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Spectral analysis of one class of the integro-differential operators

Let ℋ = L2[0,ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on the closed interval [0,ω]. In this paper, we study the spectral properties of a second-order differential operator which perturbed by an integral operator. The integral operator belong to the two-sided ideal S 2 (ℋ) of Hilbert-Schmidt operator in ℋ. We use similar operators method for studying this operator. The similar operators method was developed by A. G. Baskakov and his collaborators. This method allows us to reduce the study of the operator to the one with a diagonal and a block-diagonal matrix. Asymptotic estimates of eigenvalues, eigenvectors, and spectral projections of an integro-differential operator are obtained.


Introduction
Let H = L 2 [0, ω] be the Hilbert space of equivalence classes of square integrable Lebesgue measurable complex functions on closed interval [0, ω]. The inner product on this space is given, as usually, by the formula (x, y) = 1 ω ω 0 x(t)y(t)dt, x, y ∈ H.
The norm in H is induced by this inner product. By W 2 2 [0, ω] we denote the Sobolev space, {y ∈ L 2 [0, ω]: y is absolutely continuous and y ∈ L 2 [0, ω]}, W 2 2 [0, ω) ⊂ H. We consider an integro-differential operator L : D(L) ⊂ H → H generated by the integrodifferential expression The domain D(L) of the operator L is defined by the periodic boundary conditions We assume the integral operator B will have the following form ω 0 ω 0 |K(t, s)| 2 dtds < ∞.
The class of the operators under consideration arises in the problems of the chemical kinetics (see [1,2]).
In this paper we study a spectral properties of the operator L defined by (1), such these the asymptotic estimates of eigenvalues, eigenvectors and spectral projectors.
We shall call the operator A, which defined by (2), unperturbed of free. The operator A is self-ajoined and has a compact resolvent. Its spectrum σ(A) will have the following form where λ n = −( πn ω ) 2 are simple isolated eigenvalues, E n = Span{e n }, n ∈ Z + , is the eigenspace for an eigenvalue λ n = −( πn ω ) 2 , where e n (t) = √ 2 cos πn ω t, n ∈ Z + . The spectral projections P n = P (σ n , A), σ n = {λ n }, n ∈ Z + are given by P n (x) = (x, e n ), e n , x ∈ H. We will use the notation H k = ImP k , k ∈ Z + , P (m) = i≤m P i , H (m) = ImP (m) .
Operator A − B is well-defined by condition D(A) ⊂ D(B). Operator B will play the role of the perturbation.

Matherials and methods
In this section we introduce the necessary notation following the standard scheme of the similar operators method.
Given a function v ∈ L 2 [0, ω] its Fourier series are given by where the Fourier coefficients arê and P n (x) =v(n)e n , n ∈ Z + .
We also will use the ideal of nuclear operators S 1 (H). Recall that in S 1 (H) the norm is given by ||X|| 1 = ∞ n=1 s n . We refer to [13,14] for the standard properties of these ideals used in this paper.
We begin with the following definition. Definition 1. [13] Two linear operators A m : D(A m ) ⊂ H → H, m = 1, 2, are called similar, if there exists a continuously invertible operator U ∈ End H such that The operator U is called the similarity transform of A 1 into A 2 .
Directly from Definition 1, we have the following result about the spectral properties of similar operators. 2) If λ is an eigenvalue of the operator A 2 and x is a corresponding eigenvector, then y = U x is an eigenvector of the operator A 1 corresponding to the same eigenvalue λ.
3) If P σ is the spectral projection of A 2 that corresponds to the spectral component σ ⊆ σ(A 2 ) = σ(A 1 ), then P σ = U P σ U −1 is the spectral projection of A 1 that corresponds to the same spectral component σ.
The method of similar operators uses the commutator transform ad A : The domain D(ad A ) consists of all X ∈ End H such that the following two properties hold: 2) The operator ad A X : D(A) → H admits a unique extension to a bounded operator Y ∈ End H; we then let ad A X = Y .
The key notion of the method of similar operators is that of an admissible triplet. Once such a triplet is constructed, achieving the goal of the method becomes a routine task.
Definition 2. [4,7] Let J, G ∈ End S 2 (H). The collection (S 2 (H), J, G) is an admissible triplet for the operator A, and the space S 2 (H) is the space of admissible perturbations, if the following five properties hold.
, and there is a constant γ > 0 such that For every X ∈ S 2 (H) and ε > 0 there exists a number λ ε ∈ ρ(A), such that To formulate the main theorem of the similar operators method we will use the map Φ : S 2 (H) → S 2 (H), given by Theorem 1. [9] Assume that (S 2 (H), J, G) is an admissible triplet for an unperturbed operator A. Assume also that where γ comes from Property 3 of Definition 2 and ||J|| = 1. Then the map Φ : S 2 (H) → S 2 (H), given by (4) is a contraction and has a unique fixed point X * in ball which can be found as a limit of simple iterations: Proof. We use the Banach fixed-point theorem to prove existence of X * . We show that We have Now the first part of the Theorem 1 is prooved. Next, we need to verify the following We have JX * = J(BGX * ) − JB, Then if X is the solution of the equation then the formulae (5) takes place. Next, we need to show that I + GX * is continously invertible if X * ∈ B. We get for any λ ∈ ρ(A). Using Property 5 we choose λ ∈ ρ(A) such that The theorem 1 is prooved. We note, that ussually used another version of the similar operator method. The main theorem of the similar operator method for example, in [4] is differs from the Theorem 1.

Results and discussions
Let H = L 2 [0, ω] and operators A and B be defined by (2), (3). For each operator X ∈ S 2 (H) we define his operator matrix. The entries of the matrix will be the operators X nm = P n XP m , n, m ∈ Z + , X = (X nm ). The operator X p = n−m=p X ij is the p-th diagonal of the matrix of the operator X [15].
The transform J ∈ EndS 2 (H) is supposed to pick out the main diagonal of the matrix of operator X ∈ S 2 (H), JX = i∈Z + P i XP i , where the series converge unconditionally in S 2 (H).
Observe that in this setting the matrix of the commutator ad A X satisfies (ad A X) nm = (λ n − λ m )X nm , X ∈ D(ad A ).
Therefore, in view of Property 2 of Definition 2 it is natural to define the transform G ∈ EndS 2 (H) via here X ∈ S 2 (H), GX = n,m∈Z + Xnm λn−λm , where the series converge unconditionally in S 2 (H). We also define a family of transformers J k X and G k X, k ∈ Z + , X ∈ S 2 (H) as follows: We note, that the operators G k , J k ∈ EndS 2 (H) differ from the operators G, J ∈ EndS 2 (H) on operator of finite rank. Proof. Since the projectors P n , n ∈ Z + are orthogonal, it follows that the norm in S 2 (H) can be defined by the formula Obviously, ||J k || = 1, because for X ∈ S 2 (H) If the matrix of operator X is diagonal, i. e. X = i∈Z + P i XP i , then J k X = X and ||J k X|| 2 = ||X|| 2 . Then, It follows that ||G k X|| 2 ≤ ω π 2 1 2k + 1 ||X|| 2 , and γ = γ k = ω π 2 1 2k+1 . If operator X, Y ∈ S 2 (H) then X(G k Y ), (G k X)Y also belong to S 2 (H) and If follows that property 3 of Definition 2 is hold with Operator J is idempotent because The fact that is also follows from the preceding argument is that J 2 k X = J k X, k ∈ Z + , X ∈ S 2 (H) and property 1 of Difinition 2 holds true.
Property 2 is now verified by direct computation. In particular, one checks that for a given The first factor is finite and the other can be chosen arbitrarily small if λ = in, where n ∈ N is the larg natural number.
Then the operator A − B is simular to the diagonal operator A − JX * = A − i∈Z + P i X * P i , X * ∈ S 2 (H), where X * is limit of simple iteration: X 0 = 0, X 1 = Φ(X 0 ) = B, etc, with Φ given by (4). The transform operator is U = I + GX * .
Proof. Follows immediately from Theorem 1 and Theorem 2. Theorem 4. There exists a number k ∈ Z + such that the operator A − B is similar to the operator A − P (k) X * P (k) − i>k P i X * P i , where X * ∈ S 2 (H). The identity holds true and subspaces H (k) = ImP (k) and H j = Im P j , j > k are invariant with respect to the operator A and J k X * . The operator X * ∈ S 2 (H) is the solution of the equation (4) and it can be found with the simple iteration method.
Then the similarity of operator L and The operator X * ∈ S 2 (H) is unknown and we know only approximations for it. The first approximation is B, In what follows, by l p , p ≥ 1 we denote the Banach space of p-summable (p ≥ 1) sequenses of complex numbers with the norm ||y|| p = ( n∈Z + |y(n)| p ) 1/p , y ∈ l p . Lemma 2. Under the assumptions of Theorem 3 we have where the sequence {α i , i > k} belong to l 1 .
Proof. We have We note that BG k X * ∈ S 1 (H) and where the sequence {α i , i > k} belong to l 1 .
Theorem 5 follows from Lemma 1 and Lemma 2. This theorem describes the spectral properties of the operator L.
Theorem 5. There is a number k ∈ Z + such that the spectrum σ(L) of the operator L satisfies where σ (k) consists of no more than k + 1 eigervalues. The sets σ i , i > k are one-point sets The corresponding eigenvector e i , i ∈ Z + form the Riesz basis in the space H. The following estimates hold for the eigenvector e i , i > k: where the sequense {β i , i > k} belong to l 2 . Let P (k) , k ∈ Z + and P i , i > k be the spectral projections P (k) = P ( σ (k) , L), P i = P ( σ i , L) corresponding to the sets σ (k) , σ i , i > k described in Theorem 5.
From the Lemma 1 and Theorem 3 follows Theorem 6. We have where the sequence {ν i , i > k} belong to l 2 .
where m > k, N > k, N > m, N ∈ Z + , M 1 , M 2 > 0, k ∈ Z + . The spectral projections P (k) , P i , i > k also satisfy the following estimates of uniform unconditional equiconvergence of spectral expansions: We need the following definition in the monograph [16]. Definition 3.
[16] Let C : D(C) ⊂ H → H be a linear operator in the space H whose spectrum can be represented as a union of pairwise disjoint sets σ l , l ∈ c. ALso let P l , l ∈ Z + be the Riesz projection corresponding to the spectral set σ l , l ∈ Z + . An operator C is said to be spectral with respect to the expansion (7) (or generalized-spectral) if the series l P l x is convergent for any vector x ∈ H. If σ l = {λ l }, l ∈ Z + , are one-point sets and CP l = λ l P l for all l ∈ Z + , except for finitely many, then spectral with respect to the expansion (7) operator C is a spectral operator. Operator C is the spectral operator of scalar type if CP l = λ l P l for all l ∈ Z + . Theorem 7. The operator L is a spectral operator. Theorem 8. The integro-differential operator iL is a generator of a C 0 -group T : R → End H. This group is similar to the group T : R → End H, T (t) = U T (t)U −1 , t ∈ R, U = I + G k X * , X * ∈ S 2 (H). The operator group T can be writen as where sequense {α i , i > k} belong to l 2 . The above results motivates the terminology in the following definition. If ||V (t)|| ≤ e −βt , t ≥ 0, for some β > 0, then we call T 0 an exponentially basic (semi)group.
We have T (t) = T 0 (t)V (t), where T 0 (t) = e i(A (k) −X * (k) )t P (k) + l>k e −i( πl ω ) 2 t P l , t ∈ R, It follows that the group T 0 : R → End H is basic for the group T : R → End H if ImK l < 0, l > k, k ∈ Z + . The existence of the operator group T guaranteed by Theorem 8 is important because it allows one to use the results, for example, from [17] on exponential dichotomy and the estimates for Green's function.

Conclusion
In this paper we studied the spectral properties of a second order integro-differential operator in the close interval [0, ω] with the periodic boundary conditiones. The considered class of operators arises in problems of chemical kinetics. To this end, we used the similar operators method. This method allowed us to reduce the study of the operator to one with a block-diagonal matrix. This method was thoroughly described in the paper and its basic definitions, such as admissible triplet and commutator transform, were given. Main results of the paper were formulated in eight theorems and in two lemmas, proofs of some of the theorem and lemmas were also provided. The article is ten pages long and list of references consists of seventeen issues.