Some Sobolev spaces as Pontryagin spaces Текст научной статьи по специальности «Математика»

УДК 517.982.22+517.983.35

SOME SOBOLEV SPACES AS PONTRYAGIN SPACES1

V.A. Strauss2, C. Trunk3

We show that well known Sobolev spaces can quite naturally be treated as Pontryagin spaces. This point of view gives a possibility to obtain new properties for some traditional objects such as simplest differential operators.

Keywords: Function spaces, Pontryagin spaces, selfadjoint operators, differential operators

Introduction

Let H be a separable Hilbert space with a scalar product (•,•). H is said to be an indefinite metric space if it is equipped by a sesquilinear continuous Hermitian form (indefinite inner product) [•, •] such that the corresponding quadratic form has indefinite sign (i.e. [x,x] takes positive, negative and zero values). The indefinite inner product can be represented in the form [•,•] = [G-, •], where G is a so-called Gram operator. The operator G is bounded and self-adjoint. If the Gram operator for an indefinite metric space is boundedly invertible and its invariant subspace corresponding to the negative spectrum of G is finite-dimensional, lets say k -dimensional, the space is called a Pontryagin space with k negative squares. There are a lot of problems in different areas of mathematics, mechanics or physics that can be naturally considered as problems in terms of Operator Theory in Pontryagin spaces. We have no aim to give here an overview on this theory and its application. We refer only to the standard text books [1, 2, 10] and to [14] for a brief introduction.

Our scope is a modest illustration of some singular situations that shows an essential difference between Operator Theory in Hilbert spaces and in Pontryagin spaces. For this goal we use Sobolev spaces that represents a new approach.

1. Preliminaries

A Krein space (K,[-, •]) is a linear space K which is equipped with an (indefinite) inner product (i.e., a hermitian sesquilinear form) [•, •] such that K can be written as

K = G+[+]G_ (1)

where (G±,±[-, ]) are Hilbert spaces and + means that the sum of G+ and G_ is direct and [G+, G_] = 0 . The norm topology on a Krein space K is the norm topology of the orthogonal sum of the Hilbert spaces G±. It can be shown that this norm topology is independent of the particular decomposition (1); all topological notions in K refer to this norm topology and || • || denotes any of the equivalent norms. Krein spaces often arise as follows: In a given Hilbert space (G ,(•, •)), every bounded self-adjoint operator G in G with 0 e p(G) induces an inner product

[x, y]:= (Gx, y), x, y e G, (2)

such that (G,[•, ]) becomes a Krein space; here, in the decomposition (1), we can choose G+ as the spectral subspace of G corresponding to the positive spectrum of G and G_ as the spectral subspace of G corresponding to the negative spectrum of G . A subspace L of a linear space K with inner product [•, •] is called non-degenerated if there exists no x e L, x ^ 0, such that [x, L] = 0, otherwise L is called degenerated; note that a Krein space K is always non-degenerated, but it may have degenerated subspaces. An element x e K is called positive (non-negative, negative, non-positive, neutral, respectively) if [x, x] > 0 (> 0, < 0, < 0, = 0, respectively); a subspace of K is called positive (non-negative, etc.,

1 V. Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.

2 Strauss Vladimir Abramovich is Ph. D., Departamento de Matemáticas, Universidad Simóon Bolívar, Caracas, Venezuela.

E-mail str@usb.ve

3 Carsten Trunk is Dr. rer. nat., Professor, Analysis and Systems Theory Group, Technische Universität Ilmenau, Institut für Mathematik, Ilmenau, Germany.

^^-mail^carsten.trunk^iu—

respectively), if all its nonzero elements are positive (non-negative, etc., respectively). For the definition and simple properties of Krein spaces and linear operators therein we refer to [2], [13] and [1].

If in some decomposition (1) one of the components G± is of finite dimension, it is of the same dimension in all such decompositions, and the Krein space (K,[-, •]) is called a Pontryagin space. For the Pontryagin spaces K occurring in this paper, the negative component G_ is of finite dimension, say k; in this case, K is called a Pontryagin space with k negative squares. If K arises from a Hilbert space G by means of a self-adjoint operator G with inner product (2), then K is a Pontryagin space with k negative squares if and only if the negative spectrum of the invertible operator G consists of exactly k eigenvalues, counted according to their multiplicities. In a Pontryagin space K with k negative squares each non-positive subspace is of dimension < k, and a non-positive subspace is maximal non-positive (that is, it is not properly contained in another non-positive subspace) if and only if it is of dimension k. If L is a non-degenerated linear space with inner product [•, •] such that for a k -dimensional subspace

L_ we have

[x, x] < 0, x e L_, x ^ 0

but there is no (k+1) -dimensional subspace with this property, then there exists a Pontryagin space K with k negative squares such that L is a dense subset of K . This means that L can be completed to a Pontryagin space in a similar way as a pre-Hilbert space can be completed to a Hilbert space. The spectrum of a selfadjoint operator A in a Pontryagin space with k negative squares is real with the possible exception of at most k non-real pairs of eigenvalues X, X of finite type. We denote by Lx(A) the algebraic eigenspace of A at X. Then dim LX (A) = dim L (A) and the Jordan structure of A in L (A) and in Lx (A) is the same. Further the relation

k= £ k- (A) + £ dim Lx (A)

Xes0 Xes( A)nC+

holds, where s0 denotes the set of all eigenvalues of A with a nonpositive eigenvector and k- (A) denotes the maximal dimension of a nonpositive subspace of L (A).

Moreover, according to a theorem of Pontryagin, A has a k -dimensional invariant non-positive subspace L““. If q denotes the minimal polynomial of the restriction A | L“ax , then the polynomial q* q, where q* (z) = q(z), is independent of the particular choice of and one can show that

[q*(A)q(A)x,x] > 0 for xe D(Ak). As a consequence, a selfadjoint operator in a Pontryagin space possesses a spectral function with possible critical points. For details we refer to [11, 13].

The linear space of bounded linear operators defined on a Pontryagin or Krein space K1 with values in a Pontryagin or Krein space K2 is denoted by L(K1,K). If K := K = K we write L(K). We study linear relations in K, that is, linear subspaces of K2. The set of all closed linear relations in K is denoted by C(K). Linear operators are viewed as linear relations via their graphs. For the usual definitions of the linear operations with relations and the inverse we refer to [7, 8, 9]. We recall only that the multivalued part mul S of a linear relation S is defined by mul S = {y|(0)€ S}.

Let S be a closed linear relation in K. The resolvent set p(S) of S is defined as the set of all X e C such that (S _ X)-1 e L(K). The spectrum s(S) of S is the complement of p(S) in C . The extended spectrum s(S) of S is defined by s(S) = s(S) if S e L(K) and s(S) = s(S) U {¥ otherwise. We set p(S) := C \ s(S). The adjoint S + of S is defined as

S + := {(hh,)|[f',h] = [f,h'] for all (f,)e s} .

S is said to be symmetric (selfadjoint) if S c S + (resp. S = S +).

For the description of the selfadjoint extensions of closed symmetric relations we use the so-called boundary value spaces (for the first time the corresponding approach was applied in fact by A.V. Strauss [15, 16] without employing the term “boundary value space”).

Definition 1. Let A be a closed symmetric relation in the Krein space (K,[-, •]). We say that {G, r0, rx} is a boundary value space for A+ if (G, (•, •)) is a Hilbert space and there exist linear mappings r0, G : A+ ® G such that r := (jG0): A+ ® G x G is surjective, and the relation

[ f', g ] _ [f, g ] = (rf r g) _ (r„/, rg (3)

holds for all f = (f -), g = (g- )e A.

If a closed symmetric relation A has a selfadjoint extension A in K with p(A) ^ 0, then there exists a boundary value space {G, G0, Gj} for A+ such that A coincides with ker G0 (see [4]).

For basic facts on boundary value spaces and further references see e.g. [3, 4, 5] and [6]. We recall only a few important consequences. For the rest of this section let A be a closed symmetric relation and assume that there exists a boundary value space {G, G0, Gj} for A+. Then

A0 := ker r0 and A1 := ker rx (4)

are selfadjoint extensions of A . The mapping r = (jT0) induces, via

A© := r_1© = { f e A+ | rf e©}, Qe C(G),

a bijective correspondence © ^ A© between C(G) and the set of closed extensions A© c A+ of A. In particular (5) gives a one-to-one correspondence between the closed symmetric (selfadjoint) extensions of A and the closed symmetric (resp. selfadjoint) relations in G . Moreover, A© is an operator if and only if

Qnr{(h)|he mulA+} = {0}. (6)

If © is a closed operator in G , then the corresponding extension A© of A is determined by

A© = ker^ _©r0). (7)

Let NX := ker(A+ _ X) = ran(A _ X)[1] be the defect subspace of A and set

NX :={(f )lf e NX} .

Now we assume that the selfadjoint relation A0 in (4) has a nonempty resolvent set. For each X e p(A0) the relation A+ can be written as a direct sum of (the subspaces) A0 and NX (see [4]). Denote by p the orthogonal projection onto the first component of K2. The functions

X ^ y(X) := p (ro | Nx r1 e L(G, K), X e p( A,),

and

X ^M(X) := rx(r01 NNxe L(G), Xe p(A,) (8)

are defined and holomorphic on p(A0) and are called the y -field and the Weyl function corresponding to A and {G,r0,Gj} . For X,Ze p(A0) the relation (3) implies M(X)* = M(X) and

y(Z) = (1 + (Z-X)( A, _ ZT1) y(X) (9)

and

m (X) _ m (Z)*= (X _ Z) y(Z)+ y(X) (10)

hold (see [4]). Moreover, by [4], we have the following connection between the spectra of extensions of A and the Weyl function.

Lemma 2. If Qe C(G) and A© is the corresponding extension of A then a point Xe p(A0) belongs to p(A©) if and only if 0 belongs to p(Q_M(X)). A point Xe p(A0) belongs to si(A©) if and only if 0 belongs to si (Q_ M(X)), i = p, c, r.

For X e p(A©) n p(A0) the well-known resolvent formula

(A© - X)-1 = (A _ X)-1 + y(X)(Q_M(X))-1 y(X)+ (11)

holds (for a proof see e.g. [4]).

Recall, that X0 e C is called the eigenvalue of the operator pencil L(X), if there is a vector

h0 e G (h0 ^ 0) such that L(X0 )h0 = 0 . The vector h0 e G is called the eigenvector of the operator pen-

cil L(X). A system h0,h1,...,hk, is called a Jordan chain for L(X), if

m 1

Z jLj)(X0)hm_j = 0, for m = 0,1,...,k . (12)

j=0 j •

2. The Underlying Space

Let H1,2 (0,1) be the Sobolev space of all absolutely continuous functions f with f 'e L2 (0,1). Let k be a positive real number, k > 0 . We define for f, g e H1,2 (0,1)1

[f, g ]k := k (f, g\2(0,1} _ (f, g )i2(0,1) . (13)

If L is an arbitrary subset of H1,2 (0,1) we set

L[1]k :={ x e H1,2 (0,1): [x, y]k = 0 for all y e l} .

Then we have the following.

Proposition 3. For the space (H 1,2(0,1), [•, ]k) we have the following properties.

(1) If k equals for some ne N, then the function ge H1,2(0,1), defined by g(x) = cos(npx)

n P

belongs to the isotropic part of (H 1,2(0,1), [•, ]k), that is

[ f, g]k = 0 for all f e H 1,2(0,1).

(2) Ifk >P2, then (H 1,2 (0,1), [•, ]k) is a Pontryagin space with one negative square.

(3) If k < ^2 and k for all n e N, then (H 1,2(0,1), [•, ]k ) is a Pontryagin space with a finite

P n P ' '

number of negative squares. Set

h_ :=span \fj| k < -2-^ je n^| , where fj e H1,2 (0,1) is defined by fj (x) = sin( j-x). Then the number k_ of negative squares of (H 1,2(0,1), [v]k) satisfies

k_ = dim H_ +1.

Proof: Assertion (1) is an easy calculation. We assume k for ne N all. Define the operator

n —

A0 by

D(A,):={ge H1,2(0,1) | g e H1,2(0,1) and g(0) = g(1) = 0},

A) g :=_ g' for g e D( A,).

Let us note that the functions fj (x) = sin(j-x), j = 1,2... are eigen functions of A,.

For g e D(A,) n H-, where H- denotes the orthogonal complement with respect to the usual scalar product (•,but within the Hilbert space H 1,2(0,1), we have also that (f,g) 2mi = 0 for all

L (0,1 ) L (0,1 )

f e H_ . Thus, g has the representation g = Z °° aifi . This implies that there exists an e > 0 with

J >—4k

1 Let us note that the expression к ( y(t ))2 - y (t )2 with t as the time is (up to a constant) the Lagrangian for free small oscillations in one di-^ension^see_[12,^.^8]fordetails2.^romthis^ointofviewthecorres£ondingJntegralre£resentstheaction^^^^^^^^^^^^^^^^^^^^_

(A0 g, g )f2mn > (| + e)( g, g) r2(0 1) for all g e D(A0) n H- . Therefore, there exists constants c, c > 0

L (0,1) k L (0,1)

with

[ g, g ]k > c( A g,g )l2(0,1) +e( g,g )l2(0,1) >c( g, g )h1,2(0,1)

for ge D(A,)nH- , so D(A0)nH- is a uniformly positive. It is easy to see that for f e H_ we have [ f, f ]k < 0 . This shows that the closure of D(A,) with with respect to the usual scalar product in the Hilbert space H 1,2(0,1) is a Pontryagin space where the number of negative squares equals dim H_ . We define h1, h2 e H1,2 (0,1) by

h1 = sin (4k x) + cos(4k x) and h2 = sin (4k x)_ cos (4k x).

We have

[h1,h1]k =_2yfk (sinyfk ) and [h2,h2]k = 24k (sin-y/k ) .

and

(D( A,))[1]k = sp{h1, h2},

This proves (3). If k > ^2, then H_ = {0} and the above considerations imply (2). □

—

3. A Symmetric Operator Associated to the Second Derivative of Defect Four

For the rest of this paper, we assume that k is such, that

sin 4k ^ 0 .

Then, according to Proposition 3, the space (H 1,2(0,1), [ , ]k) is a Pontryagin space. We consider the following operator A, defined by

D(A) :={ge H1,2(0,1) | g'g№e H 1,2(0,1) with g(0) = g(1) = g'(0) = g'(1) = g'(0) = g'(1) = 0]

and

Ag := _g" for g e D( A).

Lemma 4. Then A is a closed symmetric operator in (H 1,2(0,1), [•, ]k).

Proof: Obviously, A is symmetric. The best way to show the closedness is via the calculation of A++ . We leave it to the reader. □

As

mul A+= (D (A,))[1]k = {x e H 1,2(0,1)|[ x, y]k = 0 for all y e D (A)}, we have g e mul A+ if and only if for all f e D(A)

0 =[f, g ]k =_( f, kg" + g)

L2(0,1)

The set D(A) is dense in L2(0,1) and this implies

mul A+= sp^^/1, f2}, (14)

where f1, f2 are defined by

f1 = sin 4k x and f2 = cos 4k x.

An easy calculation (more detailed?) shows that

A+ ={(-g- Hf+f )l g'- *' E H 1'2(0-1)- a-b C}.

Let (_f'+afij+bf2 ) and (_g'+a2gf1+fl2f2 ) be elements from A+ with f,g e D(A) and a1,a2,b1,b2 e C .

Then we have

[- f + aifi + b1f2, S \k - [ f, - g" + a2fi +P2fl\k =

= -g 10 +fg 10 -kfg 10 + kfg 10 +4k (aif2 -bifi) S10 -y/kOf-bf) 10.

We define mappings Г0, Г1 : A+ ® С4 by

G0 (

f

0 V -f’+alfl+b1f2

Г1 ( f

f ’+a1f1+bf2

) = ) =

and

( f (0)+kT(0) ^ f (i)+kf '(i)

f(0)

V f(i) )

-f (0) f '(i)

-ai 4k

л/ka cos л/k -bi sin 4k -1)

f

- f '+aifi+bif2

)e A+.

Theorem 5. The triplet (T0, Tj} is a boundary value space for A+. In particular A1 := ker G is an operator and a selfadjoint extension of A, i.e.

D(Ai):={ge H1,2(0,i)|g' ,g'e H 1,2(0,i) with g'(0) = g'(i) = 0]

and

Ajg :=-g", g e D(Aj). Moreover, for X e r(A0), the Weyl function is given by

1 ( 4X 4k1 1 1

M (X) =

4\ + -Jk

i kX V taii-v/x tan 4k ) i kX V sin 4\ sinyfk ) 4k tan 4k 4k sin 4k

4X 4k- л

4X + 4k

1

4k tan 4k

-1

4k sin 4k -

i-kx V tan^ tan 4k - J 4k sin 4k 4k tan 4k

-1 i-kX i-kX

4k sin 4k 4k tan 4k 4k sin 4k

1 i-kX i-kX

-Jk tanVk

Vk sin 4k -

-Jk tan *Jk

Proof: The above calculations imply that {T0, T1} is a boundary value space for A+. Let X e C \ M.

Define gi, g2 e Hi,2(0,i) by

Then we have

gi = cos(л/Xx) and g2 = sin (л/Xx).

ker( A+ - X) = sp {gi, g2, fl, f2 } .

(i5)

Let f = ag1 + bg2 + gf + 8f2 for some a,b,g,8e C . Then

Г0 ( f ) = Г0

0I -f '+g(x--bfi+£(x-i)f2

( a(i-kX)

a(i-kX )cos>/x+b(i-kx)sin-s/x a+d

a cos Л+b sinVX+gsin 4k1 +s cos 4k -

and

Г

i ( x/ ) =

( -bVx-g/T1 ^

-a\/\ sin ■J\+b^/X cos\/\ + g/k cos ■Jk -d4k sin 4k —1

-r4k (X -1)

g4k (X —1) cos 4k —d4k (X—1)sinVk

Now, by (8), it is follows that M is of the above form. □

Now, via (5) we can parameterize all selfadjoint extensions of A via all selfadjoint relations 0 in

C4.

Theorem 6. Let 0 be a selfadjoint relation in C4 . Then A0 is a selfadjoint extension of A . If for all

a,be C

i

i

й mul 0 \ {0}

(16)

b sin 4k —acos 4k

holds, then A0 is an operator. If, in particular, 0 is a selfadjoint matrix, then A0 is a selfadjoint operator and an extension of A with domain

D( A ):=

g e H 1,2(0,1) | g', g e H 1,2(0,1),

( -g (0) ^ ( g(0)+kg'(0) V

g '(1) = 0 g (1)+kg' (1)

0 g (0)

V 0 J V g(1) Jj

Proof: Relation (16) follows from (6), (14) and the definitions of T0 and T1. If 0 is a matrix, (16) is satisfied and the description of D(A0) follows from (7). □

4. A Symmetric Operator Associated to the Second Derivative of Defect Two

We start this Section opposite to Section 3. For this we put

D(A):={ge H 1,2(0,1)| g',g'e H 12(0,1)

1

(17)

and

Ag :=-g', g e D( A).

Thus, the operator A corresponds to the same formal differential expression as the operator considered in the previous section, but with a different domain which is in some sense maximal. Let us calculate A+ . For f, g e D(A) we have

[ Af, g ]k = -k |0 f'(t) g' (t )dt +10 f" (t) g (t )dt =

= -k (f(t) g '(t) - f '(t) g '(t) )| o + (f (t) g (t) - f (t) g '(t))| ^ - k Jf (t) gm (t )dt + J^f (t) g" (t )dt =

= -(kf "(1) + f (1))g(I) + f '(1)(kg(X) + g(1))-f '(0)(kg(0) + g(0)) + (kf '(0) + f (0))g(0) + [f,Ag]k.

Note that the maps f(t) ^ (f(1) + f(1)), f(t) ^ f(1), f(t) ^ (f(0) + f(0)) and f(0) ^ f(0) represent unbounded linear functionals on H 1,2(0,1). Thus, the expression [Af, g]k gives a continuous linear functional (with respect to f) on H1,2 (0,1) if and only if

g '(1) = (kg "(1) + g (1)) = (kg "(0) + g (0)) = g '(0) = 0 and by the definition of the adjoint operator the latter

conditions restrict the domain of A+. For brevity below we set A: = A+ . Thus, we have the following operator A, defined by

D(A) :={g e H1,2 (0,1) | gf ,g'e H 1,2(0,1) with g'(0) = g'(1) = 0, g(0) + kg'(0) = 0 and g(1) + kg'(1) = 0] and

Ag :=- g', g e D( A).

Then A is a closed symmetric operator in (H 1,2(0,1), [ , ]k), which is, in contrast to Section 3, densely defined. In particular

A+= A ={(- p )|g', g" e H 1,2(0,1)]

is an operator and therefore all selfadjoint extensions of A are operators.

We define mappings T0, T1: A+ ® C 2 by

G0 (-f )=( f!,°f0) and G (-f)=(-;f:(<0)>) for (-f)e A+.

Theorem 7. The triplet {r0, T1} is a boundary value space for A+. The Weyl function is given by

a

M (X) =

-VX

(1-kX)tan-v/X (1-kX ) sin

-VX _______________VX______

, Xep(A) .

(1-kX )sin^ (1-kX)tan^

Proof: The above calculations imply that {r0, T1} is a boundary value space for A+. Let X e C \ M and g1,g2 e H1,2(0,1) as in (15). Then we have

ker( A+- X) = sp{g1, g2}.

Let f = ag1 + bg2 for some a,be C . Then

T ( f ) = T ( f ) = ( a(1-kX) )

0 \ Xf ) 0 \ - f') \ a(1-kX) cos ■JX+b(1-kX)si^ VX )

and

1 ( Xf ) 1 (- f ) (-aVXsin-v/x+Zb/X oo^a/x )

Now, by (8), it is follows that M is of the above form. □

Lemma 8. The operator A, = kerT0 is a selfadjoint extension of A with a compact resolvent and

s(Aq) = CTp(AQ) = {k_1,p2,4p2,9p2,...}.

Proof: The operator A1 = kerT1 is selfadjoint in the Hilbert space H , (0,1). We have for f e D(A)

(( A, +1 ) f, f ) 12 =|| f ll22 +

W 1 , J /я1.2(0,1) n J 11£2(0Д)

1.2/

where H2,2(0,1) is the Sobolev space of all functions f e H 1,2(0,1) with fe H 1,2(0,1). This gives

1.2/

2

H2,2 (0.1) .

2 2 2 <|| (A +1 ) f || 12

H2.2(0.1) 11 v llH1.2(0.1)

H 2.2(0.1)

Therefore, as the embedding of H2,2(0,1) into H 1,2(0,1) is compact, the selfadjoint operator A1 has a compact resolvent. By (11) the difference between the resolvents of A0 and A1 is of finite rank, hence A0 has a compact resolvent. We have s(A0) = sp(A0). Now (18) follows from a simple calculation. □

QUESTION: Is A simple? That is H1,2(0,1) = clsp{ker(A+-X): X e p(A0)} Give a simple prooffor it Proposition 9. Let ae M, a ^ 0 and

|a|< 24k . (19)

Then the operator Aa defined by

D( Aa):={ g e Hx’2 (0,1) | g, g" e H1,2 (0,1) with a g (0) = g (0) + kg" (0) and a g (1) = g (1) + kg' (1) = 0} and

Aag :=-g" , g e D(Aa).

is a selfadjoint extension of A with non-real eigenvalues.

In the case a = 24k we have that the selfadjoint extension A2^ of A has a Jordan chain of length

two corresponding to the eigenvalue -1.

Proof: Set

0 =

-a 1 0

-1

0 a~

Then A© = Aa, hence, by Lemma 2 and the fact that s(A0) e M (see Lemma 8), we have for all non-real X that Xe sp(Aa) if and only if

0 = det(M (X) -0) =

k2

{ 2 . Xa2

a2 (1 - kX)2

X2 +-

k

2

2 k k2

\

(20)

J

Hence,

= 1 a2 a a2

1,2 = k 2k2 _ k2 V 4

are the solutions of Equation (20). Assertion (19) implies now the existence of two non-real eigenvalues

of Aa.

In the case a = 24k we have that the functions h0, \ e D(A2^ ) given by

1 „ . i /x _ x xfk 1

satisfy

h0( x) = ex'Jk and h1( x) = — ex

A2Vk + ^ 1 h1 = h0 and f A2Vk + ^ I h0 = 0 •

i.e. {Z0, is a Jordan chain of A2fk corresponding to the eigenvalue -1. D

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3. Derkach V.A. On Weyl Function and Generalized Resolvents of a Hermitian Operator in a Krein Space. Integral Equations Operator Theory. 1995. Vol. 23. pp. 387-415.

4. Derkach, V.A. On Generalized Resolvents of Hermitian Relations in Krein Spaces. J. Math. Sci. (New York). 1999. Vol. 97. pp. 4420-4460.

5. Derkach V.A., Malamud M.M. Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps. J. Funct. Anal. 1991. Vol. 95. pp. 1-95.

6. Derkach V.A., Malamud M.M. The Extension Theory of Hermitian Operators and the Moment Problem. J. Math. Sci. (New York). 1995. Vol. 73. pp. 141-242.

7. Dijksma A., de Snoo H.S.V. Symmetric and Selfadjoint Relations in Krein Spaces I. Operator Theory: Advances and Applications (Birkhäuser Verlag Basel). 1987. Vol. 24. pp. 145-166.

8. Dijksma A., de Snoo H.S.V. Symmetric and Selfadjoint Relations in Krein Spaces II. Ann. Acad. Sci. Fenn. Math. 1987. Vol. 12. pp. 199-216.

9. Haase M. The Functional Calculus for Sectorial Operators. Basel, Boston, Berlin, Birkhäuser Verlag, 2006. 392 p.

10. Iohvidov I.S., Krein M.G., Langer H. Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric. Berlin: Akademie-Verlag, Mathematical Research, 1982. Vol. 9. 120 p.

11. Krein M.G., Langer H. On the Spectral Functions of a Self-Adjoint Operator in a Space with Indefinite Metric. Dokl. Akad. Nauk SSSR. 1963. Vol. 152, no. 1. pp. 39-42.

12. Landau L.D., Lifshitz E.M. Course Of Theoretical Physics. Vol 1: Mechanics (3rd ed.). Oxford, UK, Butterworth-Heinemann, 2007. 170 p.

13. Langer H. Spectral Functions of Definitizable Operators in Krein Spaces. Lecture Notes in Mathematics. Springer Verlag, Berlin-Heidelberg-New York, 1982. Vol. 948. pp. 1-46.

14. Langer H., Najman B., Tretter C. Spectral theory of the Klein-Gordon equation in Pontryagin spaces. Comm. Math. Phys. 2006. Vol. 267, no. 1. pp. 159-180.

15. Strauss A.V. On Selfadjoint Extensions in an Orthogonal Sum of Hilbert Spaces. Dokl. Akad. Nauk SSSR. 1962. Vol. 144, no. 5. pp. 512-515.

16. Strauss A.V. Characteristic Functions of Linear Operators. Izv. AN SSSR, Serija mate-maticheskaja. 1960. Vol. 24, no. 1. pp. 43-74.

Received 22 march 2011.

НЕКОТОРЫЕ ПРОСТРАНСТВА СОБОЛЕВА КАК ПРОСТРАНСТВА ПОНТРЯГИНА

В.А. Штраус1, К. Труни2

Показано, что известные пространства Соболева могут быть естественно снабжены структурой пространства Понтрягина. Такой подход позволяет получить новые свойства у таких традиционных объектов как, например, простейшие дифференциальные операторы.

Ключевые слова: функциональные пространства, пространства Понтрягина, самосопряжённые операторы, дифференциальные операторы.

Литература

1. Azizov, T.Ya. Linear Operators in Spaces with an Indefinite Metric / T.Ya. Azizov, I.S. Iokhvidov. - Chichester: John Wiley & Sons, Ltd. - 1989. - 304 с.

2. Bognar, J. Indefinite Inner Product Spaces / J. Bognar. - New York-Heidelberg: Springer Verlag, 1974. - 224 с.

3. Derkach, V.A. On Weyl Function and Generalized Resolvents of a Hermitian Operator in a Krein Space / V.A. Derkach // Integral Equations Operator Theory. - 1995. - Т. 23. - С. 387-415.

4. Derkach, V.A. On Generalized Resolvents of Hermitian Relations in Krein Spaces / V.A. Derkach // J. Math. Sci. (New York) - 1999. - Т. 97. - С. 4420-4460.

5. Derkach, V.A. Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps / V.A. Derkach, M.M. Malamud // J. Funct. Anal. - 1991. - Т. 95. - С. 1-95.

6. Derkach, V.A. The Extension Theory of Hermitian Operators and the Moment Problem / V.A. Derkach, M.M. Malamud // J. Math. Sci. (New York). - 1995. - Т. 73. - С. 141-242.

7. Dijksma, A. Symmetric and Selfadjoint Relations in Krein Spaces I / A. Dijksma, H.S.V. de Snoo // Operator Theory: Advances and Applications (Birkhäuser Verlag Basel). - 1987. - Т. 24. - С. 145166.

8. Dijksma, A. Symmetric and Selfadjoint Relations in Krein Spaces II // A. Dijksma, H.S.V. de Snoo // Ann. Acad. Sci. Fenn. Math. - 1987. - Т. 12. - С. 199-216.

9. Haase, M. The Functional Calculus for Sectorial Operators / M. Haase. - Basel: Birkhäuser Verlag, 2006. - 392 с.

10. Iohvidov, I.S. Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric / I.S. Iohvidov, M.G. Krein, H. Langer. - Berlin: Akademie-Verlag, Mathematical Research, 1982. - Т. 9. - 120 с.

11. Krein, M.G. On the Spectral Functions of a Self-Adjoint Operator in a Space with Indefinite Metric / M.G. Krein, H. Langer // Докл. АН СССР. - 1963. - Т. 152, № 1. - С. 39-42.

12. Landau, L.D. Mechanics (3rd ed.) / L.D. Landau, E.M. Lifshitz. - Oxford, UK: ButterworthHeinemann, 2007. - 170 с.

13. Langer, H. Spectral Functions of Definitizable Operators in Krein Spaces / H. Langer // Lecture Notes in Mathematics (Springer Verlag: Berlin-Heidelberg-New York). - 1982. - Т. 948. - C. 1-46.

14. Langer, H. Spectral theory of the Klein-Gordon equation in Pontryagin spaces / H. Langer, B. Najman, C. Tretter // Comm. Math. Phys. - 2006. - Т. 267, № 1. - С. 159-180.

15. Штраус, А.В. On Selfadjoint Extensions in an Orthogonal Sum of Hilbert Spaces / А.В. Штраус // Докл. АН СССР. - 1962. - Т. 144, № 5. - С. 512-515.

16. Штраус, А.В. Characteristic Functions of Linear Operators / А.В. Штраус // Изв. АН СССР, Серия математическая - 1960. - Т. 24, № 1. - С. 43-74.

Поступила в редакцию 22 марта 2011 г.

1 Штраус Владимир Абрамович - Ph. D., кафедра математики, Университет Симона Боливара, г. Каракас, Венесуэла.

E-mail: str@usb.ve

2 Трунк Карстен - Dr. rer. nat., профессор, институт математики, Технический Университет Ильменау, г. Ильменау, Германия.