Original Research ARTICLE
Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator
- 1Department of Mathematics, Faculty of Arts and Sciences, Çankaya University, Ankara, Turkey
- 2The Theoretical Physics Laboratory, Institute of Space Sciences, Magurele-Bucharest, Romania
In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential expression and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal non-selfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.
2000 Mathematics Subject Classification: Primary 47A45, 47E05; Secondary 47A20
A model operator may be regarded as an equivalent operator to another operator in a certain sense. Such an equivalent representation has been constructed by Szokefalvi-Nagy and Foiaş  for a contractive operator. The main idea for this construction is to obtain the unitary dilation of the contraction. In fact, if the following equality holds
where T is a contraction on the Hilbert space H and U is the operator on y ∈ H, n ≥ 0 and P is the orthogonal projection of onto H, then U is called a dilation of T. U is called unitary provided that U is a unitary operator and in this case U is called unitary dilation of T. There exists a geometric meaning of the dilation space. This meaning has been given by Sarason . Sarason showed that U is a dilation of T if and only if has the representation
where UG ⊂ G and This representation is closely related with incoming and outgoing spaces in the scattering theory . In the case that
then G and G∗ are uniquely determined and U is called minimal. If U is unitary minimal dilation of T then one may consider the decomposition 
where and DT are so called defect operators of T defined by
𝔇T and are the defect spaces defined by
V is a partial isometry with the initial space 𝔇T and final space E = G ⊖ UG and V∗ is a partial isometry with initial space and the final space
Now consider the transformations
where υ and υ∗ are the unitary mappings defined by
In the literature the operators π and π∗ are called functional embeddings. The function acting from E into E∗ is called the characteristic function of the contraction T. If υ and υ∗ are chosen as the unitary identifications between 𝔇T and E, and, and E∗, respectively, then the characteristic function can be introduced as
Nagy and Foiaş introduced the characteristic function of a contraction as 
and this can be obtained from the previous equation by choosing E = 𝔇T and
Solomyak  using the connection between dissipative operators and their Cayley transforms introduced an effective way to obtain the characteristic function of both dissipative operators and related contractions generated by Cayley transforms. By a dissipative operator it is meant an operator A with a dense domain D(A) acting on a Hilbert space K satisfying
An immediate result on dissipative operators is that all eigenvalues lie in the closed upper half-plane. If a dissipative operator does not have a proper dissipative extension then A is called maximal dissipative. The Cayley transform of a dissipative operator
is a contraction from (A + iI)D(A) onto (A − iI)D(A), i.e.,
It is known that a dissipative operator is maximal if and only if C(A) is a contraction such that domain of C(A) is the Hilbert space K and 1 can not belong to the point spectrum of C(A). Solomyak used these connections and boundary spaces associated with A to construct the characteristic function SA(λ) with the rule
where P and P∗ are the natural projections. To be more precise we should note that for a maximal dissipative operator A the Hermitian part AH of A is defined as the restriction of A to the following subspace
The natural projection P is defined by
where D(A)/GA is the quotient space. Similarly P∗ is defined by
On the quotient spaces the following inner products are defined
Let F(A) be the completion of the quotient space D(A)/GA with respect to the norm
In a similar way one may define and F∗(A) is equipped with the norm
These spaces F(A) and F∗(A) are called boundary spaces. Solomyak showed for a maximal dissipative operator A and its Cayley transform C(A) that there exist isometric isomorphisms
with the rules
Then fixing arbitrary isometric isomorphisms Ω : E → 𝔇C, the characteristic function ΘC of the Cayley transform C(A) can be introduced by
Finally taking Ω = ρ, Ω∗ = ρ∗, E = F(A), E∗ = F∗(A) one obtains (1.1).
In this paper using the results of Solomyak we investigate some spectral properties of a regular third-order dissipative operator. We should note that such an investigation with the aid of Solomyak's approach has not been introduced for the third-order case. In fact, the literature has less works on odd-order operators than on even-order equations even if there exists some results in the literature [6–13]. The main reason is the confusion of imposing the boundary conditions because as Everitt says in  that it is impossible to impose separated boundary conditions for the solutions of a third-order equation. Consequently, this paper may give an idea to use Solomyak's method for the odd-order dissipative or accumulative operators.
2. Maximal Dissipative Operator
Throughout the paper we consider the following third-order differential expression
where qj, pj, j = 0, 1, w are real-valued and continuous functions on [a, b] and q0 > 0 or q0 < 0 and w > 0 on [a, b].
The quasi-derivative y[r] of the function y is defined by
Let H denote the Hilbert space with the usual inner product
and with the norm
Now consider the subspace D of H which consists of the functions y ∈ H such that y[r], 0 ≤ r ≤ 2, is locally absolutely continuous on [a, b] and ℓ(y) ∈ H. The maximal operator L is defined on D by
For y, z ∈ D following Lagrange's formula can be introduced
(2.1) particulary implies the meaning of [y, z](a) and [y, z](b) for y, z ∈ D.
Let be a set of D that consists of those functions y ∈ D such that y has a compact support on [a, b]. The operator which is the restriction of L to is a densely defined symmetric operator and therefore it admits the closure. Let L0 be the closure of L0 then becomes a densely defined, symmetric operator with domain D0 that consists of the functions y ∈ D satisfying
For the symmetric operators there exists a useful theory called deficiency indices theory to construct the extensions. In fact, let M be a symmetric operator on a Hilbert space B and Rλ denoted the range of M − λI, where λ is a parameter and I is the identity operator in B. The deficiency spaces Nλ and are defined by Naimark 
The deficiency indices (m, n) of the operator M are defined by
Note that the deficiency indices of L0 are (3, 3).
To describe the extensions of a closed, symmetric operator with equal deficiency indices one may use the boundary value space. Boundary value space of the closed symmetric operator M is a triple (K, σ1, σ2) such that σ1, σ2 are linear mappings from D(M*) (domain of M*) into K and following holds:
(i) for any f, g ∈ D(M*)
(ii) for and F1, F2 ∈ K, there exists a vector f ∈ D(M*) such that σ1f = F1 and σ2f = F.
Now for y ∈ D consider the following mappings
Then we have the following Lemma.
Lemma 2.1. Fory, z ∈ D
Proof: Let y, z ∈ D. Then
This completes the proof.□
One of our aim is to impose some suitable boundary conditions for the solution y of the equation
where λ is the spectral parameter and y ∈ D. We should note that the Equation (2.2) has a unique solution χ(x,λ) satisfying the initial conditions
where lr is a complex number. This fact follows from the assumptions on the coefficients q0, q1, p0, p1, w, and following representation
Then the theory on ordinary differential equations may be applied to the first-order system (2.3), where the elements of A are integrable on each compact subintervals of [a, b].
Now the next Lemma can be introduced with the aid of Naimark's patching Lemma .
Lemma 2.2. There existsy ∈ D satisfying
where αr, βr are arbitrary complex numbers.
Now we may introduce the following.
Theorem 2.3. is a boundary value space for L0.
Proof: Since we obtain for y, z ∈ D that
Therefore, Lemma 2.1 and Lemma 2.2 complete the proof.□
Let S be a contraction and N be a selfadjoint operator on ℂ3. Then using the Theorem of Gorbachuks' , p. 156, the following abstract Theorem can be introduced.
Theorem 2.4. Let f ∈ D. Then the conditions
describe, respectively, the maximal selfadjoint, maximal dissipative, and maximal accumulative extensions of L0.
Since we will investigate the spectral properties of the maximal dissipative extension of L0 we shall introduce the following.
Corollary 2.5. For y ∈ D the maximal dissipative extension of L0 is described by
Corollary 2.6. For y ∈ D the conditions
where h2 = h2,∗/2, describe the maximal dissipative extension of L0.
Remark 2.7. As may be seen in the next sections, the caseh2 = i may give rise to some complications. Therefore, we exclude this case.
Now let be a set consisting of all functions y ∈ D satisfying the conditions (2.4). Let us define the operator on with the rule
Then is a maximal dissipative operator on H.
The adjoint operator of is given by
where is the domain of consisting of all functions y ∈ D satisfying
Theorem 2.8. is totally dissipative (simple) in H.
Proof: This follows from choosing h2 and h3 with positive imaginary parts. Indeed, for one gets
If had a selfadjoint part in Hs ⊂ H then from (2.5) one would get
and therefore y ≡ 0. This completes the proof.□
3. Contractive Operator
There exists a connection between dissipative operator and the contractive operator . This connection can be given by the following relation
Since is maximal dissipative the domain of is the whole Hilbert space H.
An important class of contractions on a Hilbert space consists of completely non-unitary (c.n.u.) contractions. A contraction C is said to be c.n.u. if there exists no non-zero reducing subspace H0 such that C ∣ H0 is a unitary operator.
From the simplicity of we have the following.
Theorem 3.1. is a c.n.u. contraction onH.
Proof: Let where and f ∈ H. Then we get
(3.1) implies that
and this completes the proof.□
Now we define the defect operators of as
and the defect spaces of as
The numbers 𝔡C and defined by
are called the defect indices of
Proof: Consider the equation
where f ∈ H and Then
Equation (3.3) implies that is spanned by two independent solutions. In fact, let φ(x, λ) and be two solutions of (2.2) satisfying
where c is a constant and
(2.5) needs the solutions of (2.2) satisfying the condition
Clearly φ and satisfies (3.5) and φ can not be represented by a constant of If there exists any other solution ψ(x, λ) of (2.2) satisfying
where c1 is another constant different from c then becomes a solution of (2.2) satisfying (3.5) and ψ(x, λ) may be introduced by φ(x, λ).
where d1 and d2 are constants and is spanned by φ(x, i) and
With a similar argument one may see that
and therefore is spanned by φ(x, −i) and
This completes the proof.□
Definition 3.3.  The classesC0. andC.0 are defined as
C00 is defined by C00 = C0. ∩ C.0.
Proof: This follows from (3.2), and the equalities
The class C0 consists of those c.n.u. contractions T for which there exists a non-zero function u ∈ H∞ (Hp denotes the Hardy class) such that u(T) = 0. Since C belongs to the class C00 with finite defect numbers this implies the following .
4. Characteristic Function
We shall consider the inner product on the quotient space as follows
where P is the natural projection with The completion of is denoted by with respect to the corresponding norm. Similarly and P∗ is defined by One has
and are the boundary spaces of From (4.1) we get
From (4.2) and (4.3) we may set
Setting we define the following isometric isomorphisms
where Then we may introduce the characteristic function of
Theorem 4.1. The characteristic matrix-function of is given by
Proof: Consider the equation
(4.4) implies that y ∈ P−1 Ψc with and therefore
Using (4.6) and (4.7) we obtain
From (4.8) we should find a solution u = z − y of the Equation (2.2) satisfying (3.5). Therefore, we may set , where φ is the solution of (2.2) satisfying the conditions in (3.4). Consider the equation
Since and we get from (4.10)
Similarly the equation
(4.6) and (4.9) show that
Consequently (4.11) – (4.13) complete the proof.□
Remind that a function Θ(ζ) whose values are bounded operators from a Hilbert space ℍ to a Hilbert space ℍ∗, both separable and which has a power series expansion
whose coefficients are bounded operators from ℍ to ℍ∗. Moreover assume that
Such a function with the spaces ℍ and ℍ∗ is called bounded analytic function. If const = 1 then it is called contractive analytic function. The contractive analytic function Θ is said to be inner if Θ(eit) is isometry from ℍ into ℍ∗ for almost all t.
Since there exists a connection between the characteristic function of and the characteristic function of with the rule
we have the following.
Corollary 4.2. The characteristic function of C is given by
Since is a c.n.u. contraction belonging to the class C.0 we have the following.
Theorem 4.3. ΘC(μ) is inner.
Corollary 4.4. det ΘC(μ) is inner.
An operator A ≥ 0 on a Hilbert space is said to be of finite trace if A is compact and its eigenvalues is finite. This sum is called the trace of A [1, 18]. A contraction C on a Hilbert space H is called weak contraction if
(i) its spectrum does not fill the unit disc D,
(ii) I − C*C is of finite trace.
Since every C0 contraction with finite-multiplicity is a weak contraction , p. 437, we may introduce the following.
Theorem 4.5. is of finite trace.
The following Theorem is obtained from Nikolskiĭ , p. 134.
Theorem 4.6. The followings are satisfied:
(i) The root functions of are complete in H,
(ii) The roots functions of are complete in H,
where μ belongs to the point spectrum of and d(μ) is the rank of the Riesz projection at a point μ in the set of point spectrum.
Proof: The proof follows from the fact is a Blashke product. So we shall prove this fact.
By Corollary 4.4 we may write
where b > 0, Im λ > 0 and 𝔹(λ) is a Blashke product in the upper half-plane. Hence
For λs = is we have from (4.15) that the following possibilities may occur:
(i) as s → ∞,
(ii) as s → ∞,
(iii) and as s → ∞.
In fact (iii) is possible because in this case λ is an eigenvalue of and this implies that λ∞ is an eigenvalue of the operator or equivalently 1 is an eigenvalue of the c.n.u. contraction However the latter one is not possible. Therefore, this completes the proof.□
Definition 4.7. Let all root functions of the operatorL span the Hilbert space. Such an operator is called complete operator. If everyL−invariant subspace is generated by root vectors of L belonging to the subspace then it is said L admits spectral synthesis.
Since every complete operator in C0 admits spectral synthesis , we obtain the following.
Theorem 4.8. admits spectral synthesis.
Since the root functions of span H then those of must span H  (p. 42). Consequently we may introduce the following.
Theorem 4.9. Root functions of associated with the point spectrum of in the open upper half-planeIm λ > 0 span the Hilbert space H.
5. Dilation Operator and Its Eigenfunctions
In this section we investigate the properties of selfadjoint dilation of the operator and eigenfunctions of selfadjoint dilation.
5.1. Selfadjoint Dilation of the Maximal Dissipative Operator
Following theorem gives the selfadjoint operator with free parameters .
Theorem 5.1.1. The minimal selfadjoint dilation of the maximal dissipative operator in the space
has the form
and the domain of is given by the conditions
where denotes the Sobolev space.
The isometries are the free parameters. In the case that then one may consider the boundary spaces and instead of and Then following Lemma gives a direct approach for the dilation .
Lemma 5.1.2. The minimal selfadjoint dilation in the space of the maximial dissipative operator inH with finite defects has the form
where and are the isometric isomorphisms and the domain of is given by the conditions
If is dense in H one may consider and
The following Corollary now may be introduced .
Corollary 5.1.3. The selfadjoint dilation of the maximal dissipative operator with finite defects such that is dense in H has the form
and the domain of is given by the conditions
Now using Corollary 5.1.3 we may introduce the following.
Theorem 5.1.4. The selfadjoint dilation of the maximal dissipative operator in the space
is given by the rule
whose domain is given by the conditions
Proof: Let y ∈ with and with Then if and only if
and (5.2) implies
Similarly if and only if
(5.3) shows that
and (5.4) shows
Therefore the proof is completed.□
5.2. Eigenfunctions of the Dilation
As is pointed out in Solomyak  the generalized eigenfunctions of the dilation may be introduced by incoming eigenfunctions
and outgoing eigenfunctions
where r ∈ ℝ−, s ∈ ℝ+, c ∈ E, and λ ∈ ℝ.
Therefore we may introduce the following.
Theorem 5.2.1. The incoming and outgoing eigenfunction of can be introduced by
wherer ∈ ℝ−, s ∈ ℝ+, λ ∈ ℝ.
Proof: Consider the equation
Therefore the left-hand side of (5.5) can be introduced as
Now consider the equation
where y − z = B(λ)φ(x,λ), and A similar argument completes the proof.□
6. Conclusion and Remarks
This paper provides a new method to analyze the spectral properties of some third-order dissipative boundary value problems and it seems that such a method has not been introduced previously for third-order case. This method is very effective and can be applied for other odd-order dissipative operators generated by suitable odd-order differential equation and boundary conditions.
Finally we should note that the differential expression ℓ can also be handled as the following
where r is a suitable function. Then with some modifications a similar boundary value problem as (2.2), (2.4) can be analyzed.
All datasets generated for this study are included in the manuscript and the supplementary files.
All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.
Conflict of Interest Statement
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
4. Nikolskii NK, Vasyunin VI. A Unified approach to function models, and the transcription problem. In: Dym H, Goldberg S, Kaashoek MA, Lancaster P editors. The Gohberg Anniversary Collection. Operator Theory: Advances and Applications, Vol 41. Basel: Birkhäuser (1989). p. 405–34.
5. Solomyak BM. Functional model for dissipative operators. A coordinate-free approach, Journal of Soviet Mathematics 61 (1992), 1981–2002; Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR.(1989) 178:57–91.
Keywords: coordinate-free approach, model operator, characteristic function, spectral analysis, dissipative operator
Citation: Uğurlu E and Baleanu D (2019) Coordinate-Free Approach for the Model Operator Associated With a Third-Order Dissipative Operator. Front. Phys. 7:99. doi: 10.3389/fphy.2019.00099
Received: 05 April 2019; Accepted: 21 June 2019;
Published: 10 July 2019.
Edited by:Cosmas K. Zachos, Argonne National Laboratory (DOE), United States
Reviewed by:Ebenezer Bonyah, University of Education, Winneba, Ghana
Daniel Luiz Nedel, Universidade Federal da Integração Latino-Americana, Brazil
Copyright © 2019 Uğurlu and Baleanu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Ekin Uğurlu, email@example.com