On Singular Dissipative Fourth-Order Differential Operator in Lim-4 Case

In 1910, Weyl showed that [1] the singular second-order differential operators can be fallen into two classes: operators in limit-circle case and operators in limit-point case.The operators in limit-circle case have the solutions that are all in square integrable space. However, in limit-point case, only one linearly independent solution can be in square integrable space. The development of this theory belongs to Titchmarsh [2]. After these fundamental works, second-order singular differential operators have been developed by many authors (e.g., see [3–6]). Following the methods of Weyl and Titchmarsh, the theory for higher order equations and Hamiltonian systems was constructed in [7–19]. In [7], a regular self-adjoint fourth-order boundary value problem was investigated. Further Green’s function and the resolvent operator were constructed. In [8, 13], such a construction was done for the singular self-adjoint fourth order boundary value problem. In [9, 10], higher-order differential equations whose coefficients are complex-valued were studied. In 1974, Walker [20] showed that an arbitrary order selfadjoint eigenvalue problem can be represented as an equivalent self-adjoint Hamiltonian systems. Further, the developments in the theory of singular self-adjoint Hamiltonian systems were given in [14–19]. On the other hand, an important class of the nonselfadjoint operators is the class of the dissipative operators [21]. It is known that all eigenvalues of the dissipative operators lie in the closed upper half-plane, but this analysis is so weak. There are somemethods to complete the analysis of the dissipative operators. Some of them are Livšic’s, Krein’s, and Lidskĭı’s theorems and functional model [21, 22].These methods were used in the literature for the second-order differential operators (see [22–26]). In this paper a singular dissipative fourth order differential operator in lim-4 case is investigated. In particular, using Lidskĭı’s theorem, it is shown that the system of all eigenand associated functions is complete in


Introduction
In 1910, Weyl showed that [1] the singular second-order differential operators can be fallen into two classes: operators in limit-circle case and operators in limit-point case.The operators in limit-circle case have the solutions that are all in square integrable space.However, in limit-point case, only one linearly independent solution can be in square integrable space.The development of this theory belongs to Titchmarsh [2].After these fundamental works, second-order singular differential operators have been developed by many authors (e.g., see [3][4][5][6]).
In [7], a regular self-adjoint fourth-order boundary value problem was investigated.Further Green's function and the resolvent operator were constructed.In [8,13], such a construction was done for the singular self-adjoint fourth order boundary value problem.In [9,10], higher-order differential equations whose coefficients are complex-valued were studied.In 1974, Walker [20] showed that an arbitrary order selfadjoint eigenvalue problem can be represented as an equivalent self-adjoint Hamiltonian systems.Further, the developments in the theory of singular self-adjoint Hamiltonian systems were given in [14][15][16][17][18][19].
On the other hand, an important class of the nonselfadjoint operators is the class of the dissipative operators [21].
It is known that all eigenvalues of the dissipative operators lie in the closed upper half-plane, but this analysis is so weak.There are some methods to complete the analysis of the dissipative operators.Some of them are Livšic's, Krein's, and Lidskiȋ's theorems and functional model [21,22].These methods were used in the literature for the second-order differential operators (see [22][23][24][25][26]).In this paper a singular dissipative fourth order differential operator in lim-4 case is investigated.In particular, using Lidskiȋ's theorem, it is shown that the system of all eigen-and associated functions is complete in  2 (Λ).

Preliminaries
Let  denote the linear nonself-adjoint operator in the Hilbert space  with the domain ().The element  ∈ (),  ̸ = 0, is called a root function of the operator  corresponding to the eigenvalue  0 , if all powers of  are defined on this element and ( −  0 )   = 0 for some  > 0.
The completeness of the system of all eigen-and associated functions of  is equivalent to the completeness of the system of all root functions of this operator.
Theorem 1 (see [26]).Let  be an invertible operator.Then, − is dissipative if and only if the inverse operator  −1 of  is dissipative.
It is known that a kernel (, ) of an integral operator A defned by where A kernel satisfying the property (, ) = (, ) is called a Hermitian kernel.Properties of Hermitian kernels and related integral operators can be found in [27] (further see [28]).Now we shall remind some results.Let us consider the equation which is related with (1) as The function  which differs from zero is called a characteristic function of A which corresponds to the characteristic value .If a Hermitian Hilbert-Schmidt kernel is not null then it posseses at least one characteristic value and this characteristic value (every characteristic value) is real.Further there is a finite orthonormal base of characteristic functions for each characteristic value  of A. Using the union of these bases one obtains an orthonormal system of characteristic functions of the kernel (, ).If  1 ,  2 , . . .,   are distinct members of such a system belonging respectively to the characteristic value  1 ,  2 , . . .,   (not necessarily all different) then for 1 ≤ V ≤ ,  V ⊗  V =  V () V () is an orthonormal system of two variables.For such a system the equivalence holds.Moreover if  = A,  ∈  2 , then the equivalence holds.Further the series in ( 5) are relatively uniformly absolutely convergent.However, equivalence can be replaced by equality.Before showing this, let us consider the equality Following the same idea of [27, pages 22] if (, ) is an  2kernel such that (, ) is continuous and  is an  2 -function then () is continuous.Hence if (, ) is a continuous Hermitian  2 -kernel (see [27, pages 127]) and  = A,  ∈  2 , then the equality holds and the series are uniformly absolutely convergent.In this case taking () = (, (, ⋅)) the series given in (4) converge to the continuous kernel (, ) uniformly in the variable  for every .Hence from (4) one can get Integrating both side from  to  we obtain Consequently if (, ) is integrable on (, ) then the series converge.This implies that A is of trace-class (nuclear).For the definition and properties of trace-class operators see, for example, [21].Above arguments given in [27] hold for Hermitian kernels.However from arbitrary  2 -kernels one can pass to the Hermitian kernels.For example  1 (, ) = (, ) + (, ) and  2 (, ) = ((, ) − (, )) are Hermitian kernels.These Hermitian kernels are continuous and  2 -kernels if (, ) is so.Hence above arguments hold for  1 (, ) and Hence if  1 and  2 are trace-class kernels then so is .However this result has been given in [29, pages 526] as a definition.
Lidskiȋ's Theorem (see [21, page 231]).If the dissipative operator  is the trace class operator, then its system of root functions is complete in the Hilbert space .

Statement of the Problem
The fourth order differential expression is considered as where −∞ <  <  ≤ ∞,  is the regular point and  is the singular point for , and  is real-valued, Lebesgue measurable, and locally integrable function on Λ.
Let  2 (Λ) be the Hilbert space consisting of all functions  such that Let where AC loc (Λ) denotes the set of all locally absolutely continuous functions on Λ.For arbitrary ,  ∈ Ω, Green's formula is obtained as where  ≤  The latter also follows from Green's formula.In fact, it is sufficient to get the second factors with their complex conjugates on the left-hand side in Green's formula.
In this paper it is assumed that () satisfies the lim-4 case conditions at  (see [30] and references therein).Lim-4 case is also known as Weyl's limit-circle case [8].
It should be noted that for any solutions (, ) of ( 14), conditions ( 16) and (17), respectively, can be written as The operator  is defined on () as  = (),  ∈ (),  ∈ Λ.The main aim of this paper is to investigate the spectral analysis of the operator  (the boundary value problem ( 14)-( 19)).
It is known that the following equality holds [7,8,13]: This equation also shows that { 1 ,  2 ,  3 ,  4 } of any four solutions of () =  is independent of  and depends only on  [7,8].Now consider the solutions  1 (, ) and  2 (, ) of the equation () = ,  ∈ Λ, satisfying the conditions For the existence of these solutions given with the previous conditions, see [13].Clearly these solutions satisfy conditions (18) and (19), respectively.Now let us set Then () becomes an entire function, and the zeros of () coincide with the eigenvalues of the operator  [13].This implies that all zeros of  (all eigenvalues of ) are discrete and that possible limit points of these zeros (eigenvalues of ) can only occur at infinity.Using (22), we immediately have
Theorem 2 shows that all eigenvalues of  lie in the closed upper half-plane.
From Theorem 3 it is obtained that all eigenvalues of  lie in the open upper half-plane.In particular zero is not an eigenvalue of .Hence, the operator  −1 exists.
The equation  = (),  ∈ (),  ∈ Λ, is equivalent to the nonhomogeneous differential equation subject to the boundary conditions (compare the boundary conditions at  with ( 20) and ( 21)).Using Everitt's method [7] (further see [8,13]) the general solution is obtained as where  ∈  2 (Λ), is the inverse operator of .Hence the completeness of the system of all eigen-and associated functions of  is equivalent to the completeness of those of  in  2 (Λ).
Since (, ) is a continuous Hilbert-Schmidt kernel and (, ) is integrable on [, ), the operator  is of trace class.
Let us consider the operator −.Since  is dissipative in  2 (Λ), − is also dissipative in  2 (Λ).Thus all conditions are satisfied for Lidskiȋ's theorem.So we have the following.Theorem 4. The system of all root functions of − (also ) is complete in  2 (Λ).
Since the completeness of the system of root functions (eigen-and associated functions) of  is equivalent to the completeness of those of , is obtained that the following.Theorem 5.All eigenvalues of the problem (14)- (19) lie in the open upper half-plane and they are purely discrete.The system of all eigen-and associated functions of the problem (14)-( 19) is complete in  2 (Λ).