OPERATORS IN DIVERGENCE FORM AND THEIR ...

Opuscula Mathematica • Vol. 31 • No. 4 • 2011 http://dx.doi.org/10.7494/OpMath.2011.31.4.501

OPERATORS IN DIVERGENCE FORM AND THEIR FRIEDRICHS AND KRE˘IN EXTENSIONS Yury Arlinski˘ı, Yury Kovalev Abstract. For a densely defined nonnegative symmetric operator A = L∗2 L1 in a Hilbert space, constructed from a pair L1 ⊂ L2 of closed operators, we give expressions for the Friedrichs and Kre˘ın nonnegative selfadjoint extensions. Some conditions for the equality (L∗2 L1 )∗ = L∗1 L2 are obtained. Applications to 1D nonnegative Hamiltonians, corresponding to point interactions, are given. Keywords: symmetric operator, divergence form, Friedrichs extension, Kre˘ın extension. Mathematics Subject Classification: 47A20, 47B25, 47E05, 34L40, 81Q10.

1. INTRODUCTION Let H be a separable Hilbert space and let A be a densely defined closed, symmetric, and nonnegative operator, i.e., (Af, f ) ≥ 0 for all dom (A). As is well known, the operator A admits at least one nonnegative self-adjoint extension AF called the Friedrichs extension, which is defined as follows. Denote by A[·, ·] the closure of the sesquilinear form A[f, g] = (Af, g), f, g ∈ dom (A), and let D[A] be the domain of this closure. According to the first representation theorem [16] there exists a nonnegative self-adjoint operator AF associated with A[·, ·], i.e., (AF h, ψ) = A[h, ψ], ψ ∈ D[A], h ∈ dom (AF ). Clearly A ⊂ AF ⊂ A∗ , where A∗ is adjoint to A. It follows that dom (AF ) = D[A] ∩ dom (A∗ ). By the second representation theorem the equalities 1/2

D[A] = dom (AF )

1/2

1/2

and A[φ, ψ] = (AF φ, AF ψ),

φ, ψ ∈ D[A]

hold. 501

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M.G. Kre˘ın in [19] via fractional-linear transformation and parametrization of all contractive self-adjoint extensions of a non-densely defined Hermitian contraction discovered one more nonnegative self-adjoint extension of A having extremal property to be a minimal (in the sense of corresponding quadratic forms) among others nonnegative self-adjoint extensions of A. This extension we will denote by AK and call it the Kre˘ın extension of A. When A is positively definite, i.e., the lower bound of A is a positive number, it is shown in [19, 20] that ˙ ker(A∗ ). dom (AK ) = dom (A)+ Thus, in that case the Kre˘ın extension coincides with selfadjoint extension constructed by J. von Neumann. That’s why AK is often called the Kre˘ın–von Neumann extension of A. For the case of zero lower bound of A Ando and Nishio [2] proved that AK can be defined as follows. Let Pran (A) be the orthogonal projection onto ran (A) in H, and let Q be the operator in ran (A) given by Q(Af ) = Pran (A) f, f ∈ dom (A). Then Q is symmetric, nonnegative and densely defined in ran (A). Let QF be the Friedrichs extension of Q. Its inverse Q−1 F exists and the relation AK = Q−1 F Pran (A) holds. One more intrinsic construction of the Kre˘ın extension AK by means of the Friedrichs extension AF has been proposed in [9] and [10]. Another approach to nonnegative selfadjoint extensions is connected with boundary triplets (boundary value spaces) and corresponding Weyl functions [3, 12–14, 18, 23]. In concrete situations the intrinsic characterizations of the Friedrichs and Kre˘ın extensions for symmetric operator with zero lower bound is a non-trivial problem. In this paper we study this problem for operators of the form A = L∗2 L1 ,

(1.1)

where L1 and L2 are closed operators in H taking values in a Hilbert space H and possessing the condition L1 ⊂ L2 . (1.2) Such kind of operators A we call operators in divergence form. A particular case is A = L20 , where L0 is symmetric operator in H. Here L1 = L0 , L2 = L∗0 . In [26] and [27] (see also [8,28]) it is shown that each nonnegative symmetric operator by artificial way can be represented in divergence form and descriptions of Friedrichs, Kre˘ın, and all other extremal extensions have been obtained. Similar approach for representations of extremal extensions has been proposed in [15] for the case of nonnegative linear relations. Sturm-Liouville differential operators have natural representation in divergence form [11]. We establish here (see Theorem 3.1) that under the condition A∗ = L∗1 L2 the Friedrichs and Kre˘ın extensions of A are given by the operators L∗1 L1 and

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L∗2 Pran (L1 ) L2 , respectively. The equality A∗ = L∗1 L2 holds true if, for example, dim (dom (L2 )/dom (L1 )) < ∞ [7] or if L1 = L0 , L2 = L∗0 , where L0 maximal symmetric operator, cf. [22]. It is proved in [21, Theorems 6.4, 6.5] that (L20 )F = L∗0 L0 for a closed symmetric operator L0 provided L20 is densely defined and (L20 )∗ = L∗2 0 , while (L20 )K = L0 L∗0 , if, additionally, ker(L∗0 ) = {0}. For a pair L0 ⊂ L∗0 we prove ∗ ∗ ∗ ∗ (see Theorem 3.4) that if (L20 )∗ = L∗2 0 , then (LL0 ) = L0 L and (L0 L) = LL0 for an arbitrary selfadjoint extension L of L0 . Our main results are applied to the following differential operators in the Hilbert space L2 (R):  dom (A0 ) = f ∈ W22 (R) : f (y) = 0, y ∈ Y ,

d2 , dx2 d2 ˚ A := − 2 , dx d2 H0 := − 2 . dx A0 := −

 dom (˚ A) = g ∈ W22 (R) : g 0 (y) = 0, y ∈ Y ,  dom (H0 ) = f ∈ W22 (R) : f (y) = 0, f 0 (y) = 0, y ∈ Y ,

(1.3) (1.4) (1.5)

Here W21 (R) and W22 (R) are Sobolev spaces, Y is finite or infinite monotonic sequence of points in R satisfying the condition inf{|y 0 − y 00 |, y 0 , y 00 ∈ Y, y 0 6= y 00 } > 0.

(1.6)

The operators A0 , ˚ A, and H0 are densely defined and nonnegative with finite (the set Y is finite) or infinite defect indices (the set Y is infinite) and are basic for investigations of Hamiltonians on the real line corresponding to the δ, δ 0 and δ − δ 0 interactions, respectively [1]. Note that in [7, 8] Theorem 3.1 has been applied to the case of one point interaction in R. 2. PRELIMINARIES 2.1. UNIQUENESS AND TRANSVERSALNESS Let Nz (A) = H ran (A − zI) = ker(A∗ − zI) be the defect subspace of A. By von Neumann formulas ˙ z (A)+N ˙ z¯(A), dom (A∗ ) = dom (A)+N

Im z 6= 0.

M.G. Kre˘ın [19] established that A has a unique nonnegative selfadjoint extension if and only if |(f, ϕ−a )|2 = ∞ for all f ∈dom (A) (Af, f ) inf

ϕ−a ∈ N−a (A) \ {0},

a > 0.

(2.1)

Let Ae be a nonnegative selfadjoint extension of A. It is established by M.G. Kre˘ın e = dom (Ae1/2 ) admits the decomposition [19] that the domain D[A] 1/2 ˙ dom (Ae1/2 ) = dom (AF )+(dom (Ae1/2 ) ∩ N−a (A))

(2.2)

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for arbitrary a > 0 and 1/2 kAe1/2 vk2 = kAF vk2 ,

1/2

v ∈ dom (AF ).

Since (Ae − zI)(Ae − λI)−1 Nz (A) = Nλ (A),

z, λ ∈ C \ [0, ∞),

we get also the decomposition 1/2 ˙ dom (Ae1/2 ) = dom (AF )+(dom (Ae1/2 ) ∩ Nz (A)),

z ∈ C \ [0, ∞).

(2.3)

The Kre˘ın extension AK of A, possesses the properties [19] 1/2

1/2

dom (Ae1/2 ) ⊆ dom (AK ), kAe1/2 uk ≥ kAK uk2 ,

u ∈ dom (Ae1/2 )

1/2 for each nonnegative selfadjoint extension Ae of A. The domain dom (AK ) can be characterized as follows [2] ) ( |(Aϕ, u)|2 1/2 0. Proposition 3.3. Let L0 be a densely defined closed symmetric operator in H. The following conditions are equivalent: (i) (L20 )F = L∗0 L0 , (ii) dom (L0 ) ∩ ran (L0 − λI) is dense in ran (L0 − λI) for at least one non-real λ. Proof. Clearly, (L0 f, L0 g)+(f, g) = ((L0 +iI)f, (L0 +iI)g) = ((L0 −iI)f, (L0 −iI)g), f, g ∈ dom (L0 ) One can easily proof that dom (L20 ) = (L0 − λI)−1 (ran (L0 − λI) ∩ dom (L0 )) ,

Im λ 6= 0.

(3.4)

The equality (L20 )F = L∗0 L0 is equivalent to the condition: dom (L20 ) is dense in dom (L0 ) w.r.t. graph norm in dom (L0 ). The latter is equivalent to that there is no nontrivial vector g ∈ dom (L0 ) such that (L0 f, L0 g) + (f, g) = 0 for all f ∈ dom (L20 ). From (3.4) it follows the equivalence of (i) and (ii). Theorem 3.4. Let L0 be a densely defined closed symmetric operator with equal deficiency indices in H. Suppose L20 is densely defined and (L20 )∗ = L∗2 0 . Then for an arbitrary selfadjoint extension L of L0 the equalities (LL0 )∗ = L∗0 L

(3.5)

Operators in divergence form and their Friedrichs and Kre˘ın extensions

509

and (L0 L)∗ = LL∗0

(3.6)

hold. ∗ ∗ Proof. Denote by Nλ (L0 ) the deficiency subspace of L0 . Since L∗2 0 +I = (L0 +iI)(L0 − iI), we have the equality

˙ ker(L∗2 0 + I) = Ni (L0 )+N−i (L0 ). Taking into account that (L20 )∗ = L∗2 0 and applying statement 2) of Theorem 3.1 to ˙ −i (L0 ) is a subspace in H. Let a selfadjoint the pair L0 ⊂ L∗0 , we get that Ni (L0 )+N extension L is given by ˙ + U)Ni (L0 ), dom (L) = dom (L0 )+(I where U is an isometric mapping of Ni (L0 ) onto N−i (L0 ). Let us show that (LL0 )∗ = L∗0 L. Statement 1) of Theorem 3.1 for the pair L0 ⊂ L∗0 ∗ and the equality (L20 )∗ = L∗2 0 imply that the operator L0 L0 is the Friedrichs extension 2 ˙ −i (L0 ) is a subspace in H, the of the operator L0 . In addition, because Ni (L0 )+N linear manifold (I + U)Ni (L0 ) is a subspace in H as well. Clearly, ker(L∗0 L + I) = (I + U)Ni (L0 ). Since LL0 ⊇ L20 and dom (L20 ) is dense in dom (L0 ) (w.r.t. the graph norm in dom (L0 )), the domain dom (LL0 ) is also dense in dom (L0 ), i.e., (LL0 )F = L∗0 L0 . Applying statement 2) of Theorem 3.1 to the pair L0 ⊂ L, we obtain that (LL0 )∗ = L∗0 L. Next we equip dom (L∗0 ) by the inner product (f, g)+ = (f, g) + (L∗0 f, L∗0 g). Then dom (L∗0 ) becomes a Hilbert space, which we denote by H+ . Then (+)-orthogonal decomposition H+ = dom (L0 ) ⊕ Ni (L0 ) ⊕ N−i (L0 ) holds. Let N = (I + U)Ni (L0 ) and M = (I − U)Ni (L0 ). We have (+)-orthogonal decompositions dom (L) = dom (L0 ) ⊕ N ,

H+ = dom (L) ⊕ M.

Clearly LN = M, and

L∗0 M = N ,

LL∗0 h = −h,

h ∈ M,

L∗0 Le

e ∈ N.

= −e,

Let Le be one more selfadjoint extension of L0 given by e = dom (L0 ) ⊕ M = dom (L0 )+(I ˙ − U)Ni (L0 ), dom (L)

e Le = L∗0  dom (L).

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e we conclude that (LL e 0 )F = L∗ L0 , i.e., dom (LL e 0) Then, considering the pair L0 ⊂ L, 0 is dense in dom (L0 ) in (+)-norm. In addition, the linear manifold e ∗0 + I) = (I − U)Ni (L0 ) = M ker(LL e = −h for all h ∈ M, LLe e = −e for all e ∈ N , is a subspace in H. In addition LLh and e e)+ = −(h, Le)+ , h ∈ M, e ∈ N . (Lh, + Let us describe dom (L0 L). Denote by PM the (+)-orthogonal projection in H+ onto M. Let f ∈ dom (L). Then

f = ϕ0 + e, ϕ0 ∈ dom (L0 ),

e ∈ N,

Lf = L0 ϕ0 + Le.

e Because Le ∈ M we have that Lf ∈ dom (L0 ) if and only if L0 ϕ0 = Lf −Le ∈ dom (L) e 0 ) and P + L0 ϕ0 = −Le. Finally, ⇐⇒ ϕ0 ∈ dom (LL M e + L0 )dom (LL e 0 ). dom (L0 L) = (I + LP M Let us show now that dom (L0 L) is dense in dom (L) w.r.t. (+)-norm. Suppose there is g ∈ dom (L) such that g is (+)-orthogonal to dom (L0 L), e + L0 )h0 , g)+ = 0 ((I + LP M

e 0 ). for all h0 ∈ dom (LL

In particular, taking h0 ∈ dom (L20 ), we get that the vector g is (+)-orthogonal to dom (L20 ). But dom (L20 ) is (+)-dense in dom (L0 ). It follows that g ∈ N . Since e 0 ) ⊂ dom (L0 ), we have dom (LL e + L0 h0 , g)+ = 0, (LP M

e 0 ). h0 ∈ dom (LL

Further e + L0 h0 , g)+ = (P + L0 h0 , Lg)+ . 0 = (LP M M Let Lg = x. Then x ∈ M and e + L0 h0 , g)+ = (L0 h0 , x)+ = (L0 h0 , x) + (LL0 h0 , Lx) e = 0 = (LP M e + (LL0 h0 , Lx) e = ((LL e 0 + I)h0 , Lx). e = (h0 , Lx) It follows that e ∈ ker((LL e 0 )∗ + I). Lx e 0 )∗ = L∗ L. e Hence, Lx e ∈ M. Applying equality (3.5) to Le instead of L we get that (LL 0 e = −g ∈ N . Hence g = 0. Thus, dom (L0 L) is (+)-dense in On the other hand Lx dom (L) and, therefore, (L0 L)F = L2 . Applying statement 2) of Theorem 3.1, we arrive at (3.6).

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4. APPLICATIONS Let Y be a finite or infinite monotonic sequence of points in R satisfying condition (1.6). Let A0 , ˚ A and H0 be given by (1.3), (1.4), (1.5), respectively. Notice that (see [1]): dom (A∗0 ) = W21 (R) ∩ W22 (R \ Y ), A∗0 = −

d2 , dx2

˚∗ = − dom (˚ A∗ ) = {g ∈ W22 (R) : g 0 (y+) = g 0 (y−), y ∈ Y }, A 2

dom (H0∗ ) = W22 (R \ Y ), H0∗ = −

d . dx2

d2 , dx2

(4.1)

Let Z be the set of all integers and let Z− = {j ∈ Z, j ≤ −1}, Z+ = {j ∈ Z, j ≥ 1}. For infinite Y it is possible three cases Y = {yj , j ∈ Z},

if

inf{Y } = −∞ and

sup{Y } = +∞,

Y = {yj , j ∈ Z− }, if y−1 = sup{Y } < +∞, Y = {yj , j ∈ Z+ },

if y1 = inf{Y } > −∞.

By J we will denote one of the sets Z, Z− , Z+ for infinite Y . Consider in the Hilbert space L2 (R) the following operators dom (L0 ) = {f ∈ W21 (R) : f (y) = 0, y ∈ Y }, dom (L) = W21 (R),

d , dx d L=i . dx

L0 = i

(4.2) (4.3)

From (4.2) it follows that L0 is a densely defined symmetric operator and its adjoint L∗0 is given by d dom (L∗0 ) = W21 (R \ Y ), L∗0 = i . (4.4) dx The operator L is a selfadjoint extension of L0 . So, we have L0 ⊂ L ⊂ L∗0 . In addition A0 ⊃ H0 , ˚ A ⊃ H0 . If Y consists of N points, then the deficiency indices of L0 are hN, N i, and the deficiency indices of H0 , A0 , ˚ A are h2N, 2N i, hN, N i, and hN, N i, respectively. Let dk = |yk − yk+1 |, k ∈ J, L0k = i

d , dom (L0k ) = {f ∈ W21 ([yk , yk+1 ]) : f (yk ) = f (yk+1 ) = 0}, dx

k ∈ J.

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The operator L0k is symmetric with deficiency indices (1, 1) in the Hilbert space L2 [yk , yk+1 ]. Hence, (L20k )∗ = L∗2 0k (see Theorem 2.2). Clearly, M M dom (L0 ) = dom (L0k ), L0 = L0k , k

dom (L∗0 )

=

k

M

dom (L∗0k ),

L∗0

=

M

k

L∗0k .

k

Hence, ker(L∗0k ) = {f (x) = const, x ∈ [yk , yk+1 ]}, and ker(L∗0 ) =

M

ker(L∗0k ).

k

Observe that dom (H0 ) =

M

dom (H0∗ )

M

dom (L20k ),

H0 = L20 =

dom (L∗2 0k ),

H0∗

M

k

=

L20k ,

k

=

L∗2 0

=

M

k

L∗2 0k .

(4.5)

k

From Theorem 3.4 (and also from (4.1), (4.3) (1.3), (1.4), (1.5)) it follows that A0 = LL0 ,

H0 = L20 ,

˚ A = L0 L,

A∗0 = L∗0 L,

˚ A∗ = LL∗0 ,

H0∗ = L∗2 0 .

(4.6)

Denote by χk the characteristic function of the interval [yk , yk+1 ]. Then the functions ) ( χk √ dk k∈J

form an orthonormal basis of ker(L∗0 ). Therefore,  yk+1  Z X 1  Pker(L∗0 ) L∗0 f = if 0 (x)dx χk = dk k

yk

X 1 (f (yk+1 − 0) − f (yk + 0))χk , =i dk

(4.7) f ∈ dom (L∗0 ),

k

and Pran (L0 ) L∗0 f = if 0 − i

X 1 (f (yk+1 − 0) − f (yk + 0))χk , dk

f ∈ dom (L∗0 ).

(4.8)

k

If f ∈ W21 (R), then f (y ± 0) = f (y), y ∈ Y . From (4.6) it follows that conditions (1.1) and (3.1) are fulfilled for the pairs hL0 , Li, hL, L∗0 i, and hL0 , L∗0 i and we can apply Theorem 3.1.

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4.1. THE FRIEDRICHS AND KRE˘IN EXTENSIONS OF THE OPERATOR A0 Let A0 be given by (1.3). Then as has been mentioned above one has A0 = LL0 ,

A∗0 = L∗0 L,

where L0 , L, and L∗0 are given by (4.2), (4.3), and (4.4), respectively. Since L is a selfadjoint extension of L0 we can apply Theorem 3.1 by setting L1 = L0 , L2 = L. Hence, the Friedrichs extension A0F is the operator A0F = L∗0 L0 , i.e.,  dom (A0F ) = f ∈ W21 (R) : f 0 ∈ W21 (R \ Y ), f (y) = 0, y ∈ Y ,

A0F f = −

d2 f . dx2

From Theorem 3.1 and (4.8) we get  dom (A0K ) = f ∈ dom (L) : Pran (L0 ) Lf ∈ dom (L) = ( ) X 1 = f ∈ W21 (R) : f 0 − (f (yk+1 ) − f (yk ))χk ∈ W21 (R) , dk k

and A0K f = −

d2 f , dx2

f ∈ dom (A0K ).

It follows that the boundary conditions for f ∈ dom (A0K ) are f 0 (yk − 0) −

1 1 (f (yk ) − f (yk−1 )) = f 0 (yk + 0) − (f (yk+1 ) − f (yk )) , dk−1 dk

k ∈ J,

or in equivalent form 0

0

f (yk + 0) − f (yk − 0) =

1 dk−1

f (yk−1 ) −

1

1 + dk−1 dk

! f (yk ) +

1 f (yk+1 ), dk

k ∈ J.

Additional conditions arise in the cases inf{Y } > −∞, sup{Y } < +∞. In particular, if Y is an infinite set, −∞ < y1 = inf{Y }, then f 0 (y1 − 0) − f 0 (y1 + 0) =

1 (f (y1 ) − f (y2 )), d1

and if +∞ > y−1 = sup{Y }, then f 0 (y−1 + 0) − f 0 (y−1 − 0) =

1 d−1

(f (y−1 ) − f (y0 )).

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For a finite Y = {y1 , y2 , . . . , yN } we get 1 (f (y1 ) − f (y2 )), d1 1 (f (yN ) − f (yN −1 )), f 0 (yN + 0) − f 0 (yN − 0) = dN −1 f 0 (y1 − 0) − f 0 (y1 + 0) =

0

1

0

f (yk + 0) − f (yk − 0) =

dk−1 +

f (yk−1 ) −

1

1 + dk−1 dk

! f (yk )+

1 f (yk+1 ), k = 2, . . . , N − 1. dk

For A0K [f, g] we get D[A0K ] = W21 (R) and Z   X1 A0K [f, g] = f 0 (x)g 0 (x)dx− (f (yk+1 ) − f (yk )) g(yk+1 ) − g(yk ) , f, g ∈ W21 (R). dk k

R

4.2. THE FRIEDRICHS AND KRE˘IN EXTENSIONS OF THE OPERATOR ˚ A ˚∗ = LL∗0 . Put Now we consider the operator ˚ A given by (1.4). Then ˚ A = L0 L, A ∗ L1 = L, L2 = L0 . Applying Theorem 3.1 we get that 2

d f dom (˚ AF ) = dom (L2 ) = W22 (R), ˚ AF f = L2 f = − 2 , f ∈ W22 (R). dx Since ker(L) = {0} we get ˚ AK = L0 L∗0 , i.e.,  d2 f dom (˚ AK ) = f ∈ W21 (R \ Y ) : f 0 ∈ W21 (R), f 0 (y) = 0, y ∈ Y , ˚ AK f = − 2 . dx In addition D[˚ AK ] = W21 (R \ Y ) and Z ˚ AK [f, g] = f 0 (x)g 0 (x)dx, f, g ∈ W21 (R \ Y ). R

4.3. THE FRIEDRICHS AND KRE˘IN EXTENSIONS OF THE OPERATOR H0 Let H0 be given by (1.5), then H0 = L20 , H0∗ = L∗0 2 . Put L1 = L0 , L2 = L∗0 . Applying Theorem 3.1 we obtain the Friedrichs extension dom (H0F ) = {f ∈ dom (L0 ) : L0 f ∈ dom (L∗0 )} = = {f ∈ W21 (R), f 0 ∈ W21 (R \ Y ), f (y) = 0, y ∈ Y }.

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Notice that A0F = H0F . For dom (H0K ) we have dom (H0K ) = {f ∈ dom (L∗0 ) : Pran (L0 ) L∗0 f ∈ dom (L0 )} = n X 1 = f ∈ W21 (R \ Y ) : g = f 0 − (f (yk+1 − 0) − f (yk + 0))χk ∈ W21 (R), dk k o g(y) = 0, y ∈ Y . The boundary conditions for f ∈ dom (H0K ) we can write in the form: 1 (f (yk+1 − 0) − f (yk + 0)) , dk 1 f 0 (yk − 0) = (f (yk − 0) − f (yk−1 + 0)) dk−1

f 0 (yk + 0) =

for all

yk ∈ Y,

and additionally f 0 (y−1 + 0) = 0

if

+ ∞ > y−1 = sup{Y },

f 0 (y1 − 0) = 0

if

− ∞ < y1 = inf{Y },

or

and if Y = {y1 , . . . , yN }, then f 0 (y1 − 0) = 0, f 0 (yN + 0) = 0, 1 (f (yk+1 − 0) − f (yk + 0)) , dk 1 f 0 (yk − 0) = (f (yk − 0) − f (yk−1 + 0)) , dk−1

f 0 (yk + 0) =

k = 1, . . . , N − 1, k = 2, . . . , N.

Clearly, D[H0K ] = W21 (R \ Y ) and Z H0K [f, g] = f 0 (x)g 0 (x)dx− R

−

  X 1 (f (yk+1 − 0) − f (yk + 0)) g(yk+1 − 0) − g(yk + 0) , dk k

f, g ∈ W21 (R \ Y ). Notice that due to (4.5) and according to [25, Corollary 5.5] we have H0F =

M k

L20k


F

,

H0K =

M k

L20k


K

.

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Yury Arlinski˘ı yury [email protected] East Ukrainian National University Department of Mathematical Analysis Kvartal Molodyozhny 20-A, Lugansk 91034, Ukraine Yury Kovalev [email protected] East Ukrainian National University Department of Mathematical Analysis Kvartal Molodyozhny 20-A, Lugansk 91034, Ukraine Received: January 26, 2011. Revised: February 21, 2011. Accepted: February 22, 2011.