Factorisation of Non-Negative Fredholm Operators ... - Semantic Scholar

Factorisation of Non-Negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators Sergio Albeverio, Rostyslav Hryniv, Yaroslav Mykytyuk no. 385

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungsgemeinschaft getragenen Sonderforschungsbereichs 611 an der Universität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2008

FACTORISATION OF NON-NEGATIVE FREDHOLM OPERATORS AND INVERSE SPECTRAL PROBLEMS FOR BESSEL OPERATORS S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK Abstract. We study the problem of factorisation of non-negative Fredholm operators acting in the Hilbert space L2 (0, 1) and its relation to the inverse spectral problem for Bessel operators. In particular, we derive an algorithm of reconstructing the singular potential of the Bessel operator from its spectrum and the sequence of norming constants.

1. Introduction In the Hilbert space L2 (0, 1), we consider an operator S = I + F , where F is of the Hilbert–Schmidt class. It is well known [9,16] that if S is positive, then it can uniquely be factorised in the form (1.1)

S = (I + K)−1 (I + K ∗ )−1 ,

where K is a Volterra integral operator of the Hilbert–Schmidt class with uppertriangular kernel k (i.e., k(x, t) = 0 for a.e. (x, t) satisfying 0 ≤ t ≤ x ≤ 1). The kernel k can be found from the equation Z 1 (1.2) k(x, t) + f (x, t) + k(x, s)f (s, t) ds = 0, 0 ≤ x < t ≤ 1, x

where f is the kernel of F . In the special case where f has the form f (x, t) = φ(2 − x − t) ± φ(|x − t|) for some function φ ∈ L2 (0, 2), equation (1.2) is called the Gelfand–Levitan–Marchenko (GLM) equation, and if f (x, t) = φ(|x − t|), then (1.2) is the Krein equation. These equations naturally arise while solving the inverse spectral problems for the Sturm–Liouville operators d2 +q dx2 on the interval (0, 1) subject to suitable boundary conditions; then F is constructed from the spectral data for T , and I + K is the transformation operator between the unperturbed (q = 0) and perturbed (q 6= 0) Sturm–Liouville operators, see Subsection 4.2. Moreover, under suitable regularity assumptions on the potential q the kernel k is continuous in the domain {(x, t) | 0 ≤ x ≤ t ≤ 1}, and the potential q is related to k T =−

Date: 04 February 2008. 2000 Mathematics Subject Classification. Primary 47A68, 34A55; Secondary 34B24, 34B30, 47E05. Key words and phrases. Factorisation in operator algebras, nonnegative operators, Bessel operators, inverse problems. 1

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S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK

via dk(x, x) . dx On the other hand, there are Sturm–Liouville operators with singular potentials, for which the above reconstruction procedure is impossible because the corresponding operator I + F is only non-negative and has a non-trivial nullspace. One such an example is given by the so called Bessel operators arising as follows. It is well known (see, e.g., Example 4 of Appendix to X.1 of [14]) that a radial Schr¨odinger operator −∆ + q(|x|) considered in the unit ball of R3 with q supported on (0, 1) decomposes into the direct sum of the Bessel operators Tm corresponding to the angular momenta m ∈ Z+ , i.e., the operators generated in L2 (0, 1) by the differential expressions d2 m(m + 1) Tm := − 2 + +q dx x2 and suitable boundary conditions. If one tries to follow the same classical approach to solve the inverse spectral problem for the Bessel operators Tm with m > 0 as for the Sturm–Liouville case m = 0, then one immediately encounters the problem that the operator S constructed via the spectral data for Tm is non-negative and has a non-trivial nullspace. Therefore the representation (1.1) is clearly impossible, and the very existence of the transformation operators is questionable. However, such an operator S (i.e., nonnegative and with finite-dimensional nullspace) might still be factorisable as q(x) = −2

(1.3)

S = S+ S+∗ ,

where S+ is an upper-triangular operator, see definitions in Section 3. This question is studied in detail in Section 3, and the main result there (Theorem 3.7) states that such a representation is always possible and, moreover, it is even unique if S+ is to belong to some smaller subalgebra of operators. In Section 4 we study properties of the factors S± in the case where S is constructed from the spectral data for some Bessel operator Tm , m > 0, as explained above. We prove in Theorem 4.2 that then the factor S+ might be used to reconstruct the potential q, as in the regular case of Sturm–Liouville operators. This leads to the new algorithm of solving the inverse spectral problems for Bessel operators with momenta m ∈ N described in Section 4. 2. Factorisation in operator algebras 2.1. Definitions. In the Hilbert space H := L2 (0, 1), we introduce the chain of orthoprojectors P (t), t ∈ [0, 1], given by
P (t)f (x) := χt (x)f (x), where χt is the characteristic function of the interval [0, t]. A bounded operator S+ (resp. S− ) is called an upper-triangular (resp. lower-triangular ) operator if, for all t ∈ [0, 1],

I − P (t) S+ P (t) = 0, (resp., P (t)S− I − P (t) = 0. All bounded upper-triangular (resp. lower-triangular) operators constitute a closed subalgebra B + (resp. B − ) in the Banach algebra B = B(H) of all bounded linear operators in H. It is clear that the involution A 7→ A∗ maps B + (resp. B − ) into B − (resp. B + ).

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Definition 2.1. Let A be a subalgebra of B. We say that an operator S ∈ A admits factorisation, or is factorisable in A if there exist S+ ∈ A ∩ B + and S− ∈ A ∩ B − such that S = S+ S− . Clearly, this notion is an infinite-dimensional generalisation of the Gauss method of inverting a square matrix using its LU -decomposition, i.e., the representation as the product of lower- and upper-triangular matrices. M. Krein [10, 11] was seemingly the first to consider the factorisation problem in an infinite-dimensional algebra—namely, in the algebra of operators of the form zI + K, where z ∈ C, I is the identity operator, and K is an integral operator with continuous kernel. A more general algebra A = {zI + B | B ∈ B∞ }, with B∞ denoting the ideal of all compact operators was studied in detail in the book by I. Gokhberg and M. Krein [9], and the case of a Banach space H was considered in [3]. L. Sakhnovich in [15, 16] investigated factorisation in the group Binv of all invertible operators; he required that the factors S+ and S− in S = S+ S− be in addition invertible in B + and B − and called such a factorisation the special factorisation. We observe that also in [3, 9–11] only invertible operators were considered. 2.2. The necessary condition for special factorisability. In all cases above, there is a simple necessary condition for special factorisability. Indeed, assume that an operator S admits a factorisation S = S+ S− , in which S+ and S− are invertible in B + and B − respectively. We set, for every t ∈ [0, 1],
S+ (t) := P (t) + I − P (t) S+ ∈ B + ,
S− (t) := P (t) + S− I − P (t) ∈ B − ,

S(t) := P (t) + I − P (t) S I − P (t) . It is easy to see that S(t) = S+ (t)S− (t) and that the operators S+ (t) and S− (t) are boundedly invertible, with
−1
S+ (t) = P (t) + I − P (t) S+−1 ,
−1
S− (t) = P (t) + S−−1 I − P (t) . Hence if an invertible operator S ∈ Binv admits a special factorization, then the following condition holds: (I) for every t ∈ [0, 1], the operator S(t) is boundedly invertible in the algebra B. For a generic algebra A condition (I) is by no means sufficient in order that an operator S ∈ A be factorisable in A . However, (I) is sufficient, e.g., in the algebras Ap := CI + Bp := {zI + B | z ∈ C, B ∈ Bp }, where Bp is the Schatten–von Neumann ideal. In particular, we have the following statement: Proposition 2.2. Assume that F is a Hilbert–Schmidt operator such that S := I + F is positive. Then S admits special factorisation in the algebra A2 . Some other non-classical operator algebras in which (I) is sufficient for factorisability were given in [12,13], where also methods for constructing such algebras were presented.

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2.3. Uniqueness. Assume that S ∈ A admits two special factorisations in the algebra A , S = S+ S− = S˜+ S˜− . By definition, the factors S+ and S˜+ are invertible in A ∩ B + , and S− and S˜− are invertible in A ∩ B − ; therefore the operator S˜−1 S+ = S˜− S −1 −

+

belongs to the subalgebra D := B ∩ B of diagonal operators. Every diagonal operator D is in fact the operator of multiplication by a function d ∈ L∞ (0, 1). Every diagonal operator D that is invertible in A causes therefore non-uniqueness of factorisation in A via S = S+ S− = (S+ D)(D−1 S− ). +

−

Conversely, if the only diagonal operators in A that are invertible in A are the scalar operators (i.e., the operators zI for z ∈ C \ {0}), then every S ∈ A can admit at most one special factorisation in A (modulo scalar factors). These scalar factors can actually be fixed by requiring that the factors are in the set BI of operators having diagonal equal to the identity operator I. We say that an operator A ∈ B has diagonal D ∈ D if (2.1)

s-lim d(τ )→0

n X

∆j P A∆j P = D,

j=1

where τ = {0 =: t0 < t1 < · · · < tn := 1} is a partition of the interval [0, 1], d(τ ) := max{tj − tj−1 | j = 1, . . . , n} is its diameter, ∆j P := P (tj ) − P (tj−1 ), and the limit is taken over all possible partitions. It can be shown that every compact operator has diagonal equal to zero: we establish this for rank one operators first, then for their finite linear combinations, and then use density of finite-rank operators in B∞ . Corollary 2.3. In Proposition 2.2 there is a unique factorisation with factors belonging to the sets BI± := B ± ∩ BI . Indeed, in view of the above elements of A2 that belong to BI have the form I + K with K ∈ B2 . Moreover, if I + K ∈ A2 ∩ BI± and I + K is invertible in B, then the inverse (I + K)−1 =: I + K1 also belongs to A2 ∩ BI± . The inclusion K1 ∈ B2 follows from the equality K1 = −K(I + K1 ), and that K1 ∈ B ± can be proved directly by studying the integral equation Z 1 k(x, t) + k1 (x, t) + k(x, s)k1 (s, t) ds = 0 0

for the corresponding kernels. Therefore every factorisation in A2 of a positive operator is a special one, and the claim follows from the above considerations. 3. Factorization of non-negative Fredholm operators In this section, we consider the factorisation problem for non-invertible Fredholm operators, which, to the best of our knowledge, has not yet been studied. We denote by F the algebra of all Fredholm operators in B and set F ± := F ∩B ± , FI := F ∩BI , and FI± := F ± ∩ FI . We shall start with the simplest situation of factorisation of orthogonal projectors with one-dimensional nullspace.

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3.1. Factorisation of orthogonal projectors with one-dimensional nullspace. Let φ be a function in L2 (0, 1) of norm 1 whose support contains 0. We set Z x a(x) := |φ(t)|2 dt 0

and introduce an operator Vφ in L2 (0, 1) via Z

1

(Vφ y)(x) := y(x) − φ(x) x

φ(t)y(t) dt. a(t)

Lemma 3.1. The operator Vφ is isometric and Vφ Vφ∗ = I − Pφ , where Pφ := (·, φ)φ is the orthogonal projector onto φ. Proof. Direct calculations give Z 1 Z (Vφ f, Vφ g) = f (t)g(t) dt − 0

1

1

dxf (x)φ(x)

0

Z

1

−

Z

x 1

φ(t)g(t) dt a(t)

φ(t)f (t) dt a(t) x Z 1 Z 1 φ(t)f (t) φ(t)g(t) 2 dx|φ(x)| dt dt. a(t) a(t) x x dxφ(x)g(x)

0

Z + 0 2

Z

1

0

Observing that |φ(x)| = a (x) and integrating by parts in the last integral, we see that the last three integrals sum up to zero, thus showing that Vφ is isometric. The equality Vφ∗ y = 0 implies that Z a0 (x) x φ(x)y(x) = φ(t)y(t) dt, a(x) 0 i. e., by integrating, that Z x

φ(t)y(t) dt = ca(x) 0

for some constant c, and finally, upon differentiating, that y = cφ. It thus follows that ker Vφ∗ is spanned by φ. Since Vφ Vφ∗ is the orthogonal projector onto ker Vφ∗ , we get Vφ Vφ∗ = I − Pφ as claimed.  To prove that Vφ is the only isometric operator from BI+ that factorises I − Pφ in the above sense, we establish first two auxiliary results. Lemma 3.2. Assume that U ∈ B and that φ is a unit vector in L2 (0, 1) whose support contains 0. If Vφ U belongs to B + , then U belongs to B + , too. Proof. According to the definition of the subalgebra B + we have to show that, for every t ∈ [0, 1],
I − P (t) U P (t) = 0. Since Vφ U ∈ B + by assumption, we have for every t ∈ [0, 1]
I − P (t) Vφ U P (t) = 0. Using the fact that Vφ belongs to B + , we rewrite the above equality as h i

ih
(3.1) I − P (t) Vφ I − P (t) I − P (t) U P (t) = 0.

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The operator At :=

h



i I − P (t) Vφ I − P (t)

, L2 (t,1)

being equal to the identity operator plus a Volterra one, is invertible in L2 (t, 1); hence (3.1) yields
I − P (t) U P (t) = 0 for every t ∈ [0, 1], and U ∈ B + as required.



Lemma 3.3. Under the assumptions of the above lemma, the operator A := Vφ U − U has diagonal equal to zero. Proof. For B ∈ B and f ∈ H, we set n

X

∆j P B∆j P f , S(B, f ) := lim sup d(τ )→0

j=1

with notations explained after the displayed formula (2.1). It is easily seen that for every B and C in B and f ∈ H, one gets S(B, f ) ≤ kBkkf k, (3.2)

S(B + C, f ) ≤ S(B, f ) + S(C, f ), S(P (t)B, f ) = S(B, P (t)f ),

t ∈ [0, 1].

Fix an arbitrary t ∈ [0, 1) and observe that the operator (I −P (t))(Vφ −I) is compact. Therefore for such t compact is also the operator At := (I − P (t))A, so that At has diagonal zero and S(At , f ) = 0 for every f ∈ H. Using properties (3.2), we conclude that S(A, f ) ≤ S(P (t)A, f ) = S(A, P (t)f ) ≤ kAkkP (t)f k. Since t ∈ [0, 1) is arbitrary, it follows that S(A, f ) = 0, i.e., that A has diagonal zero.  Corollary 3.4. If, under the above assumptions, one of the operators Vφ U or U belongs to BI , then so does the other one. Corollary 3.5. If U is an isometric operator in FI+ such that U U ∗ = I − Pφ , then U = Vφ . Proof. Taking into account the relations ran U = ran Vφ = ran(I − Pφ ), we conclude that the operator B := Vφ∗ U is unitary and Vφ B = (I − Pφ )U = U . By Lemma 3.2 and Corollary 3.4, the latter equality implies that B ∈ BI+ , and unitarity of B then yields that B ∈ BI+ ∩ BI− . Thus B = I, i.e., U = Vφ . 

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3.2. Factorisation of orthogonal projectors with finite-dimensional nullspace. We extend now the results of the previous subsection to Fredholm orthogonal projectors as follows: Theorem 3.6. Assume that P is an orthoprojector in L2 (0, 1) of finite rank. Then there exists a unique isometric operator UP in FI+ such that UP UP∗ = I − P. Proof. The proof is by induction on rank P . Lemma 3.1 and Corollary 3.5 handle the case where rank P = 1. We assume next that, for some n ∈ N, the theorem is already proved whenever rank P ≤ n and let P be an orthoprojector of rank n + 1. Fix an element φ ∈ ran P of unit norm and set Q := Vφ∗ P Vφ . Since Q = Q∗ and Q2 = Vφ∗ P (I − Pφ )P Vφ = Q by Lemma 3.1, Q is an orthoprojector on ker Q. Taking into account that Vφ is injective and ran Vφ = L2 (0, 1) φ, we see that Q is of rank n. We set now UP := Vφ UQ , where UQ is a unique isometric operator in FI+ that satisfies UQ UQ∗ = I − Q, whose existence is guaranteed by the hypothesis of induction. The operator UP belongs to F + and is obviously isometric. Since UQ has diagonal equal to I, so does UP by Corollary 3.4, i.e., UP ∈ FI+ . Next we see that UP UP∗ = Vφ UQ UQ∗ Vφ∗ = Vφ (I − Q)Vφ∗ = Vφ Vφ∗ (I − P )Vφ Vφ∗ = (I − Pφ )(I − P )(I − Pφ ) = I − P, as required. Assuming that there is another isometry U˜P in FI+ such that U˜P U˜P∗ = I − P and putting U˜Q := Vφ∗ U˜P , we see that U˜Q U˜Q∗ = Vφ∗ U˜P U˜P∗ Vφ = Vφ∗ (I − P )Vφ = I − Q. Since ran U˜P = ran(I − P ), the equalities Vφ U˜Q = (I − Pφ )U˜P = U˜P and Lemma 3.2 imply that U˜Q belongs to BI+ ; moreover, U˜Q is an isometry as U˜Q∗ U˜Q = U˜P∗ (I − Pφ )U˜P = I. We showed above that Q is an orthoprojector of rank n, so that by the assumption of induction we have U˜Q = UQ resulting in U˜P = Vφ U˜Q = Vφ UQ = UP . The proof is complete.  3.3. Factorisation of non-negative Fredholm operators. Theorem 3.7. Assume that F is a self-adjoint operator of Hilbert–Schmidt class such that I + F is non-negative. Then I + F admits a factorisation (3.3)

I + F = S+ S−

with S± ∈ BI± ; moreover, such a factorisation is unique and S− = S+∗ . Denote by P the orthoprojector on ker(I + F ); then S+ = UP + K+ , where UP was introduced in Theorem 3.6 and K+ ∈ B + is a Hilbert–Schmidt operator.

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Proof. By Theorem 3.6, we have I − P = UP UP∗ , so that I + F = UP UP∗ (I + F )UP UP∗ . It is easily seen that the operator U˜ = UP∗ (I + F )UP = I + UP∗ F UP is strictly positive and thus admits a unique factorisation as U˜ = S˜+ S˜− ˜ + := S˜+ − I ∈ B2 and S˜− = S˜∗ . Now we conclude with S˜± ∈ BI± ; moreover, we have K + that I + F = S+ S− with S+ := UP S˜+ ∈ BI+ and S− := S˜− UP∗ = S+∗ ∈ B − . If a factorisation of the form (3.3) were not unique, then U˜ could be factorised non-uniquely, which is impossible. The theorem is proved.  4. Application to the inverse spectral problem for Bessel operators 4.1. Special property of Vφ . A careful examination of the transformation Vφ studied in Subsection 3.1 reveals that it is inverse to the well known double commutation transformation used in the spectral analysis of the Sturm–Liouville and Dirac operators [1,2,7,8,19]. This suggests that the factorisation problem discussed in the previous section is intimately related to Sturm–Liouville and Dirac differential operators and might be of much use in the spectral analysis of the latter. We demonstrate this for the Sturm–Liouville case. Assume that q0 ∈ L2 (0, 1) is real-valued and that a real-valued function φ0 of unit norm in L2 (0, 1) obeys the terminal condition φ0 (1) = 0 and is a solution of the differential equation −y 00 + q0 y = λ20 y with a real λ0 . We set q1 := q0 − 2(log a)00 ,

(4.1) where a(x) := equation

Rx 0

|φ0 (t)|2 dt, and, for every λ ∈ C, denote by u(·, λ) a solution of the

−u00 + q1 u = λ2 u satisfying the terminal conditions u(1) = 0 and u0 (1) = 1. Lemma 4.1. Under the above assumptions, for every λ ∈ C the function Z 1 φ0 (t)u(t, λ) (4.2) v(·, λ) := Vφ0 u(·, λ) = u(x, λ) − φ0 (x) dt a(t) x verifies the relations v(1) = 0 and v 0 (1) = 1 and solves the equation −y 00 (x) + q0 y(x) = λ2 y(x). Proof. We start by observing that the function φ0 /a is collinear to u(·, λ0 ). Indeed, direct calculations give  φ 00 φ00 a0 φ00  a0 0 φ0  a0 2 φ0 0 = 0 −2 − + , a a a a a a a a

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which in view of the relations 2a0 φ00 = a00 φ0 and φ000 = (q0 − λ20 )φ0 implies that  φ 00   φ0 0 00 2 φ0 = q0 − 2(log a) − λ0 = (q1 − λ2 ) . a a a It remains to observe that φ0 /a satisfies along with φ0 the Dirichlet condition at x = 1. Next, it is clear that v(·, λ) obeys the stated conditions at x = 1. Differentiating (4.2), we get Z 1 φ0 (t)u(t, λ) a0 (x) 0 0 0 v (x, λ) = u (x, λ) − φ0 (x) dt + u(x, λ), a(t) a(x) x and then, using the fact that u(·, λ) and φ0 satisfy the corresponding differential equations, we find that Z 1 φ0 (t)u(t, λ) 00 00 00 v (x, λ) = u (x, λ) − φ0 (x) dt a(t) x  a0 (x) 0 1 a00 (x) a0 (x) 0 + u(x, λ) + u(x, λ) + u (x, λ) 2 a(x) a(x) a(x) Z 1

φ0 (t)u(t, λ) 2 2 dt = q1 (x) − λ u(x, λ) − q0 (x) − λ0 φ0 (x) a(t) x  a0 (x) 0 1 a00 (x) a0 (x) 0 + u(x, λ) + u(x, λ) + u (x, λ) 2 a(x) a(x) a(x) Z 1 φ0 (t)u(t, λ) 2 2 2 = (q0 − λ )v(x, λ) + (λ0 − λ )φ0 (x) dt a(t) x  a0 (x) 0 a0 (x) 0 1 a00 (x) u(x, λ) − u(x, λ) + u (x, λ). + 2 a(x) a(x) a(x) Recalling that φ0 /a is collinear to u(·, λ0 ) and using the Lagrange identity, we derive the relation Z 1  φ (x) 0 φ0 (t)u(t, λ) φ0 (x) 0 0 2 2 (λ0 − λ ) dt = u(x, λ) − u (x, λ) a(t) a(x) a(x) x which, on account of the Riccati identity (a0 /a)0 = a00 /a − (a0 /a)2 , shows that the last four summands above cancel out. Therefore v 00 (x, λ) = (q0 − λ2 )v(x, λ), and the proof is complete.  4.2. Solution of the classical Sturm–Liouville inverse spectral problem. We recall first in some more detail the classical method of reconstruction of a Sturm– Liouville operator d2 S(θ, q) := − 2 + q dx subject to the boundary conditions cos θy(0) − sin θy 0 (0) = y(1) = 0 from its spectral data, the sequences of eigenvalues and norming constants. To be specific, we shall consider the cases θ = 0 or θ = π/2 and shall assume that the potential q is real-valued and belongs to L2 (0, 1). For every λ ∈ C, we denote by y(·, λ) a solution to the equation −y 00 + qy = λ2 y subject to the terminal conditions y(1) = 0 and y 0 (1) = λ. Then y(·, λ) has the

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S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK

representation Z (4.3)

1

k(x, t) sin λ(t − 1) dt,

y(x, λ) = sin λ(x − 1) + x

where k is the kernel of the corresponding transformation operator I + K. The kernel k has the property that, for every fixed x ∈ [0, 1), the function k(x, ·) belongs to the Sobolev space W21 (x, 1). The eigenvalues λ2n of the operator S(0, q) are then squared zeros of the characteristic function Z 1 k(0, t) sin λ(1 − t) dt ∆(λ) := sin λ + 0

and have the asymptotics ˜n, λ2n = π 2 n2 + A + λ

(4.4)

R1 ˜ n ). For the Neumann boundary condition with A := 0 q(t) dt and an `2 -sequence (λ at x = 0 (θ = π/2) the above asymptotics is shifted by −π/2. For the eigenvalue λ2n , we denote the norming constant αn as the squared L2 -norm of the eigenfunction y(·, λn ), i.e., Z 1 |y(x, λn )|2 dx. (4.5) αn := 0

Using the properties of the operator K, it is easy to prove that the αn have the form ˜n 1 α + , 2 n where the numbers α ˜ n form a sequence from `2 . Now the resolution of the identity for the operator S(0, q) reads (4.6)

αn =

I = s-lim N →∞

N X


αn−1 ·, yn yn ,

n=1

with yn := y(·, λn ). Setting sn (x) := sin λn (x − 1) and using the representation (4.3), we conclude that N h X
i −1 I = (I + K) s-lim αn ·, sn sn (I + K ∗ ). N →∞

n=1

The operator in the square brackets is uniformly positive in L2 (0, 1) and has the form I + F , where F is a Hilbert–Schmidt operator with kernel (4.7)

f (x, y) = ϕ(2 − x − y) − ϕ(|x − y|),

where (4.8)

ϕ(s) :=

i 1 Xh 2 cos πks − αk−1 cos λk s 2 k∈N

is a function from W21 (0, 2). Setting I + L := (I + K)−1 , we get I + F = (I + L)(I + L∗ );

FACTORISATION OF NON-NEGATIVE FREDHOLM OPERATORS

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since I + L belongs to BI+ , by Corollary 2.3 this is a unique factorisation of I + F with factors in BI+ . Finally, we recall that the potential q of the operator S(0, q) can be reconstructed through dl(x, x) (4.9) q(x) = 2 , dx where l is the kernel of L; it is proved in the classical inverse theory that the function l(x, x) belongs to W21 (0, 1) and thus the above formula yields a function from L2 (0, 1) as required. Summarising, we arrive at the following reconstruction algorithm: first, given the spectral data, we construct the function ϕ and the operator F , then factorise the operator I + F , call the factor I + L, and finally use the kernel l of L to determine q via (4.9). It is worth noting that any two sequences of real numbers λ2n and αn such that λ2n strictly increase and satisfy the asymptotics of (4.4) and αn are positive and satisfy (4.6), are sequences of eigenvalues and norming constants of a unique Sturm– Liouville operator S(0, q) with q ∈ L2 (0, 1). The method of finding this q is precisely the same as above. 4.3. Reconstruction in the Bessel case. Assume now that m ∈ N and q ∈ L2 (0, 1) is real-valued and consider the Bessel operator T (m; q) given by the differential expression d d2 m(m + 1) m  d m t(m; q) := − 2 + +q =− − + +q dx x2 dx x dx x subject to the Dirichlet boundary condition at x = 1. The differential expression t(m; q) is well defined on the set of functions y that together with their quasi-derivatives y [1] := y + (m/x)y are absolutely continuous on [ε, 1] for every ε ∈ (0, 1). It is well known [1, 4–6, 17] that being considered on the domain dom T (m; q) := {y ∈ dom t(m; q) ∩ L2 (0, 1) | t(m; q)y ∈ L2 (0, 1), y(1) = 0} the operator T (m; q) becomes self-adjoint, bounded below, and has a discrete spectrum. As earlier, for a nonzero λ ∈ C, we denote by y(·, λ) a solution of the equation m(m + 1) y + q(x)y = λ2 y x2 satisfying the terminal conditions y(1) = 0 and y 0 (1) = λ. The function y(x, λ) either vanishes at x = 0 or has there a pole. In the former case λ2 is an eigenvalue of T (m; q) (and thus is real) and y(·, λ) is a corresponding eigenfunction. We enumerate the eigenvalues λ21 < λ22 < . . . in increasing order and recall [4–6, 17] the asymptotic relation
2 ˜n, (4.10) λ2 = π 2 n − m + A + λ −y 00 +

n

2

˜ n ). Without loss of generality we assume that λ2 > 0 with A ∈ R and an `2 -sequence (λ 1 as otherwise we can shift the spectrum by adding a suitable constant to q. Slightly modifying the arguments of [1, Lemma 2.2], we can show that the eigenfunctions y(·, λn ) have the form (4.11)

y(x, λn ) = xm+1 un (x),

where un is a function from W22 (0, 1) that does not vanish at x = 0. Next, we introduce the norming constants αn corresponding to λ2n via (4.5); then [1, 6] αn have the same asymptotics as in the case m = 0.

12

S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK

Keeping the notations of the previous subsection, we introduce the function ϕ of (4.8), the kernel f of (4.7), and the corresponding integral operator F . Then I + F = s-lim N →∞

N X


αn−1 ·, sn sn

n=1

is nonnegative but in view of the asymptotic behaviour of λn it has a nontrivial kernel of dimension [(m + 1)/2], [a] denoting the integral part of a number a. By Theorem 3.7 there is a unique operator S+ ∈ BI+ such that I + F = S+ S+∗ . The operator S+ has the form I + L with L an “almost” Hilbert–Schmidt operator in the sense that L(I − P (t)) is of Hilbert–Schmidt class for every t > 0. The properties of the kernel l of the operator L are given in the following theorem. Theorem 4.2. Under the above assumptions, (I + L)y(x, λ) = sin λx, i.e., I + L is the transformation operator between the Bessel operator T (m, q) and the unperturbed Sturm–Liouville operator T (0, 0). Moreover, the kernel l is continuous if x > 0, the function l(x, x) belongs to W21 (ε, 1) for every ε > 0, and m(m + 1) dl(x, x) + q(x) = 2 . 2 x dx Proof. The proof is by induction on m ∈ N; moreover, we have to consider separately the cases of even and odd m. Even m: base of induction. We start with the case m = 2 and fix arbitrary positive λ20 and α0 such that λ20 < λ21 . By the results of Subsection 4.2 the sets {λ2n }n≥0 and {αn }n≥0 are sets of eigenvalues and norming constants for a unique Dirichlet Sturm– Liouville operator d2 T (0, 0, q0 ) := − 2 + q0 , dx −1 with some q0 ∈ L2 (0, 1). Put F0 := F + α0 (·, s0 )s0 , with s0 (x) := sin λ0 (x − 1); then I + F0 = (I + L0 )(I + L∗0 ), where I + L0 ∈ BI+ , the kernel l0 of L0 is continuous on Ω, and dl0 (x, x) . dx The operator I + L0 is the transformation operator between T (0, q0 ) and T (0, 0), i.e., if v(·, λ) is the solution of the equation `(0, q0 )v = λ2 v subject to the terminal conditions v(1) = 0 and v 0 (1) = 1, then (I + L0 )v(x, λ) = λ−1 sin λ(x − 1). Next we find that   I + F = (I + L0 ) I − (·, φ0 )φ0 (I + L∗0 ), q0 (x) = 2

where φ0 := α−1/2 (I +L0 )−1 s0 is an eigenfunction of the operator T (0, q0 ) of norm 1 corresponding to the eigenvalue λ20 . Therefore I − (·, φ0 )φ0 = Vφ0 Vφ∗0 (see Subsection 3.1), and, setting I + L := (I + L0 )Vφ0 , we get the required factorisation I + F = (I + L)(I + L∗ )

FACTORISATION OF NON-NEGATIVE FREDHOLM OPERATORS

13

of I + F . The kernel l of L equals φ0 (x)φ0 (t) l(x, t) = l0 (x, t) − − a(t) with a(x) :=

Rx 0

(4.12)

Z

t

l0 (x, s) x

φ0 (s)φ0 (t) ds a(t)

|φ0 (t)|2 dt and thus it has the stated smoothness properties; moreover, 2

dl(x, x) = q0 (x) − 2 (log a)00 (x) =: q1 (x). dx

Applying Lemma 4.1, we conclude that the operator I + L transforms the solutions of the equation −y 00 + q1 y = λ2 y

(4.13)

satisfying the terminal conditions y(1) = 0 and y 0 (1) = λ into such solutions sin λ(x−1) for zero potential (i.e., for q1 ≡ 0). Next we observe that the function a has the form a(x) = x3 b(x) for some function b ∈ W22 (0, 1) that is positive on [0, 1]; this follows from the behaviour of the eigenfunction φ0 at the origin and the properties of the Hardy operators, see details in [1, App. A]. Therefore, 2(log a)00 = −2 · 3/x2 + 2(log b)00 ; it follows that the function q1 has the form q1 (x) =

2·3 + q˜(x) x2

with q˜ := q0 (x) − 2(log b)00 ∈ L2 (0, 1), and it remains to show that q˜ = q. To this end we recall that the operator Vφ0 maps isometrically L2 (0, 1) onto its range L2 (0, 1) φ0 . Thus the pre-images of the eigenfunctions v(·, λn ), n ≥ 1, of the 1/2 operator T (0, 0, q0 ) have norm αn in L2 (0, 1), satisfy the Dirichlet condition at x = 1, and solve (4.13) with λ = λn . Therefore the operator T (2, q˜) has eigenvalues λ2n , n ≥ 1, and the corresponding norming constants are equal to αn . The direct spectral analysis of Bessel operators (in particular, the known asymptotics of their eigenvalues) suggests that T (2, q˜) has no other eigenvalues. By the Borg–Levinson uniqueness result [1, 4, 5], we get q˜ = q, and the case m = 2 is done. Even m: Induction step. Assume that we have already proved the theorem for all even m less than 2k and now consider the case m = 2k. We again augment the spectral data {(λ2n )n∈N , (αn )n∈N } of the operator T (2k, q) by the pair (λ20 , α0 ) with λ20 < λ21 and α0 > 0. The augmented data {(λ2n )n≥0 , (αn )n≥0 } are the spectral data for a unique operator T0 := T (2k − 2, q0 ) with some q0 ∈ L2 (0, 1). We denote by F0 the operator constructed for T0 and observe that, by the assumption of induction, I + F0 = (I + L0 )(I + L∗0 ) for some L0 ∈ BI+ . Denote by φ0 an eigenfunction of the operator T (2k − 2, q0 ) of norm 1 corresponding to the eigenvalue λ20 and satisfying the relation φ00 (1) > 0; then −1/2 (I + L0 )φ0 = α0 s0 with s0 (x) := sin λ0 (x − 1), and thus we get   I + F = I + F0 − α0−1 (·, s0 )s0 = (I + L0 ) I − (·, φ0 )φ0 (I + L∗0 ). Again the required factorisation follows with I + L := (I + L0 )Vφ0 ; moreover, by the inductive assumption, the kernel l of L satisfies 2

dl0 (x, x) d a0 (x) (2k − 2)(2k − 1) dl(x, x) =2 −2 = y + q1 dx dx dx a(x) x2

14

S. ALBEVERIO, R. HRYNIV, AND YA. MYKYTYUK

with q1 := q0 −2(log a)00 . By Lemma 4.1 and the induction assumption, I +L transforms solutions of the equation (2k − 2)(2k − 1) −y 00 + y + q1 y = λ 2 y x2 into sin λ(x − 1). The representation (4.11) with m = 2k − 2 and the properties of the Hardy operators [1, App. A] imply that a(x) = x4k−1 b(x) with some b ∈ W22 (0, 1) that is positive on (0, 1); therefore, (2k − 2)(2k − 1) 2k(2k + 1) + q1 = + q˜, 2 x x2 with q˜ := q0 − 2(log b)00 ∈ L2 (0, 1). The equality q˜ = q is justified as above, using the Borg–Levinson uniqueness theorem. The proof by induction for even m is complete. Odd m. Augmenting the spectral data for the operator T (1, q), we get spectral data for a unique Sturm–Liouville operator Th (0, q0 ) with a real-valued potential q0 ∈ L2 (0, 1) and subject to the Robin–Dirichlet boundary conditions y 0 (0)−hy(0) = y(1) = 0 for some h ∈ R. Repeating the arguments used in the case m = 2, we find an operator I + L that factorises the operator I + F constructed for T (1, q) and maps solutions of equation (4.13) with q1 given by (4.12) into sin λ(x − 1). However, now φ0 does not vanish at x = 0, and thus a has a simple zero at x = 0, which results in 2 q0 − 2(log a)00 = 2 + q˜ x with some q˜ ∈ L2 . We then justify that q˜ = q in the usual way. Finally, the induction step proceeds then as for the case of even m. The proof is complete.  4.4. Reconstruction algorithm. Assume that we are given two sequences (λ2n )∞ n=1 and (αn )∞ n=1 of real numbers satisfying the following conditions: ˜ n ) ∈ `2 ; (i) the sequence (λn ) is increasing and λ2n obey (4.10) with A ∈ R and (λ (ii) the αn are positive and obey (4.6) with (˜ αn ) ∈ `2 . The results of [17] imply that there is a unique real-valued q ∈ L2 (0, 1) such that λ2n are all eigenvalues and αn the corresponding norming constants for the Bessel operator T (m; q). To find this q, we perform the following steps: (1) construct the integral operator F as explained in the previous subsection; (2) uniquely factorise I + F as (I + L)(I + L∗ ) with an integral operator L ∈ F + with kernel l; (3) set dl(x, x) m(m + 1) q(x) := 2 − . dx x2 Thorem 4.2 implies that the function q obtained on the third step gives the required potential. We conclude with remark that similar results also hold for the Dirac operators with singular potentials appearing in the angular momentum decomposition of the radial Dirac systems in R3 , cf. [2, 18]. All constructions and proofs can be carried over by analogy with those presented here using the results of [2, 18]. Acknowledgements. The authors express their gratitude to DFG for financial support of the project 436 UKR 113/84 and thank the Institute for Applied Mathematics of Bonn University for the warm hospitality. The second author gratefully acknowledges the financial support of the Alexander von Humboldt Foundation.

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References [1] S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Inverse spectral problems for Bessel operators, J. Differential Equations 241(2007), no. 1, 130–159. [2] S. Albeverio, R. Hryniv, and Ya. Mykytyuk, Reconstruction of radial Dirac operators, J. Math. Phys. 48(2007), no. 4, 043501 (14 p.). [3] M. A. Barkar0 and I. Ts. Gohberg, Factorization of operators in a Banach space, Mat. Issled. 1(1966), no. 2, 98–129 (in Russian). [4] R. Carlson, Inverse spectral theory for some singular Sturm–Liouville problems, J. Diff. Equat. 106 (1993), 121–140. [5] R. Carlson, A Borg–Levinson theorem for Bessel operators, Pacific J. Math. 177 (1997), 1–26. [6] M. G. Gasymov, Determination of a Sturm–Liouville equation with a singularity by two spectra, Dokl. Akad. Nauk SSSR 161 (1965), 274–276 (in Russian); Engl. transl. in Soviet Math. Dokl. 6 (1965), 396–399. [7] F. Gesztesy, A complete spectral characterization of the double commutation method, J. Funct. Anal. 117, (1993) 401–446. [8] F. Gesztesy and G. Teschl, On the double commutation method, Proc. Amer. Math. Soc. 124 (1996), 1831–1840. [9] I. Gohberg and M. Krein, Theory of Volterra Operators in Hilbert Space and its Applications, Nauka Publ., Moscow, 1967 (in Russian); Engl. transl.: Amer. Math. Soc. Transl. Math. Monographs, 24, Amer. Math. Soc., Providence, RI, 1970. [10] M. G. Kre˘ın, On integral equations generating differential equations of 2nd order, Doklady Akad. Nauk SSSR (N.S.) 97(1954), no. 1, 21–24 (in Russian). [11] M. G. Kre˘ın, On a new method of solution of linear integral equations of first and second kinds, Doklady Akad. Nauk SSSR (N.S.) 100(1955), no. 3, 413–416 (in Russian). [12] Ya. V. Mykytyuk, Factorization of Fredholm operators, Mat. Stud. 20 (2003), no. 2, 185–199 (Ukrainian). [13] Ya. V. Mykytyuk, Factorization of Fredholm operators in operator algebras, Mat. Stud. 21 (2004), no. 1, 87–97 (Ukrainian). [14] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Selfadjointness, Academic Press, New York-London, 1975. [15] L. A. Sakhnovich, Factorization of operators in L2 (a, b), Funkt. Anal. Prilozh. 13(1979), 40–45. [16] L. A. Sakhnovich, Factorization of operators, theory and applications, Ukrain. Mat. Zh. 46(1994), no. 3, 293–304 (in Russian); Engl. transl.: Ukrainian Math. J. 46(1994), no. 3, 304–317. [17] F. Serier, The inverse spectral problem for radial Schr¨odinger operators on [0, 1], J. Differential Equations 235(2007), no. 1, 101–126. [18] F. Serier, Inverse spectral problem for singular Ablowitz-Kaup-Newell-Segur operators on [0, 1], Inverse Problems 22(2006), no. 4, 1457–1484. [19] G. Teschl, Deforming the point spectra of one-dimensional Dirac operators, Proc. Amer. Math. Soc. 126 (1998), 2873–2881. ¨r Angewandte Mathematik, Universita ¨t Bonn, Wegelerstr. 6, (S.A.) Institut fu D–53115, Bonn, Germany; SFB 611, Bonn, Germany; BiBoS, Bielefeld, Germany; IZKS; CERFIM, Locarno, Switzerland; and Accademia di Architettura, Mendrisio, Switzerland E-mail address: [email protected] (R.H. and Ya.M.) Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine and Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine E-mail address: [email protected] and [email protected]

Bestellungen nimmt entgegen: Institut für Angewandte Mathematik der Universität Bonn Sonderforschungsbereich 611 Wegelerstr. 6 D - 53115 Bonn Telefon: Telefax: E-mail:

0228/73 4882 0228/73 7864 [email protected]

http://www.sfb611.iam.uni-bonn.de/

Verzeichnis der erschienenen Preprints ab No. 355

355. Löbach, Dominique: On Regularity for Plasticity with Hardening 356. Burstedde, Carsten; Kunoth, Angela: A Wavelet-Based Nested Iteration – Inexact Conjugate Gradient Algorithm for Adaptively Solving Elliptic PDEs 357. Alt, Hans-Wilhelm; Alt, Wolfgang: Phase Boundary Dynamics: Transitions between Ordered and Disordered Lipid Monolayers 358. Müller, Werner: Weyl's Law in the Theory of Automorphic Forms 359. Frehse, Jens; Löbach, Dominique: Hölder Continuity for the Displacements in Isotropic and Kinematic Hardening with von Mises Yield Criterion 360. Kassmann, Moritz: The Classical Harnack Inequality Fails for Non-Local Operators 361. Albeverio, Sergio; Ayupov, Shavkat A.; Kudaybergenov, Karim K.: Description of Derivations on Measurable Operator Algebras of Type I 362. Albeverio, Sergio; Ayupov, Shavkat A.; Zaitov, Adilbek A.; Ruziev, Jalol E.: Algebras of Unbounded Operators over the Ring of Measurable Functions and their Derivations and Automorphisms 363. Albeverio, Sergio; Ayupov, Shavkat A.; Zaitov, Adilbek A.: On Metrizability of the Space of Order-Preserving Functionals 364. Alberti, Giovanni; Choksi, Rustum; Otto, Felix: Uniform Energy Distribution for Minimizers of an Isoperimetric Problem Containing Long-Range Interactions 365. Schweitzer, Marc Alexander: An Adaptive hp-Version of the Multilevel Particle-Partition of Unity Method 366. Frehse, Jens; Meinel, Joanna: An Irregular Complex Valued Solution to a Scalar Linear Parabolic Equation 367. Bonaccorsi, Stefano; Marinelli, Carlo; Ziglio, Giacomo: Stochastic FitzHugh-Nagumo Equations on Networks with Impulsive Noise 368. Griebel, Michael; Metsch, Bram; Schweitzer, Marc Alexander: Coarse Grid Classification: AMG on Parallel Computers

369. Bar, Leah; Berkels, Benjamin; Rumpf, Martin; Sapiro, Guillermo: A Variational Framework for Simultaneous Motion Estimation and Restoration of Motion-Blurred Video; erscheint in: International Conference on Computer Vision 2007 370. Han, Jingfeng; Berkels, Benjamin; Droske, Marc; Hornegger, Joachim; Rumpf, Martin; Schaller, Carlo; Scorzin, Jasmin; Urbach Horst: Mumford–Shah Model for One-to-one Edge Matching; erscheint in: IEEE Transactions on Image Processing 371. Conti, Sergio; Held, Harald; Pach, Martin; Rumpf, Martin; Schultz, Rüdiger: Shape Optimization under Uncertainty – a Stochastic Programming Perspective 372. Liehr, Florian; Preusser, Tobias; Rumpf, Martin; Sauter, Stefan; Schwen, Lars Ole: Composite Finite Elements for 3D Image Based Computing 373. Bonciocat, Anca-Iuliana; Sturm, Karl-Theodor: Mass Transportation and Rough Curvature Bounds for Discrete Spaces 374. Steiner, Jutta: Compactness for the Asymmetric Bloch Wall 375. Bensoussan, Alain; Frehse, Jens: On Diagonal Elliptic and Parabolic Systems with Super-Quadratic Hamiltonians 376. Frehse, Jens; Specovius-Neugebauer, Maria: Morrey Estimates and Hölder Continuity for Solutions to Parabolic Equations with Entropy Inequalities 377. Albeverio, Sergio; Ayupov, Shavkat A.; Omirov, Bakhrom A.; Turdibaev, Rustam M.: Cartan Subalgebras of Leibniz n-Algebras 378. Schweitzer, Marc Alexander: A Particle-Partition of Unity Method – Part VIII: Hierarchical Enrichment 379. Schweitzer, Marc Alexander: An Algebraic Treatment of Essential Boundary Conditions in the Particle–Partition of Unity Method 380. Schweitzer, Marc Alexander: Stable Enrichment and Local Preconditioning in the Particle– Partition of Unity Method 381. Albeverio, Sergio; Ayupov, Shavkat A.; Abdullaev, Rustam Z.: Arens Spaces Associated with von Neumann Algebras and Normal States 382. Ohta, Shin-ichi: Finsler Interpolation Inequalities 383. Fang, Shizan; Shao, Jinghai; Sturm, Karl-Theodor: Wasserstein Space Over the Wiener Space 384. Nepomnyaschikh, Sergey V.; Scherer, Karl: Multilevel Preconditioners for Bilinear Finite Element Approximations of Diffusion Problems 385. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Factorisation of Non-Negative Fredholm Operators and Inverse Spectral Problems for Bessel Operators