,z]e = \\u\\{X,z]e for u E [X,Z]0. (23)

Proof. Let u E [X, Z]0. By definition, there exists a holomorphie function $(z) defined in the band domain G = {z E C; 0 < Re z < 1} with values in X, which is continuous and bounded on the closed domain G, which takes its values on the straight line 1 = {z = 1 + iy; —to < y < to} in Z with \\$(z)\\z < to, and which takes a

value $(9) = u at the point z = 9. Then, the function ^(z) = $(z) also possesses similar properties but ^(9) = u. This then means that u also belongs to [X,Z]0. Conversely, if u E [X, Z]0, then u = u E [X, Z}0. Hence, (20) holds true. We remember that

\\u\x zh =inf { sup \$(iy)\x + sup \$(1 + iy)\\z; $(z) is

iy&R 1+iyee

G }.

Then, (21) also follows immediately from this definition.

□

This theorem means that, when XX has a conjugation f M f which is consistent with

zx

interpolation space [XX, Z]0, too. We can then apply Theorems 1 and 2 to [XX,Z]0. Let XX = X + iX and Z = Z + iZ be the decompositions for XX and Z respectively, into real part and imaginary part. We naturally define

[X, Z]0 = [X, Z]0 f X for any 0 < 9 < 1. (24)

Then, it holds true that

[X, Z]0 = [X, Z]0 + i[X, Z]0 for any 0 < 9 < 1.

4. Triplet and Real Subspaces

In this section, we want to consider a triplet of complex spaces Z c XX c Z* ( [12]). Here, X is a complex Hilbert space, its inner product and norm being denoted by (•, •) and \ • \, respectively. The space Z is a complex reflexive Banach space, its norm being denoted

\ • \ ZZ XZ ZZ *

a complex adjoint space of Z with norm \\ • ||*. There is a scaler product (•, •) between Z and Z* which is sesquilinear and satisfies

\\u\\ = sup \(u,$)\ for u E Z, (25)

IMI*<i

\\$\\* = sup \(u,$)\ for $ E Z*, (26)

IMI<i

(u,f) = (u,f) for u E Z,f E X. (27)

We assume that a conjugation f M f is defined on XX. Corollary 1 and Theorem 2 yield that XX = X + iX with a real Hilbert space X. We assume in addition that the

ZZ Z

zz

on the space Z*, too. In fact, due to (9) we have

= suP \(u,f)\ = suP \(u,f)\ = suP \(u,f)\ IMI<i iMI<i IMI<I

= sup \(u,l)\ = sup \(u,l) = \\7\\* for f E x,

IMI<i INI<i

which shows that f M f is continuous in the norm \\ • \\*, too. Density of XX in Z* then provides that the conjugation is extended on Z* continuously. Of course, it holds true that

for all $ E Z* Moreover, from (9) and (27) it is verified that

(u, $) = (u, $) for u E Z, $ E Z*. (28)

Let Z* be the real subspace of Z* induced by the conjugation on Z*. Then, (28) shows that the scaler product is real valued on Z x Z*.

Z c X c Z*

(25) and (26) induce

\\u\\ = sup \(u,$)\, for u E Z, (29)

MU<i,<f€Z*

\\$\\* = sup \(u,$)l for $ E Z*. (30)

IIuII<i,uGZ

Proof. For f E X, there exists a sequence un E Z such that un M f in XX. Sinee f M Ref

is continuous, it follows that Re un E Z and Re un M f in X; hence, Z is dense in X.

X Z*

For u E Z, there exists an element p E Z* such that \\p\\* = 1 and \\u\\ = (u,p). In view of (28), \\u\\ = (u,p) = (u,p). Therefore, we see that \\u\\ = (u, Re p) together with \\Rep\\* < \\p\\* < 1- Hence, (29) is proved.

For p E Z *, there exists a sequence un E Z such th at W^W < 1 and (un,p) M \\p\\*. It is easy to see that it is the same for the sequence u^. Then, (Reun, p) M \\p\\* together with \\Reun\\ < \\un\\ < 1. Hence, (30) is proved.

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This proposition has thus proved the following result for Z C X C Z*.

Z X Z* Z C X C Z*

make a triplet.

We finally remark an important property

[Z *Z ] 1 = X.

In fact, it is known ( [4,12]) that

[Z*,Z] 2 = X. (31)

Then, by (24), we have

[Z*, Z]i = [Z*, Z]i f z* = X f Z* = X.

5. Real Sectorial Operators Determined from Sesquilinear Forms

5.1. Real Sesquilinear Forms

Let Z and X be two complex Hilbert spaces with inner products ((•, •)) and (•, •) and norms \\ • ^d | • ^ respectively, such that Z C XX densely and continuously. Then, there

ZX* \ • \ *

triplet

Z c X c Z*.

The scaler product between Z and Z* is denoted by (•, •)zxz*-

We assume that XX has a conjugation f M f on it which is consistent with a conjugation on Z. As seen in Section 4, the conjugation induces a conjugation on Z*. het Z = Z + iZ, XX = X + iX and Z* = Z* + iZ* be the decompositions of Z, XX and Z*, respectively. We know by Theorem 8 that these real subspaces also make a triplet

Z C X C Z*.

Consider a sesquilinear form a(u, v) defined on Z. We assume that a(u, v) is continuous

and coercive on Z, i.e.,

\a(u,v)\< MIHHMI for u, v E Z, (32)

Rea(u,u) > $||u||2 for u E Z, (33)

with some constants M > 0 and S > 0.

We furthermore assume that

a(u, v) is real valued for u, v G Z. (34)

Such a sesquilinear form is called a real sesquilinear form. It is possible to characterize the definition in terms of conjugate.

Proposition 3. a(u,v) satisfies (34) if and only if

a(u,v) = a(u,v) for all u, v E Z. (35)

Proof. Let (34) be satisfied. By sesquiliniarity, we have

a(u, v) = a(Re u, Re v) + ia(Imu, Re v) — ia(Re u, Im v) + a(Imu, Im v). Therefore,

a(u, v) = a(Re u, Re v) — ia(Im u, Re v) + ia(Re u, Im v) + a(Im u, Im v) = a(Re u — iIm u, Re v — iIm v) = a(u,v).

Conversely, let (35) be satisfied. If u, v E Z, then u = v,v = v; therefore, a(u,v) = a(u,v); hence, a(u,v) is real.

□

According to the theory of variational methods ( [12]), the sesquilinear form a(-, •) satisfying (32) and (33) defines a linear operator from Z into Z* by the formula

a(u,v) = (Au,v)z f°r u, v E Z. (36)

It is also known as a linear operator of Z* that A is a sectorial operator of angle < n with the domain D(A) = Z. In addition, its part in X, denoted by A, is defined by

I D(A) = {u E Z; Au E x

1 Au = Au,

and is a sectorial operator of XX of angle <

The condition (34) in fact implies the following fact.

A A zz *

XZ A A

Proof. It follows from (36) that

a(u,v) = (Au,v)% = {v, Au)z^z* u, v G Z.

Then, it is obtained by (28) and (35) that

a(u,v) = (v, Au)= (v, Au)= (Au,v)^ for u, v E Z.

Meanwhile, by definition,

a(u,v) = (Au,v)% for u, v E Z.

Since v M v is onto Z, we must have Au = Au for any u E Z. Hence, A fulfills (13). Similarly, since

a(u,v) = (Au,v) for u E D(A), v E Z,

we have

a(u,v) = (Au,v) for u E D(A), v E Z.

A

(13).

□

We therefore arrive at the following result.

A A Zz *

xz

A Z Z* A

linear operator from D(A) H X onto X. In addition, these operators are nothing more than the operators A\Z* and A\x, respectively.

It is known that A satisfies ||(A — A)"1 < 1 /1A^ for A < 0 and this implies that A is maximal accretive, i.e., Re(Au,u) > 0 for u E D(A). Then, A possesses bounded purely imaginary powers Aiy (—to < y < to), and consequently the domains of its fractional powers Ae coincide with the interpolation spaces, that is,

D(Ae) = [X, D(A)]e for any 0 < 9 < 1. (38)

Thereby,

D(Ae) H X = [X, D(A)]o H X for any 0 < 9 < 1. In view of (19) and (24), it then follows that

D([A|xf) = [X, D(A) H X]e = [X, D(A|xfor any 0 < 9 < 1.

5.2. Real Elliptic Operators

We conclude this section with presenting an example of real sectorial operator which is determined from a real sesquilinear form.

Let Q be a bounded domain in Rra. Let L2(Q; C) (resp. H 1(H; C)) be the complex ¿2-space (resp. the complex Sobolev space of first order) in Q with norm || • Hl2 (resp. || • ||). We consider a complex triplet

H 1(Q; C) c L2(Q; C) C H1 (Q; C)*,

where H 1(Q; C)* is the adjoint space of H1 (Q; C). Let f M / be the complex conjugation on L2(Q) which obviously satisfies (1)~(4) and is consistent with the conjugation on H 1(Q; C). Thereby, this induces a conjugation on H 1(Q; C)*, too.

According to Theorem 2, the conjugation yields the decomposition of functions in H 1(Q; C), L2(Q; C) and H 1(H; C)* into real part and imaginary part. But, it is nothing more than

H 1(Q; C) = H 1(Q; R) + iH 1(Q; R), L2(Q; C) = L2(Q; R) + iL2(Q; R), (39)

H 1(Q; C)* = H 1(Q; R)* + iH 1(Q; R)*,

here L2(Q; R) (resp. H 1(Q : R)) is the real L2-space (resp. the real Sobolev space of first order) in Q and H 1(Q; R) is the adjoint space of H 1(Q; R). As shown by Theorem 8, we have a real triplet

H 1(Q; R) C L2(Q; R) C H 1(Q; R)*.

We then set Z = H 1(Q; C) and X = L2(Q; C). Consequently, Z = H 1(Q; R) and X = L2(H; R).

Consider a sesquilinear form

(u, v)= ajk(x)Dju(x) Dkv(x) dx + c(x)u(x)v(x) dx (40)

j,k=1 Jn

defined on Z = H 1(Q; C). We assume that

ajk E L^(Q; R) for 1 < j,k < n, and c E L^ (Q; R); (41)

n

ajk(x)jCk > S|C|2 for almost Vx E ^d VC = (C1,..., Cn) E Rn; and (42)

j,k=1

c(x) > S for almost Vx E Q, (43)

S > 0

a(u, v)

nn

ajk(x)(0 + inj)(6 + ink) = Y^ ajk(x)[0Ck + njnk + i(Cknj— Cknj)], j,k=1 j,k=1

(42) yields that Re

Y ajk(x)(Cj + inj)(Ck + ink) j,k=1

> S(|CI2 + Inl2)

for VC + in = (C1 + in1,... ,Cn + inn) E Cn

a(u, v)

define sectorial operators.

Let A be the associated linear operator in H 1(Q; C)*. Then, A is a sectorial operator of H 1(Q; C)* with domain D(A) = H 1(Q; C) and of angle < n .For v E C§°(Q) (c H 1(Q; C)), (40) is written as

n

a(u,v) = ( — ^^ Dk[ajk(x)Dju] + c(x)u, v ) j,k=1

Hence, (36) implies in the sense of distribution that

n

Au = — ^^ Dk[ajk(x)Dju] + c(x)u in Q. j,k=1

In Q, A is thus a realization of the elliptic differential operator —J2,n k=1 Dk[ajk(x)Dj] + c(x).

Next, let A denote the part of A in L2(Q; C). Then, A is a sectorial operator of L2(Q; C) of angle < f ■ If u E D(A), then, since (40) is written as

n r.

a(u, v) = (Au, v) + 2_, ajk (x)v (x)k Dj u(x) Dk v(x) dx, j, k=iJ

where v(x) = (v1(x),..., vn(x)) is the outer normal vector of dQ at x E dQ, u must implicitly satisfy the boundary conditions

n

ajk (x)vk (x)Dju = 0 on dQ.

j, k=1

In this sense, A is a realization of — jk=1 Dk [ajk(x)Dj] + c(x) under the Neumann type boundary conditions Yjk=1 ajk(x)vk(x)Dju = 0 on dQ.

It is immediate to verify that (41) yields (34). Hence, Theorem 9 and Corollary 2

A A A

H 1(Q; C)* and A is a real sectorial operator of L2(Q; C). Furthermore, in view of (39), A\hi(n;R)* is a densely defined, dosed linear operator of H 1(Q; R)* having the domain H 1(Q; R) and A|L2(q;R) is a densely defined, closed linear operator of L2(Q; R) having the domain D(A) n L2(Q; R).

In applications, it is often very important to know the domains of fractional powers Ae or Ae for 0 < 9 < 1. Especially, for 9 = 1, we wonder if D(A2) = L2(Q; C) or if D(A1) = H 1(Q; C). Such a problem is called the square root problem, however, the answer is already known to be no in general (although (31) and (38) are the case). We have to restrict the class of sesquilinear forms to handle to that of, for example, symmetric forms. So, in addition to (41) and (42), let us assume that

ajk (x) = akj (x) for 1 < j, k < n, (44)

which implies that a(u,v) = a(v,u) for u,v E H 1(Q; C). Then, A* = A and hence A* = A; in this way, A is a positive definite self-adjoint operator of L2(Q; C). Therefore, a(u,u) = (Au,u) = ||A2u\\l2 f°r u E D(A)- Furthermore, £||u||Hi < \\A 2 u\\L2 < m|| u\Hi for u E D(A). Finally, we conclude that D(A2) = H 1(Q; C). Due to (38), it is obtained that

D(A&) = [D(A0), D(A1 )U = [L2(Q; C), H 1(Q; C)]2» = H2e(Q; C) for 0 < 9 < 2.

Consequently, taking intersections with L2(H; R) for both hand sides, we verify by (19) and (24) that ЩЛ^)]') = H2*(П; R) for 0 < 0 < 2, (45)

where H2'(П; R) is the real Sobolev spaces with the exponent 29.

Under (44), A1 is an isomorphism from H 1(Q; C) onto L2(Q; C). So, the purely imaginary powers of A are expressed by Aiy = AA- 2 AiyA2A-1 (—to < y < to) and are bounded operators on H1 (Q; C)* (since Aiy E L(L2(Q; C)) for any —to < y < to). Hence, D(A1) = [D(A0), D(A)] 1 = L2(Q; C) due to (31). Furthermore, for 2 < 0 < 1,

D(A0) = [D(A2), D(A)]2— = [L2(Q; C), H 1(Q; C)]^ = H20-1(Q; C).

Consequently, taking intersections with H 1(Q; R)* for both hand sides, we verify by (19) and (24) that

D([A,hi(fi;r)*]0) = H20-1 (Q; R) for 2 < 0 < 1. (46)

It is equally possible to set Z = H(Q; C) (instead of H 1(Q; C)), where H(Q; C) is a completion of the space C^(Q; C) by the H 1-Sobolev norm. Then, we have a triplet

H1(Q; C) C L2(Q; C) C H-1(Q; C) (= H^Q; C)*).

The sesquilinear form (40) is considered on H(Q; C) under the same assumptions (41),

A

of H-1 (Q; C) of angle < and its part A determined by (37) is a real sectorial operator of L2(Q; C) of angle < In the meantime, A is a realization of the elliptic differential operator — ^jjk=1 Dk [ajk (x)Dj] + c(x) under the Dirichlet boundary conditions u = 0 on dQ. In addition, A|h-i(q;r) is a densely defined, closed real linear operator of H-1(Q; R), and Al(q;r) is a densely defined, closed real linear operator of L2(Q; R).

A* = A A

L2(Q; C)

to (45) and (46) which characterize the domains D([A|H-i(n;r)]0) or D([A|l2(q;r

)]0) of

fractional powers of A\H-i(n;R) or A^(q;r), respectively.

6. Real Sectorial Operators Obtained by Complexfication

This section is devoted to considering how to construct real sectorial operators from real lineax operators.

Let XX be a complex Banach space with norm \\ • \\ and with conjugation f M f, and let XX = X + iX be the decomposition into real and imaginary parts. Let a real linear operator A: D(A) m X be given with do main D(A) C X. By the formula

A(u + iv) = Au + iAv for u + iv E X + iX = X,

we can extend A to a complex linear operator in X with the domain D(A) + iD(A). Indeed, we verify that

A(u1 + iv1 + u2 + iv2) = A(u1 + u2) + iA(v1 + v2)

= A(u1 + iv1) + A(u2 + v), uk + ivk E XX, k = 1, 2,

and

A((a + bi)(u + iv)) = A(au — bv + i(bu + av)) = A(au — bv) + iA(bu + av)

= (a + bi)A(u + iv), a + bi E C, u + iv E XX.

Theorem 10. If A is a densely defined, dosed linear operator of X, then it is the same for the extended operator A in XX.

Proof. The proof is quite direct if we notice (8).

with some constant M > 1.

These conditions are then shown to be sufficient conditions in order that A is a real sectorial operator.

Theorem 11. If a densely defined, closed real linear operator A of X satisfies (47) and (48), then A is a real sectorial operator of XX.

Proof. For given £ + in G C and f + ig G X + iX, consider the equation

[(£ + in) - A](u + iv) = f + ig for u + iv G D(A) + iD(A). This is rewritten in the form

i.e., £ + in E p(A). Moreover, we verify that, if the estimate (48) holds true for (£, n) E pR(A), then the estimate (16) holds for the corresponding £ + in■ Hence, (48) implies (16).

Acknowledgements.The author is supported by Grant-in-Aid for Scientific Research (No. 26400166) of the Japan Society for the Promotion of Science.

Therefore, if (£,n) E pR(A), then this equation has a unique solution given by

□

References

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Received, December 14, 2016

УДК 517.9 Б01: 10.14529/ттр17010б

ДЕЙСТВИТЕЛЬНЫЕ СЕКТОРИАЛЬНЫЕ ОПЕРАТОРЫ

А. Яги

Секториальные операторы, которые действуют в комплексных банаховых пространствах и отображают действительные подпространства в себя, называются действительными векториальными операторами. Эти операторы уже неявно используются при изучении различных диффузионных уравнений. Между тем, в теории Лоясевича -Саймона, которая обеспечивает сходимость решений к стационарным решениям, действительнозначные функции Ляпунова играют важную роль. Для того чтобы создать общие методы изучения задач сходимости на основе теории Лоясевича - Саймона, целесообразно дать явное определение действительных секториальных операторов и показать их основные свойства, которые наследуются от комплексных секториальных операторов.

Ключевые слова: секториальные операторы; дробные степени оператора; дифференциальные операторы.

Атсуши Яги, кафедра прикладной физики, Университет Осаки (г. Суйта, Осака,

Япония), [email protected].

Поступила в редакцию Ц декабря 2016 г.