Relationships among Three Multiplicities of a Differential Operator’s Eigenvalue ()
1. Introduction
The study of spectral problems for linear ordinary differential equations (more generally, quasi-differential equations, to be abbreviated as QDE) originated from a series of seminal papers of Sturm and Liouville in [1] -[3] , while the singular case started with the celebrated work of Weyl in 1910 introducing the limit-point (LP) and limit-circle (LC) dichotomy [4] . Another important milestone in this area is the Glazman-Krein-Naimark (GKN) theorem [5] in 1950, see also [6] for generalizations (which will be included in the theorem). This theorem gives a one-to-one correspondence between the self-adjoint differential operators in a Hilbert function space representing a given QDE and the unitary isometries on an appropriate finite-dimensional subspace (or equivalently, certain Lagrange subspaces of some finite dimensional quotient space, see [7] ). In both regular case and singular case, the GKN theorem also yields a characterization of the self-adjoint operators in terms of linear complex boundary conditions (BC). So, the spectral problem of a linear ordinary differential equation (QDE) with boundary conditions maybe turn to study it of a linear ordinary differential operator [5] [8] [9] . It is well-known that the investigation of different self-adjoint extensions of symmetric operators and the estimation of the location and multiplicity of their point spectra are among fundamental mathematical problems arising in any quantum mechanical model ([10] , Chapter VIII, Section 11). While the eigenvalues of a linear ordinary differential operator were studied, three kinds of multiplicity (analytic multiplicity, geometric multiplicity and algebraic multiplicity) of an eigenvalue were defined and accompanied. Three kinds of multiplicity are often confused in some paper, and a problem how about the relationships among three kinds of multiplicity of an eigenvalue has arisen.
The differential equation
(1.1)
with boundary conditions
(1.2)
will be studied in present paper, where 
and 
a.e. on
, the functions 
are real-valued, measurable over 
and Lebesgue integrable on all compact subset of
.
The endpoint 
is said to be regular if 
and each of the functions 
is integrable in every interval
; otherwise 
is said to be singular. Similar definitions apply to endpoint
. The differential expression 
is said to be regular if it is regular at both endpoints, and otherwise is said to be singular.
We assume throughout that (1.1) is regular, and the functions 
are sufficiently smooth and Lebesgue integrable on
, then the boundary conditions may be written as
:
In the regular case, the GKN characterization of self-adjointness in terms of the complex boundary conditions can be simply expressed as the algebraic equation
(1.3)
and
, where 
and 
which come from the coefficient matrix
of the boundary conditions are 
complex matrices, while 
is a fixed matrix for Lagrange bilinear form 
of differential expression 
in (1.1).
Let
, and 
be the transpose of
, then, for
(the domain of maximal operator generated by
),
and following results are true (see [5] [6] [8] ):
1)
;
2)
.
It is well-known that the spectrum of such a problem consists of an infinite number of real eigenvalues and has no finite accumulation point. The eigenvalues are precisely the zeros of an entire function
, called the characteristic function of the problem. The analytic multiplicity of an eigenvalue is the order of the eigenvalue as a zero of
, the geometric multiplicity of an eigenvalue is the number of linearly independent eigenfunctions for the eigenvalue, and the algebraic multiplicity of an eigenvalue is the dimension of its root subspace (a subspace is spanned by the eigenvectors and its associated vectors).
The analytic multiplicity of an eigenvalue gives the maximum number of new eigenvalues into which the original eigenvalue can split when the spectral problem involved. So, it is natural to use the analytic multiplicity to count eigenvalues, and the analytic multiplicity plays an important role in the study of the dependence of the eigenvalues of a spectral problem on the differential equation boundary value problem (see, for example, [5] [11] -[14] ). The geometric multiplicity is always defined and is more widely used in spectral theory (see [4] [5] [8] -[10] [15] ). However, the algebraic multiplicities and associated vectors (functions) of an eigenvector (eigenfunction) are used for the completeness of eigenvectors (eigenfunctions) of non self-adjoint operators (see [15] - [17] ). Therefore, it is of fundamental interest to compare the three multiplicities of an eigenvalue for differential operators.
Naimark studied the relationship between the algebraic and analytic multiplicities of an eigenvalue of highorder linear differential operators in [5] , and obtained the equivalence of the algebraic and analytic multiplicities of an eigenvalue of high-order linear differential Equation (1.1) with linear boundary conditions (1.2). From then, the relationships among the three multiplicities have been payed a good deal of attentions, and have had a strong appeal to studying.
Over the last decades, the fact that the analytic and geometric multiplicities of an eigenvalue of self-adjoint Sturm-Liouville problems are equal has been solved ([5] [11] -[14] [18] -[22] ). Sturm-Liouville problems (SLP) are differential equation:
(1.4)
with boundary conditions
(1.5)
where 
a.e. on
, 
, 
and 
are 
matrices,
.
For the regular SLP, i.e. both endpoint 
and 
are regular
, the problem (1.4)-(1.5) is a self-adjiont SLP if the coefficients matrixes in (1.5) satisfy (1.3) and
.
The equality of the analytic and geometric multiplicities in the case of separated boundary conditions (BC) was proved in [13] , while the case of coupled BC’s was settled in [18] . The equality of the two multiplicities in the case of singular self-adjoint SLP with LC non-oscillatory end points was shown in [19] using a regularization; the equality in the case of all singular self-adjoint SLP with LC end points was recently established in [21] , based on the equality of the regular self-adjoint SLP and certain regular approximations.
The proof in [18] uses some sophisticated identities involving 
(the function 
in [18] differs from 
by a constant) and certain values of a fundamental set of solutions of (1.4). It seems to us that it is very hard to find similar identities for a higher order QDE by this way.
The basic idea of proof in [19] is as follows: for any eigenvalue 
of geometric multiplicity 1, they can give a smooth curve in the space 
of self-adjoint BC through the BC 
involved such that the composition of 
with a continuous eigenvalue branch through 
has a non-zero derivative along the curve at
, which then implies that 
has a non-zero derivative at
. Here, a continuous eigenvalue branch through 
means a continuous function defined on a neighborhood O of A in 
such that its value at A equals 
and its value at each BC 
is an eigenvalue for
.
In [22] , we generalized the proof in [19] , and gave a new and unified proof of the equality between the analytic and geometric multiplicities of any eigenvalue of self-adjoint SLP in the regular case based on the geometric classifications of self-adjoint BC.
The classifications of self-adjoint BC about higher order differential operator are more complicated in geometric [7] , and it seems rather complicated to find similar identities for a higher order QDE by the same method in [22] . Recently, this result was generalized to the higher order differential equations with self-adjoint boundary conditions in [23] , but the proof was not easy.
In order to classify three multiplicities of an eigenvalue for linear differential operators, to obtain the relationships among three multiplicities, and to have a short and non-technical presentation so that the main idea of the general proof can be made transparent, we only give the general proof for regular self-adjoint QDE in this paper. For arbitrary self-adjoint nth-order QDE in singular end points with defect index n, the proof is basically the same (with only obvious minor changes), but the introduction of the self-adjoint BC and the definition of the characteristic function are more involved (see, for example, [7] or [24] ).
It is the main purpose, therefore, in the present work, to give the definitions of three kinds of multiplicities of an eigenvalue for linear differential operators and the relationships among them. In Section 2, we give the definitions of the geometric and algebraic multiplicities and the relationship between them. The definition of the analytic multiplicity for an eigenvalue of linear differential operators and the relationship between its analytic and algebraic multiplicities is given in Section 3. In last section, we have the equalities among three multiplicities of an eigenvalue for a self-adjoint linear differential operator.
2. Geometric and Algebraic Multiplicities
The definitions of the geometric and algebraic multiplicities for an eigenvalue of a linear operator are from [15] . Recall that a complex number 
is called an eigenvalue of linear operator 
if there exists a non-zero element 
such that
; in this case, 
is called an eigenfunction of 
for
. The eigenfunctions for 
span a subspace of
, 
, called the eigenspace for
; and the geometric multiplicity of 
is the dimension of its eigenspace, denoted by
, i.e.
A non-zero element 
is called a root vector of 
for a complex number 
if 
for some
. In this case, 
must be an eigenvalue. Together with the vector
, the root vectors of 
span a linear subspace of
, 
, called the root lineal for
; and the algebraic multiplicity of 
is the dimension of its root lineal, denoted by
, i.e.
If an element 
is not an eigenvector for
, then it is a root vector for 
if and only if there is a 
such that 
is an eigenvector for 
provided 
and 
for
. A root vector is called an associated vector (or adjoint vector) if it is not an eigenvector. The theory of associated functions (vectors) of differential operator was originated by Keldysh [16] .
In general, the system of eigenvectors and associated vectors of 
is not complete in Hilbert space
. From the definition of the algebraic multiplicities and geometric multiplicity of an eigenvalue of linear operator T in H, the eigenvectors of 
belong to root lineal of 
(i.e. 
), so, we have the following result:
Theorem 2.1. The geometric multiplicity of any eigenvalue of linear operator 
in Hilbert space does not exceed its algebraic multiplicity. i.e.
(2.1)
In general, the algebraic multiplicity is grater than the geometric multiplicity of an eigenvalue of operators. For example.
Example 2.2. We consider an operator 
in
,
(2.2)
are eigenvalues of operator
, 
is an eigenvector for the eigenvalue
, and the eigenspace
. So, the geometric multiplicity of the eigenvalue 
is equal to 1, i.e.
. But, 
have solutions
, where
, 
, the associated vector of 
is 
or
, and the root lineal 
or
. Thus, the algebraic multiplicity of eigenvalue 
is equal to 2, i.e.
.
Example 2.3. We consider the differential equation
(2.3)
with boundary conditions
(2.4)
in Hilbert space
. After simple calculation, 
is an eigenvalue of boundary problem (1.3)-(1.4), 
is corresponding eigenvector, and the eigenspace of 
is
. So, the geometric multiplicity of eigenvalue 
is equal to 1, i.e.
. But, 
have solutions
, where
, but 
has not any solutions, the associated vector of 
is 
and the root lineal
. Thus, the algebraic multiplicity of eigenvalue 
is equal to 2, i.e.
.
If 
is a spacial operator in Hilbert space
, then the geometric multiplicity of any eigenvalues of linear operator 
maybe equal to its algebraic multiplicity, such as:
Theorem 2.4. If 
is a self-adjoint operator in Hilbert space
, then the geometric multiplicity of any eigenvalues of linear operator 
is equal to its algebraic multiplicity.
Proof: We only need to prove that the eigenspace for an eigenvalue 
of operator 
is equal to the root lineal of
.
From the definitions of eigenspace and root lineal of the eigenvalue
, we have
where 
and 
represent the eigenspace and the root lineal of
, 
is an (isolated) eigenvalue of
, i.e.
.
If 
, then there exist a associated function 
(
,
) and an eigenfunction 
(
,
), such that
, and
.
From the last equation, we have
(2.5)
By self-adjointness of operator
, the eigenvalue 
is a real number, and 
for any
. Then,
(2.6)
and there is a contradictory to the fact
. Therefore,
(2.7)
and the proof is complete. □
3. Analytic and Algebraic Multiplicities
We also introduce some notations here and review some basic facts about the problem of differential Equation (1.1) with boundary conditions (1.2). Let 
be the fundamental solution of (1.1) satisfying
(3.1)
and 
denote the Wronskian of (1.1) with respect to 
, then
. The determinant of matrix 
is denoted by
, i.e.
(3.2)
Theorem 3.1. The 
is an entire function of
, and a complex number 
is an eigenvalue of the boundary value problem consisting of (1.1) and (1.2) if and only if
(3.3)
Proof: Simply calculate or see [8] and [5] .
The entire function
, unique up to a non-zero constant multiple, is called the characteristic function for the boundary value problem consisting of (1.1) and (1.2). The analytic multiplicity of an isolated eigenvalue is the order of the eigenvalue as a zero of
, denoted by
.
From the definition of analytic multiplicity, only the eigenvalues of boundary value problems have analytic multiplicity. In general, the analytic multiplicity isn’t equal to the other multiplicities for an eigenvalue of some boundary value problems.
Example 3.2. We still study the boundary value problem (2.3)-(2.4) in Example 2.3. The characteristic function 
for the boundary value problem (2.3) and (2.4) can be easily obtained after calculating.
(3.4)
is a 3-order zero of the
, so, the analytic multiplicity of the eigenvalue 
is 3.
With the results in Example 2.3, we have proven that the geometric multiplicity, algebraic multiplicity and analytic multiplicity of the eigenvalue 
of boundary value problem (1.3)-(1.4) are 1, 2 and 3, and 
for the eigenvalue 
of boundary value problem (1.3)-(1.4).
But the linear differential Equation (1.1) with linear boundary conditions (1.2), Naimark had the following theorem in [5] (Chapter:
).
Theorem 3.3. The analytic multiplicity of any eigenvalues of the boundary value problem consisting of (1.1) and (1.2) is equal to its algebraic multiplicity. i.e.
(3.5)
The algebraic multiplicity of an eigenvalue of self-adjoint SLP factually is the analytic multiplicity in [11] - [14] , because the authors realized that the equivalence of the algebraic and analytic multiplicities for any eigenvalues of self-adjoint SLP is a foundational fact.
4. Equalities among Three Multiplicities of an Eigenvalue of Self-Adjoint Differential Operators
In this section, we first collect some basic statements about higher order differential operator (especially, the self-adjoint differential operator with high-order), and then prove the equalities among analytic, algebraic and geometric multiplicities.
For any
, we use 
to denote the vector space of 
matrices with complex entries and 
its open subset consisting of the elements with the maximum rank
. 
and 
are defined similarly. When a capital Latin or Greek letter stands for a matrix, the entries of the matrix will be denoted by the corresponding lower case letter with two indices. If
, then 
and 
are the transpose and the complex conjugate transpose of
, respectively. The general linear group 
is a complex Lie group under the matrix multiplication, while the special linear group 
consists of all elements of 
with determinant 1 and is a Lie subgroup of
. 
and 
are defined similarly. Let 
be an open interval, bounded or unbounded. Assume that 
is one of the spaces
, 
, 
and
. We denote by 
the space of Lebesgue integrable S-valued functions on
, and 
the space of S-valued functions which are absolutely continuous on all compact subintervals of
.
For the rest of this paper, we use 
to denote a fixed number satisfying
, and 
with
. When 
is even 
and 
is finite interval, there is a special case of the QDE (1.1). This special case was studied by Naimark [5] and Weidman [9] .
For any 
and 
a.e. on
, 
are integrable over
, and 
are 
times continuous differentiable function on
,
. We define
(4.1)
then 
also belongs to
. Let
, the quasi-derivatives of 
associated with 
are given by
and
then, ordinary differential Equation (1.1) is equivalent quasi differential equation (QDE)
(4.2)
Thus, 
and 
can be used as the coefficient matrix of (4.2), (4.2) is equivalent to its matrix form
(4.3)
The quasi-differential expression in 
associated with 
for 
or 
or 
are of special interest:
while, in general, 
is equal to
We now turn to the BVP consisting of the general QDE (1.2) and a (linear two-point) BC defined by
(4.4)
where 
such that
. Note that equivalent linear algebraic equations of the form (1.2) define the same BC. Each value of 
for which the QDE (4.2) has a nontrivial solution satisfying the BC (1.4) is called an eigenvalue of the BVP consisting of (4.2) and (4.4), and a solution to this problem is called an eigenfunction for this eigenvalue.
Since (4.2) has exactly 
linearly independent solutions, the geometric multiplicity of any eigenvalue is an integer not smaller than 1 and not greater than
.
Theorem 4.1. A number 
is an eigenvalue of the boundary value problem consisting of (4.2) and (4.4) if and only if
(4.5)
where 
is the fundamental solution of (1.3) satisfying
, 
is the identity matrix.
Proof: Simply calculate or see [7] . □
Theorem 4.2. 
Proof: From the definition of quasi-derives of 
associated with
, we have
, where
, 
, and
(4.6)
is a inverse matrix function on 
because of 
over
. Let 
denote the Wronskian of (1.1) with respect to
, 
, then 
, and
. Thus, 
and
. So, the equality in this theorem was proved.
A coefficient matrix 
is said to be symmetric if
(4.7)
where
(4.8)
The boundary condition (1.4) is said to be self-adjoint if 
and
(4.9)
Theorem 4.3. The differential operator 
associated with (1.1) and (1.2) (or (4.2) and (4.4)) is a self-adjoint operator if the boundary conditions (1.2) (or (4.4)) satisfy (1.3) and 
(or is self-adjoint).
Proof: See [7] . □
Theorem 4.4. The analytic, algebraic and geometric multiplicity of any eigenvalue for a self-adjoint differential operator associated with (1.1) and (1.2) (or (4.2) and (4.4)) are equal.
Proof: From Theorem 2.4 and Theorem 3.3, we get the conclusion immediately. □
Corollary 4.5. The analytic, algebraic and geometric multiplicity of any eigenvalue of the differential operator L associated with (1.1) and (1.2) (or (4.2) and (4.4)) are equality when the coefficient matrixes A and B satisfy (1.3) (or 
and 
satisfy (4.9)), and equal to the multiplicities of the eigenvalue as a zero point of 
in (3.3) (or 
in (4.5)).
Example 4.6. We consider the differential equation
(4.10)
with boundary conditions
(4.11)
in Hilbert space
. After simple calculation, 
are eigenvalues of boundary problem (4.10)-(4.11), 
and 
are corresponding eigenvectors for
, and the eigenspace of 
is
. So, the geometric multiplicity of eigenvalue 
is equal to 2, i.e.
. Boundary problem (4.10)-(4.11) is a self-adjoint problem because of boundary conditions (4.11) satisfying (4.9), so, the algebraic multiplicity of eigenvalue 
is also equal to 2, i.e.
. We also have that
(4.12)
are 2-order zero of the
, so, the analytic multiplicity of the eigenvalue 
is 2, i.e.
. Thus, the analytic, algebraic and geometric multiplicity of any eigenvalue 
for boundary problem (4.10)-(4.11) are equal to 2, i.e.
.
Funding
Work partially supported by the National Nature Science Foundation (11171295).
NOTES
*Corresponding author.