1. Introduction
In [1] Akhiezer and Glazman studied that the self-adjoint extension S of the minimal operator
generated by a formally symmetric differential expression
with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. In [2] - [5] , the relationship between the square-integrable solutions for real values of the spectral parameter and the spectrum of self-adjoint ordinary differential operators of even order with real coefficients and arbitrary deficiency index are studied.
The main results of Evans, Sobhy El-Sayed and others in [6] [7] concerning the general ordinary quasi-differential operators are generalized to
-spaces with an arbitrary interval
(see [8] [9] ). Also, the results includes those in [10] - [16] .
The operators which fulfill the role that the self-adjoint and maximal symmetric operators play in the case of a formally symmetric expression
are those which are regularly solvable with respect to the minimal operators
and
generated by a general ordinary quasi-differential expression
and its formal adjoint
respectively, the minimal operators
and
form an adjoint pair of closed, densely-defined operators in the underlying
-space,
that is
. Such an operator S satisfies
and for some
,
is a Fredholm operator of zero index, this means that S has the desirable Fredholm property that the equation
has a solution if and only if f is orthogonal to the solution space of
and furthermore the solution spaces of
and
have the same finite dimension. This notion was originally and due to Visik [17] - [24] .
Our objective in this paper, is to generalize the results in [11] - [16] for symmetric case and results of Sobhy El-sayed in [18] - [23] for general quasi-differential operators to
spaces in an analogue of Hilbert Frentzen in [8] [9] . A rather general class of quasi-differential expressions
with matrix-valued coefficients and the associated maximal operators
and minimal operator
as maps of a subspace of
into
for arbitrary
. Also, we have shown in the case of one singular end-point that all well-posed extensions of the minimal operator
generated by such expression
and their formal adjoint on the interval
with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. The domains of these operators are described in terms of boundary conditions involving the p-integrable solutions of the quasi-differential equation
and the adjoint equation
(see [21] ).
For (formally) symmetric differential expressions
and
space much work has been done on the problem of finding self-adjoint differential operator
with the aid of boundary conditions as in A. N. Krall and A. Zettl [13] , Naimark [14] , D. Race [15] [16] , Wang [25] , Zettl [26] and Zhikhar [27] to mention only a few. If
is not symmetric, one can ask for
which are Fredholm operators with index zero. For
space and second order scalar differential expressions this question was answered by Evans and Edmunds [4] [5] , for scalar differential expressions of general order on half-open intervals by Evans and Sobhy [6] and on open intervals by Sobhy El-sayed in [17] - [23] .
The results herein include those of D. Race [15] [16] , but also gives a description of the domain of the maximal symmetric extensions of
in the case when
is a symmetric operator of unequal deficiency indices. Another noteworthy special case of our result is that of Zhikhar [27] concerning the J-self-adjoint extensions of J-symmetric differential operators
, where J denotes complex conjugation.
The results include those of Jiong Sun [2] , R. Agarwal [3] and Zettl [26] concerning self-adjoint realizations of symmetric operators when the minimal operator
has equal deficiency indices. Also, includes those of Evans [4] [5] , D. Race [15] [16] and N. A. Zhikhar [27] for the special case that concerns the J-self-adjoint operators, where J denotes complex conjugation. If the deficiency indices are unequal the maximal differential operators
are determined by the results herein.
2. Notation and Preliminaries
We begin with a brief survey of adjoint pairs of operators and their associated regularly solvable operators; their full treatment can be found in ( [5] , Chapter III), ( [6] - [25] ). The domain and range of a linear operator T acting in a Hilbert space H will be denoted by
and
respectively and
will denote its null space. The nullity of T, written
, is the dimension of
and the deficiency of T, written
, is the co-dimension of
in H; thus if T is densely defined and
is closed, then
. The Fredholm domain of T is (in the notation of [4] ) the open subset
of
consisting of those values of
which are such that
is a Fredholm operator, where I is the identity operator in H. Thus
if and only if
has closed range and finite nullity and deficiency. The index of
is the number
, this being defined for
.
Two closed densely defined operators A and B acting in a Hilbert space H are said to form an adjoint pair if
and, consequently
; equivalently,
for all
and
, where
denotes the inner-product on H.
Definition 2.1: The field of regularity
of A is the set of all
for which there exists a positive constant
such that
for all
, (2.1)
or equivalently, on using the Closed Graph Theorem,
and
is closed.
The joint field of regularity
of A and B is the set of
which are such that
,
and both
and
are finite. An adjoint pair A and B is said to be compatible if
.
Definition 2.2: A closed operator S in Hilbert space H is said to be regularly solvable with respect to the compatible adjoint pair of A and B if
and
, where
.
Definition 2.3: The resolvent set
of a closed operator S in H consists of the complex numbers
for which
exists, is defined on H and is bounded. The complement of
in
is called the spectrum of S and written
. The point spectrum
, continuous spectrum
and residual spectrum
are the following subsets of
(see [5] [11] [12] [18] [19] [20] [23] [24] [25] ).
, i.e., the set of eigenvalues of S;
;
.
For a closed operator S we have,
(2.2)
An important subset of the spectrum of a closed densely defined operator S in H is the so-called essential spectrum. The various essential spectra of S are defined as in [5, Chapter 9] to be the sets:
(2.3)
where
and
have been defined earlier.
Definition 2.4: For two closed densely defined operators A and B acting in H, if
and the resolvent set
of S is nonempty (see [5] ), S is said to be well-posed with respect to A and B.
Note that, if
and
then
and
so that if
and
are finite, then A and B are compatible, in this case S is regularly solvable with respect to A and B. The terminology “regularly solvable” mentioned by Visik in [4] [5] [6] [23] and [24] , while the notion of “well-posed” was introduced by Zhikhar in [27] .
Theorem 2.5: (cf. ( [5] , Theorem III.3.1)). Let T be a closed, densely-defined operator in H with
. Then for any closed extension S of T and
we have
. If
and
, then
If A is symmetric operator, then with
, the operators S which conform to Definition 2.1 are the self-adjoint or maximal symmetric extensions of A. In this case when A is J-symmetric relative to complex conjugation J, A and
form an adjoint pair with
; any J-self-adjoint extension of A whose resolvent set is nonempty, is regularly solvable with respect to A and JAJ. For the above results, see (( [5] , Chapter III), [18] [19] [20] [23] [24] [25] and [27] ).
Throughout this paper, let
denote either
, the field of real numbers, or
, the field of complex numbers. For a positive integers k and m let
denote the vector space of
matrices with
-valued entries and
the subset of
consisting of all non-singular matrices. For
, let
denote the transpose and
the adjoint, i.e., the complex conjugate transpose of.
If Z is a subset of
and I is an interval,
denotes the set of Lebesgue measurable maps of I into Z and
the set of locally absolutely continuous maps. Measurable maps are regarded as equal if they are equal almost everywhere on I. Further we define:
,
for all
and
,
for all
,
and
for all
.
If
, then
is always chosen such that
. We always
assume that
. If
for some positive integer s, then
for
and
is a subspace of
, where
denotes the complex conjugate transpose. We refer to [8] and [9] for more details.
3. The Quasi-Differential Expressions
Let I be an interval with endpoints
, let
be positive integrs and
. The quasi-differential expressions are defined in terms of a Shin-Zettl matrix F on the interval I.
Definition 3.1 [7] [8] [9] : The set
of Shin-Zettl matrices on I consists of matrices is defined to be the set of all lower triangular matrices
of the form
,
whose entries are complex-valued functions on I which satisfy the following conditions:
(3.1)
For
we define
as the
matrix obtained from F by removing the first row and the last column, i.e.,
,
Definition 3.2 [8] [9] : For
, the quasi-derivatives associated with
are defined by
(3.2)
where the prime’ denotes differentiation.
The quasi-differential expression
associated with
is given by:
(3.3)
this being defined on the set:
where
, denotes the set of functions which are locally absolutely continuous on every compact subinterval of I.
For
, we define
.
Clearly the maps
and
are linear.
In analogy to the adjoint and the transpose of a matrix, there are two different “(formal) adjoint” of a quasi-differential expression, we refer to [8] [9] and [21] for more details.
In the following we always assume that
and
.
The formal adjoint
of
is defined by the matrix
which given by:
(3.4)
where
is the conjugate transpose of
and
is the non-singular
matrix
, (3.5)
being the Kronecker delta. If
then it follows that
. (3.6)
The quasi-derivatives associated with the matrix
in
are therefore
(3.7)
and
for all
, (3.8)
(3.9)
Note that:
and so
. We refer to [5] [6] [7] [8] [9] and [21] for a full account of the above and subsequent results on quasi-differential expressions.
For
,
and
, we have Green’s formula,
(3.10)
where,
(3.11)
see [5] [6] [7] [8] [9] [17] [24] and [26] . Let
be a non-negative weight function with
and
(for almost all
). Then
denotes the Hilbert function space of equivalence classes of Lebesgue measurable functions such that
for all
and
. (3.12)
The equation
on I, (3.13)
is said to be regular at the left end-point
, if for all
,
otherwise (3.13) is said to be singular at a. If (3.13) is regular at both end-points, then it is said to be regular; in this case we have,
(3.14)
We shall be concerned with the case when a is a regular end-point of Equation (3.13), the end-point b being allowed to be either regular or singular. Note that, in view of (3.14), an end-point of I is regular for (3.13), if and only if it is regular for the equation
on I. (3.15)
Definition 3.3 [5] - [21] [26] :
1) The maximal operators corresponding to
are defined as operators from a subspaces of
into
,
are arbitrary.
for all
,
for all
.
The subspaces
and
of
are the domains of the so-called maximal operators
and
respectively.
2) For the regular problem the minimal operators
and
are the restrictions of
and
to the subspaces:
(3.16)
The subspaces
and
are dense in
and
and
are closed operators (see [4] ( [5] Section 3) and [6] - [21] ).
In the singular problem we first introduce the operators
and
;
being the restriction of
to the subspace
.
and with
defined similarly. These operators are densely-defined and closable in
; and we define the minimal operators
and
to be their respective closures (see [5] - [14] [21] and [26] ). We denote the domains of
and
by
and
respectively. It can be shown that:
(3.17)
because we are assuming that a is a regular end-point. Clearly
and
are linear operators of
into
and
.
Moreover, in both regular and singular problems, we have
,
, (3.18)
see [6] - [16] and ( [26] , Section 5) in the case when
and compare with treatment in ( [5] , Section III.10.3), [6] in general case. Also, we refer to [17] - [22] for more details.
Corollary 3.4 (cf. ( [8] , Corollary 3.10) and [9] ):
a) If
is symmetric, then
is symmetric in the Hilbert space
.
b) If
is J-symmetric, then
is J-symmetric in the Hilbert space
.
4.
-Solutions
In this section, we shall concerned with
-Solutions of general ordinary quasi-differential equations, and we denote for
by
and
by
.
Denote by
and
the sets of all solutions of the equations
(4.1)
and
(4.2)
respectively. Let
be the solutions of the homogeneous equation
satisfying:
for all
for fixed
,
. Then
is continuous in
for
,
, and for fixed t it is entire in
. Let
denote the solutions of the adjoint homogeneous equation
satisfying:
for all
.
Suppose
, by [5] [21] [22] [23] [24] and [26] , a solution of the quasi-differential equation
(4.3)
satisfying
is giving by:
where
stands for the complex conjugate of
and for each
,
is constant which is independent of
(but does depend in general on t).
The next lemma is a form of the variation of parameters formula for a general quasi-differential equation is giving by the following Lemma.
Lemma 4.1: Suppose
locally integrable function and
is the solution of Equation (4.3) satisfying:
for
,
.
Then
(4.4)
for some constants
, where
and
are solutions of Equations (4.1) and (4.2) respectively,
is a constant which is independent of t.
Theorem 4.2: (cf. [5] [6] ). Let
be a regular quasi-differential expression of order n on the interval [a, b]. For
, the equation
has a solution
satisfying
If and only if f is orthogonal in
to solution space of
, i.e.,
Corollary 4.3 (cf. [19] [20] [21] ), As a result from Theorem 4.2, we have that
Lemma 4.4 [22] : (Gronwall’s inequality). Let
and
be two real-functions defined, non-negative and u,
for
, and if
for some positive constant c, then
. (4.5)
Lemma 4.5: Suppose
, and suppose that the Conditions (3.1) are satisfied. Then, given any complex numbers
,
and
, there exists a unique solutions of
which satisfies
Proof: The proof is similar to that in (14], part II, Theorem 16.2.2) and therefore omitted.
Lemma 4.6: Suppose that for some
all solutions of Equations (4.1) and (4.2) are in
. then all solutions of Equations (4.1) and (4.2) are in
for every complex number
.
Proof: The proof is similar to that in [21] [22] [23] [24] , and therefore omitted.
Lemma 4.7: If all solutions of the equation
are bounded on
and
for some
. Then all solutions of the equation
are also bounded on
for every complex number
.
Lemma 4.8: Suppose that for some complex number
all solutions of Equation (4.1) are in
and all solutions of (4.2) are in
. Suppose
, then all solutions of Equation (4.3) are in
for all
Proof: Let
,
be two sets of linearly independent solutions of Equations (4.1) and (4.2) respectively. Then for any solutions
of the equation
which may be written as follows
and it follows from (4.4) that
(4.6)
for some constants
. Hence
(4.7)
Since
and
for some
, then
for some
and
. Setting:
(4.8)
then
(4.9)
On application of the Cauchy-Schwartz inequality to the integral in (4.9), we get
(4.10)
From the inequality
, it follows that
(4.11)
By hypothesis there exist positive constant
and
such that
and
,
. (4.12)
Hence
(4.13)
Integrating the inequality in (4.13) between a and t, we obtain
(4.14)
where
(4.15)
Now, on using Gronwall’s inequality (Lemma 4.4), it follows that
(4.16)
Since,
for some
and for
, then
for all
.
Remark: Lemma 4.8 also holds if the function f is bounded on
.
Lemma 4.9: Let
. Suppose for some
that:
(i) All solutions of
are in
.
(ii)
are bounded on
.
Then all solutions
of Equation (4.3) are in
for all
.
Lemma 4.10: Let
. Suppose for some
that:
(i) All solutions of
are in
.
(ii)
are bounded on
for some
.
Then
for any solution
of the equation
for all
5. The Regularly Solvable Operators
We see from (3.18) that
and hence
,
form an adjoint pair of closed-densely operators in
.
Lemma 5.1 [17] [18] : For
,
is constant and
.
In the problem with one singular end-point,
, for all
.
In the regular problem,
, for all
.
Proof: The proof is similar to that in ( [4] [5] [6] ) ( [17] [18] [19] [20] ) and ( [22] [23] [24] ) and therefore omitted.
For
, we define r, s and m as follows:
, (5.1)
then,
and by Lemma 5.1, m is constant on
and
. (5.2)
For
the operators which are regularly solvable with respect to the minimal operators
and
are characterized by the following theorem which proved for a general quasi-differential operator in ( [5] , Theorem 10.15).
Theorem 5.2 ( [6] , Theorem 3.2): For
. Let r and m be defined by (5.1), and let
be arbitrary functions satisfying:
(i)
are linearly independent modulo
and
are linearly independent modulo
.
(ii)
,
.
Then the set
(5.3)
is the domain of an operator S which is regularly solvable with respect to
and
and the set
(5.4)
is the domain of the operator
; moreover
.
Conversely, if S is regularly solvable with respect to
and
and
, then with r and s defined by (5.1) there exist functions
which satisfy (i) and (ii) and are such that (5.3) and (5.4) are the domains of S and
respectively.
S is self-adjoint if, and only if,
,
and
; S is J-self-adjoint if
(J is a complex conjugate),
and
.
Proof: The proof is entirely similar to that in [6] . We refer also to [17] - [24] for more details.
6. The Spectra of General Differential Operators
In this subsection we deal with the various components of the spectra of quasi-differential operators
and
.
We see from (3.18) and Theorem 4.2 that
and hence
and
form an adjoint pair of closed, closed-densely operators in
.
We shall now investigate in the case of one singular end-point that the resolvent of all well-posed extensions of the minimal operator
and we show that in the maximal case, i.e., when
, for all
that these resolvents are integral operators, in fact they are Hilbert-Schmidt integral operators by considering that the function f to be in
, i.e., is p-integrable over the interval
.
Theorem 6.1: Suppose for an operator
with one singular end-point that,
, for all
,
and let S be an arbitrary closed operator which is a well-posed extension of the minimal operator
and
, then the resolvents
and
of S and
respectively are Hilbert-Schmidt integral operators whose kernels
and
are continuous functions on
and satisfy:
and
. (6.1)
where,
, for all
,
.
for all
,
.
Remark An example of a closed operator which is a well-posed with respect to a compatible adjoint pair is given by the Visik extension ( [5] , Theorem III.3.3) (see ( [18] , Theorem 1) [19] and [20] ). Note that if S is well-posed, then
and
are compatible adjoint pair and S is regularly solvable with respect to
and
.
Proof: Let
for all
, then we choose a fundamental system of solutions
,
of the equations,
,
on
, (6.2)
so that
,
belong to
and
repectively, i.e., they are p and q-integrable on the interval
. Let
be the resolvent of any well-posed extension of the minimal operator
. For
we put
then
and consequently has a solution
in the form,
(6.3)
for some constants
(see Lemma 4.5). Since
and
for some
, then
,
for some
and hence the integral in the right-hand of (6.3) will be finite.
To determine the constants
, let
be a basis for
, then because
, we have from Theorem 5.2 that,
,
on
(6.4)
and hence from (6.3), (6.4) and on using Lemma 4.1, we have:
(6.5)
By substituting these expressions into the Conditions (6.4), we get:
This implies that the system
(6.6)
in the variable
. The determinant of this system does not vanish (see [9] and [12] ). If we solve the System (6.6) we obtain:
(6.7)
where
is a solution of the system:
(6.8)
Since, the determinant of the above System (6.8) does not vanish, and the functions
are continuous in the interval
, then the functions
are also continuous in the interval
. By substituting in Formula (6.3) for the expressions
we get,
(6.9)
Now, we put
(6.10)
Formula (6.9) then takes the form
for all
, (6.11)
i.e.,
is an integral operator with the kernel
operating on the functions
. Similarly, the solutions
of the equation
has the form:
(6.12)
where
and
are solutions of the equations in (6.2). The argument as before leads to,
for
, (6.13)
i.e.,
is an integral operator with the kernel
operating on the function
, where
(6.14)
and
is a solution of the system
(6.15)
From definitions of
and
, it follows that
(6.16)
for any continuous functions
and by construction (see (6.10) and (6.14)),
and
are continuous functions on
and (6.16) gives us
for all
. (6.17)
Since
,
for
and for fixed s,
is a linear combination of
while, for fixed t,
is a linear combination of
. Then we have
and (6.17) implies that,
Now, it is clear from (6.8) that the functions
belong to
since
is a linear combination of the functions
which lie in
and hence
belong to
. Similarly
belong to
. By the upper half of the formula (6.10) and (6.14), we have:
for the inner integral exists and is a linear combination of the products
and these products are integrable because each of the factors belongs to
. Then by (6.17), and by the upper half of (6.14),
Hence, we also have:
,
and the theorem is completely proved for any well-posed extension.
Remark: It follows immediately from Theorem 6.1 that, if for an operator
with one singular end-point that
, for all
and S is well-posed with respect to
and
with
then
is a Hilbert-Schmidt integral operator. Thus it is a completely continuous operator, and consequently its spectrum is discrete and consists of isolated eigenvalues having finite algebraic (so geometric) multiplicity with zero as the only possible point of accumulation. Hence, the spectra of all well-posed operators S are discrete, i.e.,
, for
. (6.18)
We refer to ( [5] , Theorem IX.3.1), [14] [15] [16] [18] [19] [20] and [23] for more details.
An example of a closed operator which is a well-posed with respect to a compatible adjoint pair is given by the Visik extension ( [5] , Theorem III.3.3) (See ( [18] , Theorem 1) [19] and [20] ). Note that if S is well-posed, then
and
are compatible adjoint pair and S is regularly solvable with respect to
and
.
Lemma 6.2: The point spectra
and
of the operators
and
are empty.
Proof: Let
. Then there exists a nonzero element
, such that
In particular, this gives
From Lemma 4.2, it follows that
and hence
.
Similarly
.
Theorem 6.3: (i)
, (ii)
, (iii)
.
Proof: (i) Since
is a proper closed subspace of
, then the resolvent set
is empty.
(ii) Since
is closed, then the continuous spectrum of
is empty set, i.e.,
.
(iii) From (i) and (ii) and Lemma 6.2, it follows that
.
Corollary 6.4: (i)
,
(ii)
and
.
Proof: From Theorem 4.2 and since
, it follows that
is closed for every
, see [3, Theorem 1.3.7]. Also, we have
and
.
(i) Since
is closed and
, then
and this yields that
.
(ii) Since
for every
, then we have
. Also, it follows that
and hence
.
Lemma 6.5: (cf. ( [5] , Lemma IX.9.1). If
, with
then for any
, the operator
has closed range, zero nullity and deficiency. Hence,
. (6.19)
Proof: The proof is similar to that in [19] [20] and ( [23] , Lemma 4.9).
Corollary 6.6: Let
with
(6.20)
Then,
, for
, (6.21)
of all regularly solvable extensions S with respect to the compatible adjoint pair
and
.
Proof: Since
, for all
.
Then we have from ( [5] , Theorem III.3.5) that,
Thus S is an n-dimensional extension of
and so by ( [5] , Corollary IX.4.2),
(6.22)
From Lemma 6.2 and Lemma 6.5, we get,
(6.23)
Hence, by (6.22) and (6.23) we have that,
Remark: If S is well-posed (say the Visik’s extension, see [15] - [20] ), we get from (6.19) and (6.22) that
On applying (6.22) again to any regularly solvable extensions S under consideration, hence (6.21).
Corollary 6.7: If for some
, there are n linearly independent solutions of the equations
(6.24)
in
,
and hence,
and
,
,
where
is the joint essential spectra of
defined as the joint field of regularity
.
Proof: Since all solutions of the equations in (6.24) are in
and
respectively for some
, then,
, for some
.
From Lemma 4.2, we have that
has no eigenvalues and so
exists and its domain
is a closed subspace of
. Hence, since
is a closed operator, then
is bounded and hence
. Similarly
. Therefore
, and hence,
for all
.
From Corollary 6.6, we have for any regularly solvable extension S of
that
,
and by (6.22) we get
,
.
Similarly
,
. Hence,
Remark: If there are n linearly independent solutions of Equations (6.24) in
and
for some
then the complex plane can be divided into two disjoint sets:
We refer to [5] [6] [12] - [20] and [23] for more details.
Conclusion: It has been shown that all the well-posed extensions of the minimal operator
generated by a general ordinary quasi-differential expression
of order n with complex coefficients and their formal adjoints on the interval
with maximal deficiency indices have resolvents which are Hilbert-Schmidt integral operators and consequently have a wholly discrete spectrum. This implies that all the regularly solvable operators have all the standard essential spectra to be empty. Also, the location of the point spectra and regularity fields of these operators are investigated in the case of one singular end-point and when all solutions of the equations
and its adjoint
are p-integrable.