Real sectorial operators

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 ^ f and is decomposed into X = X + iX. where X is a, real Banach subspace of X. and when a, sectorial operator A of X maps D( A ) П 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 iproperties from A.

1.    Complex Banach Spaces with Conjugation

We begin with defining conjugation acting on a complex Banach space. Let X be a complex Banach space with norm || • | |. Assume that X is equipped with a correspondence f ^ f satisfying:

f + g = f + g for f, g G X,(1)

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

f = f for f G X,(3)

If || = If || for f G X.(4)

It is immediate to verify that the correspondence is continuous on X and one-to-one and onto. In particular, 0 = 0. The vector f is called the conjugate vector of f . Such a correspondence is called a conjugation on the space X.

For f G X. we put f + f                f - f

Re f =          and Im f =       .                    (o)

Then, it is clear that f = Re f + i Im f. The vector Re f (resp. Im f ) is called the real part (resp. imaginary part) of f G X. The vectors satisfying Im f = 0 or equiv^ralently f = f are called a real vector. By (1), (2) and (3), both Re f a nd Im f are a real vector. As noticed, 0 is also a real vector. On the other hand, the vectors satisfying Re f = 0 or equivalently f = —f are called a purely imaginary vector. Obviously, i Im f is a purely imaginary vector. It holds that

Re f + i Im f = Re f — i Im f    for f G X.                  (6)

We want to consider the space

X = {f G X ; Im f = 0 , i . e .,f = f}.

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

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

Proof. By definition, X = Im ~ 10. Mean while, f ^ Im f is continuous; therefore, X is a closed subset.

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

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We call X the real Banach subspace Induced from X.

Thereby, we have a decomposition of any f E X into the form f = Ref + iImf    vnth Ref, Imf E X.                  (7)

Indeed, such a decomposition is unique.

Proof. Since (7) gives such a decomposition, it is sufficient to prove the uniqueness. Let f = f 1 + if 2 = g 1 + ig 2 with fk, gk E X for k = 1 , 2. Then. ( f 1 - g 1) + i ( f 2 - g 2) = 0: at the same time, considering this conjugate, we have ( f 1 — g 1) — i ( f2 — g ₂) = 0. Therefore, f 1 = g 1 and f2 = g2. Hence, (7) is the only possible decomposition.

By (4), we see that

U У < У Re f У + У Im f У <  2max {У Re f У, У Im f У} <  2 \\f У, f E X. (8)

This readily yields that f ^ Re f and f ^ Im f are continuous from X onto X.

□ Corollary 1. When X is a complex Hilbert space with inner product ( ■, ■ ), its real Banach subspace X is a real Hilbert space with the same inner product.

Proof. Since

4(f,g ) = Уf + gУ2 — Уf — gУ2 + if + igУ2 — iУf — igУ2, it follows that

4f) = Уf + gУ ² — Уf — gУ ² — if + igУ ² + iУf — igУ ² .

In view of (4), we observe that

^{( f},g ) = ^{( f},g )     for f, g ^E X.                               (9)

If f, g E X. then ( f, g ) = ( f, g ). Thus. ( ■, ■ ) defines a, real irmer product 011 X which provides a Hilbert structure.

□

2.    Real Sectorial Operators

We now state a definition ofreal sectorial operators. Let X be a complex Banach space with norm У ■ ||. Assunre that X Is equipped with a, conjugation f ^ f and let X Ire Its real Banach subspace.

Let A : D( A ) ^ X be a, linear operator of X with doniain D( A ) С X. Assume that A satisfies the conditions:

u G D( A ) is equivalent to Re u, Im u G D( A ) ,                  (10)

Re Au = A (Re u) and Im Au = A (Im u) fc >r u G D( A).           (H)

These conditions can be described in terms of conjugate.

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

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

Au = Au for u G D( A ) .

Proof. Let A satisfy (10) and (H). 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 holds for u G D( A ) that

Au = Re Au + i Im Au = Re Au — i Im Au,

Au = A (Re u — i Im u ) = A (Re u ) — iA (Im u ) .

Hence, (11) implies (13).

Conversely, let A satisfy (12) and (13). Then, u G D( A ) imp>lles u, u G D( A ): then. (5) shows that Re u, Im u G D( A ); hence, (10) is verified. Moreover, under (13),

Re Au = ( Au + Au ) / 2 = ( Au + Au ) / 2 = A (Re u ) ,

Im Au = ( Au — Au ) / 2 i = ( Au — Au ) / 2 i = A (Im u ) .

Hence, (11) is verified.

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We thus observed that (10) and (11) imply that D(A) = [D(A) П X] + i[D(A) П X] and that A rmips D(A) П X Into X. In tins sense, we call A a, real linear operator of X. In addition, we are naturally led to consider a part of A restricted in the real subspace X which is defined by

( D( A|_X ) = D( A ) П X, | A| X u = Au.

By (11), we then have

Au = A|X (Re u ) + iA|_X (Im u ) for u G D( A ) .

Theorem 3. Let A : D( A ) ^ X be a densely defined, closed linear operator of X satisfying (10) and (11). Then, its part A|X in X is a densely defined, closed real linear operator of X.

Proof. First, let us prove density of D( A|X ) in X. For any f G X, there exists a sequence u_n G D( A ) such that u_n convei‘ges to f In X. T1 lern Re u_n G D( A| X ) and Re u_n ^ Re f = f In X and of course In X. He псе. D( A| X ) Is deiise In X.

Second, let us prove closedness of A| X . Consider sequences u n G D( A| X ) and f_n = Au n such that, as n ^ to. u n ^ u and f n ^ f In X. By the closedness of A. u G D( A ) and f = Au. As X Is closed In X (due to Theorem 1). u must Ire In X : thereby, u G D( A|_X ). Consequently, f = Au = A|_Xu.

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We here remember the definition of sectorial operators of X (see [4]). A densely defined, closed linear operator A is said to be a sectorial operator of X if its spectrum a ( A ) is contained in a sectorial domain

a ( A ) С Г = {A G C;

| arg A| < ш},     0 <ш < п,               (15)

M

for AG Г,                      (16)

and its resolvent satisfies the estimate

У (A - A) - 1 У with some constant M > 1.

When a sectorial operator A of X is a real linear operator, A is called a real sectorial operator of X. We can show various properties of real sectorial operators by the following theorems.

Theorem 4. Let A be a real sectorial operator of X. Then,

A G p ( A ) Д and, only if  A G p ( A );                        (17)

( A - A ) ^{- 1} f = ( A - A ) ^{- 1} f for A G p ( A ) , f G X.               (18)

In particular, when A G p ( A ) zs real. ( A — A ) ^{- 1} is a real eg)erator and A belongs to the.

resolvent set p ( A|x ) of tin.■ part A|x-

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

As seen, we have ( A — A ) u = f arid ( A — A ) u = f for A, A G p ( A ). Thereby. u = ( A — A ) ^{- 1} f and u = ( A — A ) ^{- 1} f. Hence, (18) is also shown.

When A G p ( A ) is real. (18) means that ( A — A ) ^{- 1} satisfies (13). Hence. ( A — A ) ^{- 1} is a real operator.

□ Theorem 5. Let A be a real sectorial operator of X. Let, for 0 < 9 < то, A⁶ be its fractional powers. Then, for every exponent 9, A⁶ is also a real operator.

Proof. The spectrum condition (15) implicitly means that 0 G a ( A ). So, the re exists 5 >  0 such that {A G C; |A| < 5} С p ( A ). We then introduce an integral contour Г = Г_-иГ 0 иГ + such that Г± : A = ге±^ш\ 5 < r < то and Г0 = 5е^6г, —ш < 9 < ш. Its orientation is from тое^шг tо 5е^шг. fr5е^шг tо 5е^-шг. and from 5е^-шг tо тое^-шг. By definitiorn A^-6 is given by the integral

A^-6 f = Л / A^-6 ( A — A ) ^{- 1} fdA    for f G X.

2 ni Г r

Taking the conjugate of each hand side, we obtain by (18) that

A X' = —    / A^- ( A — A ) ^{- 1} fdA.

2 ni Гг

Here. A^-6 = e^{-6 (log |л|-г arg л} ) = ( A ) ^-6. Arn 1. as A varies on Г in the positive sense. A varies on the same contour Г in the negative sense. It therefore follows that

A^-6/ = 2 1- f A '^! ( A — A ) ^{- 1} fdA = A^-6 J.

This means that A ⁶ satisfies (13). Thanks to Proposition 1, A ⁶ is a real operator (note that, as A^-6 is a bounded operator, (12) is automatically satisfied).

It is now easy to see that A⁶ is real. Indeed, if u E D( A⁶ ), then the re exists f E X for which u = A^-6f holds: therefore, u = A^-6f and u E D( A⁶ ): furthermore. A⁶u = A⁶u. It is clear that u E D( A⁶ ) converselу implies u = u E D( A⁶ ). Hence, Proposition 1 is again available to A⁶.

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Consider a real sectorial operator A of X. 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| ^ . In fact, noting that A⁶ is an operator from D( A⁶ ) П X into X, we set

[A|x]6 = [A6] |x     for any 0 < 9 < to, with the domain

D([ A ^ ] ⁶ ) = D( A⁶ ) П X.                           (19)

Then,

A⁶u = [ A| x ] ⁶ (Re u ) + i [ A| x ] ⁶ (Im u )     for u E D( A⁶ ) .

Theorem 6. Let A be a real sectorial operator XX with, angle ш < n, Then, for the analytic semigroup < -A (0 < t <  to ) genera ted by —A, e-A is a real operator for any 0 < t <  to.

Proof. Let Г be a similar integral contour used in the proof of Theorem 5. As well known, for 0 < t < to. the seinigroup e-tA is given by e-tAf = Л / e-tA(A — A)-1 fdA,    f E X.

2 ni Гг

Taking the conjugate of each hand side, we obtain by the similar arguments as in the proof of Theorem 5 that e-Af = X [ A- (A — A)-1 fdA = e-tAf.

2 ni Гг

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

□

Under the assumptions of Theorem 6, we define a semigroup on X genera ted by A \ by the formula e tA X = [ e-tA ] |x     for 0

Then, e-tAf = e-tAX (Ref) + ie-tAX (Imf)    for f E X.

Moreover, e-tA|X • e-sA|X = e-(t+s)А|х      on X.

3.    Interpolation and Real Subspaces

Let Z and X be^two complex Banach spaces such that Z С X densely and continuously. We assume that X has a conjugation f ^ f on it. In addition, we assume that this conjugation is consistent to that of Z in the sense that u G Z if and only if u G Z,                          (20)

Mz = Hull. for u ^G Z.                           (21)

For 0 < 0 < 1, let [X, Z]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 < 0 < 1, u G [X, Z]0 if and only if u G [X, Z]0,

ee llu|l[Xe,Z]6 = llu|l[Xe,Z]6       for u G [X,Z]0■

Proof. Let u G [X, Z]₀. By definition, there exists a holomorphic function Ф(z) defined in the band domain G = {z G C; 0 < Re z < 1} with vrfines in X. which is continuous and bounded on the closed domain G, which takes its values on the straight line I = {z = 1 + iy; —to< y < to} in Z with sup-^<_y<^ ||Ф(z)|_Z< to, and which takes a value Ф(0) = u at the point z = 0. Then, the function Ф(z) = Ф(z) also possesses similar properties but Ф(0) = u. Tins then means that u also belongs to [X, Z]₀. Conversely, if u G [X, Z]₀, then u = u G [X, Z]₀. Hence, (20) holds true.

We remember that

M[X,.e = inf{sup IIф(iy)h + sup IIф(1 + iy)h; ф(z)is iyEiR                1+iyEl any holomorphic function in G satisfying the proper ties mentioned above}.

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

□

This theorem means that, when X has a conjugation f ^ f which is consistent with a conjugation on Z (i.e., (20) and (21)), the conjugation induces a conjugation on any interpolation space [X, Z]0, too. We can then applу Theorems 1 and 2 to [X, Z]0. Let X = X + iX and Z = Z + iZ be the decompositions for X and Z. respectively, into real part and imaginary part. We naturally define z-^-z      z-^«z

[X, Z] 0 = [X, Z] 0 П X for any 0 < 0 < 1.

Then, it holds true that

Z-^—1      z-^z-

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

4.    Triplet and Real Subspaces

In this section, we want to consider a triplet of complex spaces Z С X C Z* ( [12]). Here. X is a complex Hilbert space, its inner product and norm being denoted by (•, •) and | • I, respective!y. The space Z is a complex reflexive Banach space, its norm being denoted z-*^z by || • ||. such that Z is densely and continuously embedded in X. The third space Z* is ee a complex adjoint space of Z with norm || • ||*. There is a scaler product (•, •) between Z and Z* which is sesquilinear and satisfies

M = sup ^|(u,ф^)| ∥φ∥∗≤1

1И1* = ^sup ku^ ∥u∥≤1

⁽u, f) = ⁽u, f)     for

for u G Z,

for ф G Z*, /-^^                  z-^z u∈Z, f ∈X.

We assume that a conjugation f ^ f is defined on X. Corollary 1 and Theorem 2 yield that X = X + iX with a real Hilbert space X. We assume in addition that the conjugation is consistent with that on Z, i.e., (20) and (21) being satisfied. Let Z be the real subspace of Z induced by this conjugation. Then, the conjugation can be extended on the space Z*, too. In fact, due to (9) we have

||^f||* = sup ^|(u,f^ = sup ^{1 (}u^,f)¹ = sup ^{1 (}uf)¹ ∥u∥≤1              ∥u∥≤1              ∥u∥≤1

—                —      —                   z^*z

=^sup ^{1 (u,f})¹ = ^sup ^|(u,^f) = IlfII* for ^{f G} X, ∥u∥≤1              ∥u∥≤1

which shows that f ^ f is continuous in the norm || • ||*, too. Density of X in Z* then provides that the conjugation is extended on Z* continuously. Of course, it holds true that z-^z

|ф|* = ||ф|*      for all Ф G Z*.

Moreover, from (9) and (27) it is verified that

''--Z                          z^«z

(и^ф¹) = (u, ф)     for u G Z, ф G Z*.

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

Proposition 2. The two embeddings Z С X C Z* are dense and continuous. Moreover, (25) and (26) induce

M = sup    ^|(u,Ф^)|,    for u ^G Z,                   (29)

∥φ∥∗≤1, φ∈Z^∗

1|ф||* = sup   ^|(u,Ф^)|,      for ф ^G Z*.                        (30)

∥u∥≤1, u∈Z

Proof. For f G X, there exists a sequence un G Z such that u_n ^ f in X. Since f ^ Ref is continuous, it follows that Re un G Z and Re un ^ f in X; hence, Z is dense in X. Similarly, we verify that X is dense in Z*.

For u G Z. there exists an element ф G Z* such that ||фП* = 1 aiid |u| = (u, ф). In view of (28). |u| = (и,ф) = (и,ф). Therefore, we see that |u| = (u, Re ф) together with |Reф|* < |ф|* < 1. Непсе. (29) is proved.

For ф G Z*. there exists a sequence un G Z such that |u_n| < 1 arк! (u_n, ф) ^ |ф|*. It is easy to see that it is the same for the sequence йф. Then, (Reu_n,ф) ^ |ф|* together with |Re uni < lun| < 1. Hence. (30) is proved.

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

Theorem 8. Let Z, X and Z* be real subspaces intreduced above. Then, Z С X C Z* make a triplet.

We finally remark an important property

• [ ZZ ] 2 = X.

• 5.    Real Sectorial Operators Determined from Sesquilinear Forms

  • 5.1.    Real Sesquilinear Forms

In fact, it is known ( [4,12]) that z-^z        z~z«z                   z~z«z

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

Then, by (24), we have

[Z*, Z] 2 = [Z*, Z] 2 П Z* = X П Z* = X.

Let Z and X be two complex Hilbert si races with inner products ((•, •)) and (-, •) arid norms | • | and | • |, respective!y, such that Z C X densely and continuously. Then, there is a unique third Banach space Z*, its norm being denoted by | • |*, which composes a triplet Z^«Z           Z^zZ           Z^zZ

Z C X C Z *.

The scaler product between Z and Z* is denoted by (•, -^e_хе..

We assume that X has a conjugation f ^ f on it which is consistent with a conjugation on Z. As seen in Section 4. the conjugation induces a, conjugation on Z*. Let Z = Z + iZ, X = X + iX and Z * = Z * + iZ * be the deco impositions of Z, X 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 arid coercive on Z. i.e..

|a(u,v)| < M|u||v|     f°^ru, v G Ze,                        (32)

Re a(u,u) > h|u|²     for u G Ze,                         (33)

with some constants M > 0 and 5 > 0.

We furthermore assume that

a(u,v) is real a^-;lined for u, v G Z.

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

.                          .                                                                                                                          z~^z

a(u,v) = a(u, v)     fiyr all u, v G Z.

Proof. Let (34) be satisfied. By sesquiliniarity, we have a (u, v) = a (Re u, Re v) + ia (Im u, Re v) — ia (Re u, Im v) + a (Im u, 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 G Z, then u = v, v = v; therefore, a(u,v) = a(u,v): heiice. 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) = (A u,v)z x

z~^z for u, v G Z.

It is also known as a linear operator of Z * th at 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

{

Z^-Z                     Z^Z

D(A) = {u g Z; Au G X},

Au = Au, and is a sectorial operator of X of angle < n. The condition (34) in fact implies the following fact.

Theorem 9. Under (32), (33) and (34), let A a nd A be sectorial operators of Z * and X. respectively, intro duccd above. Then, both, A aiul A are. a real operator.

Proof. It follows from (36) that

a(u,v) = (Au,v^e,_xz = (v, Au^z_xz*    for u,v G Z.

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

a(u, v) = (v, Au)e_xz* = (v, Au^e_xz* = (Au,v^zx    for u, v G Z.

Meanwhile, by definition,

a(u, v) = (Au,v^ e,_xz     for u, v G Z.

Since v ^ v is onto Z, we mu st have Au = Au for any u G Z. He nee, A fulfills (13).

Similarly, since

a(u, v) = (Au, v)     for u G D(A), v G Z, we have

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

Then, in view of (9), we can repeat the same argument to conclude that A also fulfills (13).

□

We therefore arrive at the following result.

Corollary 2. Under (32), (33) and (34), A a nd A are a real sectorial operator of Z * and X. respectively.

As shown. A is considered as a, rea 1 linear operator from Z onto Z*. aiid A as a, real linear operator from D(A) ПX onto X. In addition, these operators are nothing more than the operators A\_Z. aiid A|x- respectively.

It is known that A satisfies ||(A — A)~¹||L(x) < 1 /|A| fc>r A < 0 and this iruplies that A is maximal accretive, i.e., Re(Au,u) > 0 for u G D(A). Then, A possesses bounded purely imaginary powers Ai (—to< y < to), and consequently the domains of its fractional powers A⁶ coincide with the interpolation spaces, that is,

D(A⁶) = [X, D(A)]6     for any 0 < 0 < 1.                    (38)

Thereby,

D(A⁶) П X = [X-, D(A)]6 П X    for any 0 < 0 < 1.

In view of (19) and (24), it then follows that

D([A|x]⁶) = [X, D(A) П X]6 = [X, D(a|x)]6     for any 0 < 0 < 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 lie a bounded domain in Rⁿ. Let L₂(Q; C) (resp. H 1(Q; C)) lie the complex L2-space (resp. the complex Sollolev space of first order) in Q with norm || • |_L2 (resp. || • ||яi). We consider a complex triplet

H 1(Q; C) C L2(Q; C) C H 1(Q; C)*, where H 1(Q; C)* is the adjoint space of H1 (Q; C). Let f ^ f 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 indue-es a conjugation on H 1(Q; C)*. too.

According to Theorem 2, the conjugation yields the decomposition of functions in H 1(0; C), L₂(0; C) and H 1(0; C)* into real part and imaginary part. But, it is nothing more than

H 1(0; C) = H 1(0; R) + iH 1(0; R),

L₂(0; C) = L₂(0; R) + iL₂(0; R),                       (39)

H 1(0; C)* = H 1(0; R)* + iH 1(0; R)*, here L2(0; R) (resp. H 1(0 : R)) is the real L2-space (resp. the real Sobolev space of first order) in 0 and H 1(0; R) is the adjoint space of H 1(0; R). As shown by Theorem 8, we have a real triplet

H 1(0; R) C L₂(0; R) C H 1(0; R)*.

We then set Z = H 1(0; C) arid X = L₂(0; C). Consecluently. Z = H 1(0; R) arid X = L₂(0; R).

Consider a sesquilinear form

a(u, v)= ^^   kjx (x)Dju(x) D_kv (x) dx + j c(x)u(x)v (x) dx

defined on Z = H 1(0; C). We assume that ajk E L^(0; R) for 1 < j, k < n, arid c E L^(0; R);             (41)

52 aj^k(x)KjKk > 5|K|^{2 for ahuost} ^x E 0 arid VC = (К1 ,...,(„) E Rⁿ; and (42) j,k =1

c(x) > 5 for a.Imost Vx E 0,                            (43)

here 5 > 0 is some constant.

By (41), the form a(u,v) satisfies (32). In the meantime, since nn

52 ajk(x)(Cj + inj)(Ck + ink) = 52 ajk(x)[CjCk + ППк + i(Cknj - Cknj)], j,k=1 j,k=1

(42) yields that

Re 5 ajk^(x)(<j + inj⁾⁽<k + ink) > ^{5 (}К1² + Ini²⁾

j,k=1

∈

_Cn_.

for VK + in = (К1 + in I,..., Kn + inn)

This together with (43) shows that a(u,v) satisfies (33), too. So, by (36) and (37), we can define sectorial operators.

Let A be the associated linear operator in H 1(0; C)*. Then, A is a sectorial operator of H 1(0; C)* with dornain D(A) = H 1(0; C) arid of' angle < 2. For v E C^(0) (C H 1(0; C)). (40) is written as

a(u, v)

n

— E

j,k=1

D_k [ajk(x)Dju] + c(x)u, v

Cg° (Q) *x Cg° (Q)

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

n

Au = — ^2 Dk[ajk(x)Dju] + c(x)u    in 0.

j, k =1

In 0, A is thus a realization of the elliptic differential operator — V; _k=1 Dk [ajk(x)Dj] + c (x).

Next, let A denote tlre part of A iii L 2(0; C). Then. A is a, sectorial operator of L 2(0; C) of angle < n. If u e D(A), then, since (40) is written as

n

a(u, v) = (Au, v

) + V /  ajk⁽x)^{v (}x)

_kDju(x) Dkv(x) dx,

j,k=1 dQ where v(x) = (v 1(x),...,vn(x)) is the outer ncormal vector of д0 at x e д0. u must implicitly satisfy the boundary conditions

n

^2 ajk (x)vk (x)Dju = 0 on дО. j,k=1

In this sense, A is a realization of — ^j _k=1 Dk[ajk(x)Dj] + c(x) under the Neumann type boundary conditions   ”_k=1 ajk(x)vk(x)Dju = 0 оn дО.

It is immediate to verify that (41) yields (34). Hence, Theorem 9 and Corollary 2 are available to the operators A and A to conclude that A is a real sectorial operator of H 1(0; C)* and A is a real sectorial operator of L₂(0; C). Furthermore, in view of (39), AIHi(Q_;R)* N a densely defined, closed linear operator of H 1(0; R)* having the domain H 1(0; R) aiid AL₂(q_;R) is a densely defined, closed linear operator of L₂(0; R) having the domain D(A) П L₂(0; R).

In applications, it is often very important to know the domains of fractional powers A6 оr A6 fc>r 0 < 0 < 1. Especi;illy. for 0 = 2. we wonder if D(A2) = L2(0; C) or if D(A2) = H 1(0; 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,

winch implies that a (u,v) = a (v,u) fc or u, v e H 1(0; C). Tl ien. A * = A and 1 lence A* = A: in this way. A is a positive definite self-adjoint operator of L₂(0; C). Therefore. a(u,u) = (Au,u) = ||A2u\\L2 fcor u e D(A). Furthemiore. 5|u|H 1 < ||A2u\\L2 < M||u||H 1 fcor u e D(A). Finally, we conclude that D(A2) = H 1(0; C). Due to (38), it is obtained that

D(A⁶) = [D(A⁰), D(A²)]26 = [L2(0; C), H 1(0; C)]26 = H²⁶(0; C) for 0 < 0 < |.

Consequently, taking intersections with L₂(0; R) for both hand sides, we verify by (19)

and (24) that

D([A|L2(n;K)J6) = H26(0; R)     for 0 < 0 < 2, where H26(0; R) is the real Sobolev spaces with the exponent 20.

Under (44). A 2 is an isoniorphisrn from H 1(Q; C) on to L₂(Q; C). So. the purely imaginary powers of A are expressed by Aiy = A A2 AiyA 2 A -¹ (—to< y < to ) and are bounded operators on H¹ (Q; C)* (sinee Aiy G L(L₂(Q; C)) for any —to< y < to). Hence, D(A2) = [D(A⁰), D(A)] 2 = L₂(Q; C) due to (31). Furthermore, for 2 < 6 < 1,

D(A⁶) = [D(A²), D(A)]26-1 = [L2(Q; C),H 1(Q; C)bо-1 = H^26-1(Q; C).

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

D([A^ 1(Q;R)*]⁶) = H^26- 1(Q; R) for 2 < 6 < 1.                  (46)

It is equally possible to set Z = H1(Q; C) (instead of H¹(Q; C)), where H1(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) (= H1 (Q; C)*).

The sesquilinear form (40) is considered on H1(Q; C) under the same assumptions (41), (42) and (43). Then, the operator A determined by (36) becomes a real sectorial operator of H^-1 (Q; C) of angle < 2, and its part A determined by (37) is a real sectorial operator of L₂(Q; C) of angle < 2. In the meantime, A is a realization of the elliptic differential operator — Пk=1 Dk [ajk (x)Dj] + c(x) under the Dirichlet boundary conditions u = 0 on dQ. In ad(Ution. AiH-i(q_;R) is a densely defined, elosed real linear operator of H^- 1(Q; R). and AiL₂(q_;R) Is a densely defined, elosed real linear operator of L₂(Q; R).

Furthermore, assume that (44) is satisfied. Then, A * = A and A is a positive definite self-adjoint operator of L₂(Q; C). The similar arguments as above yield analogous results to (45) and (46) which characterize the domains D([A|_H-i(q_;R)]⁶) 0r D([A|L₂(q_;R)]⁶) of fractional powers of A|_H-i(q_;R) 0r A|L₂(q_;R), respectively.

6.    Real Sectorial Operators Obtained by Complexfication

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

Let X be a complex Bana ch space with norm || ■ || and with conjugation f ^ f, and let X = X + iX be the decomposition into real and imaginary parts. Let a real linear operator A: D(A) ^ X Ire given with domain D(A) C X. By the formula.

A (u + iv) = Au + iAv   for u + iv G 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 (u 1 + iv 1 + u 2 + iv 2) = A (u 1 + u 2) + iA (v 1 + v 2)

= A(u 1 + iv 1) + A(u2 + iv2),     uk + ivk G X, 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 G C, u + iv G X.

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

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

□

Consider a densely defined, closed linear operator A: D(A) ^ X of X. We set pR(A) = {(f, n) E R2; [(f — A)2 + n2] : D(A2) ^ X b one-to-one and onto, and [(f — A)2 + n2] 1 is a bounded operator on X}.

We set also aR(A) = R2 — pR(A). Assume that a r( A) С E = {(f,n) E R2; | arg( f, n) | <ш},     0 < Зш < п,          (47)

and that the inverse [(f — A)² + n2] 1 satisfies the estimate

У (f — A) [(f — A)² + n2]" 1 У + \\n [(f — A)² + n2]" 1 У

M

TM    ^{for (f},n^{)E E}, ⁽⁴⁸⁾

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 XX.

Proof. For given f + in E C arid f + ig E X + iX. consider the equation

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

(

ξ-A η

f —ⁿA   ^uv^{= f}g ^.

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

(U) = (fX f 'LAf[(f — A>2+n2]-1 (f), i.e.. f + in E p(A). Moreover, we verify that, if the estimate (48) holds true for (f,n) E pr(A), then the estimate (16) holds for the corresponding f + in. Hence, (48) implies (16). □