Some Sobolev Spaces as Pontryagin Spaces Vladimir A. Strauss Department of Pure Applied Mathematics, Simón Bolívar University, Sartenejas-Baruta, Apartado 89., Caracas, Venezuela Carsten Trun Technische Universität Ilmenau, Institut für Mathemati, Postfach 565, D-98684 Ilmenau, Germany Abstract We show that well nown 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. Key words: Function spaces, Pontryagin spaces, selfadjoint operators, spectrum MSC: 47B5, 47E5, 47B25 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] taes positive, negative 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 self-adjoint. If the Gram operator for an indefinite metric space is boundedly invertible its invariant subspace corresponding to the negative spectrum of G is finite-dimensional, lets say κ-dimensional, the space is called a Pontryagin space with κ negative squares. There are a lot of problems in different areas of mathematics, mechanics or physics that can be naturally considered as Email addresses: str@usb.ve (Vladimir A. Strauss, carsten.trun@tu-ilmenau.de (Carsten Trun.
problems in terms of Operator Theory in Pontryagin spaces. We have no aim to give here an overview on this theory its application. We refer only to the stard text boos [,2,] to [4] 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 in Pontryagin spaces. For this goal we use Sobolev spaces that represents a new approach. 2 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 ( where (G ±, ±[.,.] are Hilbert spaces [. ] means that the sum of G + G is direct [G +, G ] =. The norm topology on a Krein space K is the norm topology of the orthogonal sum of the Hilbert spaces G ± in (. It can be shown that this norm topology is independent of the particular decomposition (; all topological notions in K refer to this norm topology 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 ρ(g induces an inner product [x, y] := (Gx, y, x, y G, (2 such that (G, [.,.] becomes a Krein space; here, in the decomposition (, we can choose G + as the spectral subspace of G corresponding to the positive spectrum of G 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 L, x, such that [x, L] =, otherwise L is called degenerated; note that a Krein space K is always nondegenerated, but it may have degenerated subspaces. An element x K is called positive (non-negative, negative, non-positive, neutral, respectively if [x, x] > (, <,, =, respectively; a subspace of K is called positive (non-negative, etc., respectively, if all its nonzero elements are positive (nonnegative, etc., respectively. For the definition simple properties of Krein spaces linear operators therein we refer to [2], [3] []. If in some decomposition ( one of the components G ± is of finite dimension, it is of the same dimension in all such decompositions, 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 κ; in 2
this case, K is called a Pontryagin space with κ 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 κ negative squares if only if the negative spectrum of the invertible operator G consists of exactly κ eigenvalues, counted according to their multiplicities. In a Pontryagin space K with κ negative squares each non-positive subspace is of dimension κ, a non-positive subspace is maximal non-positive (that is, it is not properly contained in another non-positive subspace if only if it is of dimension κ. If L is a non-degenerated linear space with inner product [.,.] such that for a κ-dimensional subspace L we have [x, x] <, x L, x, but there is no (κ + -dimensional subspace with this property, then there exists a Pontryagin space K with κ 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 κ negative squares is real with the possible exception of at most κ non-real pairs of eigenvalues λ, λ of finite type. We denote by L λ (A the algebraic eigenspace of A at λ. Then dim L λ (A = dim L λ (A the Jordan structure of A in L λ (A in L λ (A is the same. Further the relation κ = λ σ R κ λ (A + λ σ(a C + dim L λ (A holds, where σ denotes the set of all eigenvalues of A with a nonpositive eigenvector κ λ (A denotes the maximal dimension of a nonpositive subspace of L λ (A. Moreover, according to a theorem of Pontryagin, A has a κ-dimensional invariant non-positive subspace L max. If q denotes the minimal polynomial of the resriction A L max, then the polynomial q q, where q (z = q(z, is independent of the particular choice of L max one can show that [q (Aq(Ax, x] for x D(A κ. As a consequence, a selfadjoint operator in a Pontryagin space possesses a spectral function with possible critical points. For details we refer to [,3]. The linear space of bounded linear operators defined on a Pontryagin or Krein space K with values in a Pontryagin or Krein space K 2 is denoted by L(K, K 2. If K := K = K 2 we write L(K. We study linear relations in K, that is, linear subspaces of K 2. 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 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 y S. 3
Let S be a closed linear relation in K. The resolvent set ρ(s of S is defined as the set of all λ C such that (S λ L(K. The spectrum σ(s of S is the complement of ρ(s in C. The extended spectrum σ(s of S is defined by σ(s = σ(s if S L(K σ(s = σ(s { } otherwise. We set ρ(s := C\ σ(s. The adjoint S + of S is defined as S + := {( h h [f, h] = [f, h ] for all ( f f S }. S is said to be symmetric (selfadjoint if S 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 [5], [6] without employing the term boundary value space. Definition Let A be a closed symmetric relation in the Krein space (K, [, ]. We say that {G, Γ, Γ } is a boundary value space for A + if (G, (, is a Hilbert space there exist linear mappings Γ, Γ : A + G such that Γ := ( Γ Γ : A + G G is surjective, the relation [f, g] [f, g ] = (Γ ˆf, Γ ĝ (Γ ˆf, Γ ĝ (3 holds for all ˆf = ( f f, ĝ = ( g g A +. If a closed symmetric relation A has a selfadjoint extension  in K with ρ(â, then there exists a boundary value space {G, Γ, Γ } for A + such that  coincides with er Γ (see [4]. For basic facts on boundary value spaces further references see e.g. [3], [4], [5] [6]. We recall only a few important consequences. For the rest of this section let A be a closed symmetric relation assume that there exists a boundary value space {G, Γ, Γ } for A +. Then A := er Γ A := er Γ (4 are selfadjoint extensions of A. The mapping Γ = ( Γ Γ induces, via A Θ := Γ Θ = { ˆf A + Γ ˆf Θ }, Θ C(G, (5 a bijective correspondence Θ A Θ between C(G the set of closed extensions A Θ A + of A. In particular (5 gives a one-to-one correspondence between the closed symmetric (selfadjoint extensions of A the closed symmetric (resp. selfadjoint relations in G. Moreover, A Θ is an operator if only if Θ Γ {( h h mul A + } = {}. (6 4
If Θ is a closed operator in G, then the corresponding extension A Θ of A is determined by A Θ = er ( Γ ΘΓ. (7 Let N λ := er(a + λ = ran (A λ [ ] be the defect subspace of A set ˆN λ := {( f λf f Nλ }. Now we assume that the selfadjoint relation A in (4 has a nonempty resolvent set. For each λ ρ(a the relation A + can be written as a direct sum of (the subspaces A ˆN λ (see [4]. Denote by π the orthogonal projection onto the first component of K 2. The functions λ γ(λ := π (Γ ˆN λ L(G, K, λ ρ(a, λ M(λ := Γ (Γ ˆN λ L(G, λ ρ(a (8 are defined holomorphic on ρ(a are called the γ-field the Weyl function corresponding to A {G, Γ, Γ }. For λ, ζ ρ(a the relation (3 implies M(λ = M(λ γ(ζ = ( + (ζ λ(a ζ γ(λ (9 M(λ M(ζ = (λ ζγ(ζ + γ(λ ( hold (see [4]. Moreover, by [4], we have the following connection between the spectra of extensions of A the Weyl function. Lemma 2 If Θ C(G A Θ is the corresponding extension of A then a point λ ρ(a belongs to ρ(a Θ if only if belongs to ρ(θ M(λ. A point λ ρ(a belongs to σ i (A Θ if only if belongs to σ i (Θ M(λ, i = p, c, r. For λ ρ(a Θ ρ(a the well-nown resolvent formula (A Θ λ = (A λ + γ(λ ( Θ M(λ γ(λ + ( holds (for a proof see e.g. [4]. 3 The Underlying Space Let H,2 (, be the Sobolev space of all absolutely continuous functions f with f L 2 (,. Let be a positive real number, >. We define for 5
f, g H,2 (, [f, g] := (f, g L 2 (, (f, g L 2 (,. (2 If L is an arbitrary subset of H,2 (, we set L [ ] := {x H,2 (, : [x, y] = for all y L}. Then we have the following. Proposition 3 For the space (H,2 (,, [.,.] we have the following properties. ( If equals for some n N, then the function g H,2 (,, defined by n 2 π 2 g(x = cos(nπx belongs to the isotropic part of (H,2 (,, [.,.], that is [f, g] = for all f H,2 (,. (2 If >, then (H,2 (,, [.,.] π 2 is a Pontryagin space with one negative square. (3 If for all n N, then (H,2 (,, [.,.] π 2 n 2 π 2 is a Pontryagin space with a finite number of negative squares. Set H := span {f j j 2 π 2, j N}, where f j H,2 (, is defined by f j (x = sin(jπx. Then the number κ of negative squares of (H,2 (,, [.,.] satisfies κ = dim H +. Proof: Assertion ( is an easy calculation. We assume n 2 π 2 for all n N. Define the operator C by D(C := {g H,2 (, g H,2 (, g( = g( = }, C g := g for g D(C. Let us note that the functions f j (x = sin(jπx, j =, 2..., are eigen functions of C. Moreover, each function g in L 2 (, can be written as g = α j f j, j= Let us note that the expression (y (t 2 y(t 2 with t as the time is (up to a constant the Lagrangian for free small oscillations in one dimension (see [2], p. 58 for details. From this point of view the corresponding integral represents the action. 6
where α j, j =, 2..., are some constants. For g D(C H, where H denotes the orthogonal complement with respect to the usual scalar product (, L 2 (, but within the Hilbert space H,2 (,, we have also that (f, g L 2 (, = for all f H. Thus, g has the representation g = j> α π j f j, where the sum converges in the norm of L 2 (,. This implies that there exists an ɛ > with (C g, g L 2 (, > ( + ɛ(g, g L 2 (, for all g D(C H. Therefore there exists constants c, c > with [g, g] > c(c g, g L 2 (, + ɛ(g, g L 2 (, c(g, g H,2 (, for g D(C H, so D(C H is a uniformly positive linear manifold. It is easy to see that for f H we have [f, f] <. Let us note, that [f j, f l ] = (j 2 π 2 (f j, f l L 2 (, = (j 2 π 2 δ jl, where δ jl is the symbol of Kronecer. This means that (D(C H [ ] H. Thus, D(C is a pre- Pontryagin space with respect to the product [, ], where the number of negative squares equals dim H. Let us note also that the subspace D(C [ ] is two-dimensional. Indeed, we define h, h 2 H,2 (, by h (x = sin( x + cos( x, h 2 (x = sin( x cos( x z, z 2 H,2 (, by z (x = cos( x, z sin 2 (x = cot( cos( x sin( x. For every g H,2 (, we have [g, z ] = g( [g, z 2 ] = g(. Thus, (D(C [ ] = sp {z, z 2 } = sp {h, h 2 }. Moreover, we have [h, h ] = 2 (sin 2 [h 2, h 2 ] = 2 (sin 2. This proves (3. If > π 2, then H = {} the above considerations imply (2. 7
4 A Symmetric Operator Associated to the Second Derivative of Defect Four For the rest of this paper, we assume that is such, that sin. Then, according to Proposition 3, the space (H,2 (,, [.,.] is a Pontryagin space. We consider the following operator A, defined by D(A := {g H,2 (, g, g H,2 (, with g( = g( = g ( = g ( = g ( = g ( = } Ag := g, g D(A. Let us calculate A +. By the definition g D(A + if only if there is g + H,2 (, such that [Af, g] = [f, g + ] for every f D(A. Let f H,2 (, be smooth f(x = for x [, δ (ɛ, ] for some δ, ɛ >. Then the expression f (tg (tdt f (tg(tdt can be considered as an action of the generalized function g IV g on the test function f. From the other h the expression [f, g + ] can be consider as an action of the generalized function (g + g + on the same probe function f. Thus, g IV (t + g (t = (g + (t + g + (t. The latter yields where f, f 2 are defined by g + (t = g (t + αf (t + βf 2 (t, (3 f (x = sin x, f 2 (x = cos x, g H,2 (,. (4 Now the direct calculation shows that the equality [Af, g] = [f, g + ] is fulfilled for every f D(A, if g is under Condition (4 g + is defined by (3. So, A + = {( g g + ( αf +βf 2 g, g H,2 (,, α, β C }. mul A + = sp {f, f 2 }, (5 Lemma 4 Then A is a closed symmetric operator in (H,2 (,, [.,.]. 8
Proof: Obviously, A is symmetric. The best way to show the closeness is via the calculation of A ++. We leave it to the reader. Let ( ( f g f +α f +β f 2 g +α 2 f +β 2 f 2 be elements from A + with f, g D(A α, α 2, β, β 2 C. Then we have [ f +α f + β f 2, g] [f, g + α 2 f + β 2 f 2 ] = = fg +f g f g +f g + + (α f 2 β f g (α 2 f 2 β 2 f. We define mappings Γ, Γ : A + C 4 by ( f(+f ( f Γ f +α f +β f 2 = f(+f ( f( f( f ( ( f f Γ f +α f +β f 2 = ( α (α cos β sin for ( f f +α f +β f 2 A +. Theorem 5 The triplet {Γ, Γ } is a boundary value space for A +. In particular A := er Γ is an operator a selfadjoint extension of A, i.e. D(A := {g H,2 (, g, g H,2 (, with g ( = g ( = } A g := g, g D(A. Moreover, for λ ρ(a, the Weyl function is given by M(λ = ( λ λ tan λ tan ( λ λ sin + λ sin tan ( λ λ λ sin + λ sin ( λ tan λ tan sin tan tan sin λ tan sin sin tan λ sin λ sin λ tan Proof: The above calculations imply that {Γ, Γ } is a boundary value space for A +. Let λ C \ R. Define g, g 2 H,2 (, by g (x = cos( λx g 2 (x = sin( λx. (6 Then we have er (A + λ = sp {g, g 2, f, f 2 }. 9
Let f = αg + βg 2 + γf + δf 2 for some α, β, γ, δ C. Then Γ ( f λf = Γ ( f f +γ(λ f +δ(λ f 2 = α( λ α( λ cos λ+β( λ sin λ α+δ α cos λ+β sin λ+γ sin +δ cos ( Γ f λf = β λ γ α λ sin λ+β λ cos λ+γ cos δ sin γ (λ γ (λ cos δ (λ sin Now, by (8, it follows that M is of the above form. Now, via (5 we can parameterize all selfadjoint extensions of A via all selfadjoint relations Θ in C 4. Theorem 6 Let Θ be a selfadjoint relation in C 4. Then A Θ is a selfadjoint extension of A. If for all α, β C ( α β sin α cos / mul Θ \ {} (7 holds, then A Θ is an operator. If, in particular, Θ is a selfadjoint matrix, then A Θ is a selfadjoint operator an extension of A with domain D(A Θ := g H,2 (, g, g H,2 (,, ( g ( g ( g(+g ( = Θ g(+g ( g(. g( Proof: Relation (7 follows from (6, (5 the definitions of Γ Γ. If Θ is a matrix, (7 is satisfied the description of D(A Θ follows from (7. 5 A Symmetric Operator Associated to the Second Derivative of Defect Two We start this Section opposite to Section 4. For this we put D(à := {g H,2 (, g, g H,2 (, } (8 Ãg := g, g D(Ã. Thus, the operator à 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 Ã+. For f, g D(Ã we have [Ãf, g] = f (tg (tdt + f (tg(tdt = = ( f (tg (t f (tg (t + ( f (tg(t f(tg (t f (tg (tdt + f(tg (tdt = = ( f ( + f( g ( + f ( ( g ( + g( f ( ( g ( + g( + ( f ( + f( g ( + [f, Ãg] Note that the maps f(t ( f ( + f(, f(t f (, f(t ( f ( + f( f(t f ( represent unbounded linear functionals on H,2 (,. Thus, the expression [Ãf, g] gives a continuous linear functional (with respect to f on H,2 (, if only if g ( = ( g ( + g( = ( g ( + g( = g ( = by the definition of the adjoint operator the latter conditions restrict the domain of Ã+. For brevity below we set A: = Ã+. Thus, we have the following operator A, defined by D(A := {g H,2 (, g, g H,2 (, with g ( = g ( =, Ag := g, g( + g ( = g( + g ( = } g D(A. Then A is a closed symmetric operator in (H,2 (,, [.,.], which is, in contrast to Section 4, densely defined. In particular A + = Ã = {( g g g, g H,2 (, } is an operator therefore all selfadjoint extensions of A are operators. We define mappings Γ, Γ : A + C 2 by ( ( Γ f f = f(+f ( f(+f ( ( ( Γ f f = f ( f ( for ( f f A +. Theorem 7 The triplet {Γ, Γ } is a boundary value space for A +. The Weyl function is given by M(λ = λ ( λ tan λ λ ( λ sin λ λ ( λ sin λ λ ( λ tan λ, λ ρ(a. Proof: The above calculations imply that {Γ, Γ } is a boundary value space
for A +. Let λ C \ R g, g 2 H,2 (, as in (6. Then we have er (A + λ = sp {g, g 2 }. Let f = αg + βg 2 for some α, β C. Then ( ( ( Γ f λf = f α( λ Γ f = α( λ cos λ+β( λ sin λ ( ( ( Γ f λf = f Γ f = β λ α λ sin λ+β λ cos λ Now, by (8, it is follows that M is of the above form. Lemma 8 The operator A = er Γ is a selfadjoint extension of A with a compact resolvent σ(a = σ p (A = {, π 2, 4π 2, 9π 2, 6π 2,...}. (9 Proof: The operator A = er Γ is selfadjoint in the Hilbert space H,2 (,. We have for f D(A ((A + If, f H,2 (, = f 2 L 2 (, + f 2 H 2,2 (,, where H 2,2 (, is the Sobolev space of all functions f H,2 (, with f H,2 (,. This gives f 2 H 2,2 (, (A + If H,2 (, f H 2,2 (,. Therefore, as the embedding of H 2,2 (, into H,2 (, is compact, the selfadjoint operator A has a compact resolvent. By ( the difference between the resolvents of A A is of finite ran, hence A has a compact resolvent. We have σ(a = σ p (A. Now (9 follows from a simple calculation. Proposition 9 Let α R, α Then the operator A α defined by D(A α := {g H,2 (, g, g H,2 (, with α < 2. (2 αg ( = g( + g ( αg ( = g( + g (} A α g := g, g D(A α. is a selfadjoint extension of A with non-real eigenvalues. 2
In the case α = 2 we have that the selfadjoint extension A 2 of A has a Jordan chain of length two corresponding to the eigenvalue. Proof: Set Θ = α.. α Then A Θ = A α, hence, by Lemma 2 the fact that σ(a R (see Lemma 8, we have for all non-real λ that λ σ p (A α if only if = det (M(λ Θ = 2 ( λ 2 + λα2 α 2 ( λ 2 2λ 2 +. (2 2 Hence, λ,2 = α2 2 ± α α 2 2 2 4 are the solutions of Equation (2. Assertion (2 implies now the existence of two non-real eigenvalues of A α. In the case α = 2 we have that the functions h, h D(A 2 given by h (x = e x h (x = x 2 ex satisfy ( A 2 + h = h ( A 2 + h =, i.e. {h, h } is a Jordan chain of A 2 corresponding to the eigenvalue. References [] T.Ya. Azizov, I.S. Iohvidov: Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Ltd., Chichester, 989. [2] J. Bognár: Indefinite Inner Product Spaces, Springer Verlag, New Yor- Heidelberg, 974. [3] V.A. Derach: On Weyl Function Generalized Resolvents of a Hermitian Operator in a Krein Space, Integral Equations Operator Theory 23 (995, 387-45. [4] V.A. Derach: On Generalized Resolvents of Hermitian Relations in Krein Spaces, J. Math. Sci. (New Yor 97 (999, 442-446. [5] V.A. Derach, M.M. Malamud: Generalized Resolvents the Boundary Value Problems for Hermitian Operators with Gaps, J. Funct. Anal. 95 (99, -95. 3
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