In [1]-[3] considered the problem of linear differential operators in Hilbert space, in [4] and [5] considered the problem of general ordinary differential operators and their adjoints in Hilbert space. In [6]-[10] Everitt considered the problem of characterizing all self-adjoint operators which can be generated by a formally symmetric Sturm-Liouville differential (quasi-differential) expression τ p , defined on a finite number of intervals I p , p = 1 , ⋯ , N in the setting of direct sum spaces. In [11]-[15] Ibrahim considered the problem of the location of the point spectra and regularity fields of general ordinary quasi-differential operators on the one interval case with one regular end-point and the other may be regular or singular.

Our objective in this paper is to investigate the location of the point spectra and regularity fields of the product operators which are generated by a general quasi-differential expression τ p on any finite number of intervals I p , p = 1 , ⋯ , N in the setting of direct sums of L w p 2 ( a p , b p ) -spaces of functions defined on each of the separate intervals. These results extend those of general ordinary quasi-differential operators in [11]-[20] with one interval case. Also, extend those of formally symmetric differential expression studied in [21]-[26].

The operators involved are no longer symmetric but direct sums as:

where T 0 ( τ j p ) is the minimal operator generated by τ j p on I p and τ j p + denotes by the formal adjoint of τ j p which form an adjoint pair of closed operators in ⊕ p = 1 N L w p 2 ( I p ) . This fact allows us to use the abstract theory developed in [1] for the operators which are regularly solvable with respect to T 0 ( τ j p ) and T 0 ( τ j p + ) . Such an operator S satisfies T 0 ( τ ) ⊂ S ⊂ [ T 0 ( τ + ) ] * and for some λ ∈ ℂ , ( S − λ I ) is a Fredholm operator with zero index; this means that S has the desirable Fredholm property that the equation ( S − λ I ) u = f has a solution if and only if f is orthogonal to the solutions of ( S * − λ ¯ I ) v = 0 and furthermore the solution spaces of ( S − λ I ) u = 0 and ( S * − λ ¯ I ) v = 0 have the same finite dimension. This notion was originally due to Visik [25].

We deal throughout with a quasi-differential expressions τ j p each of arbitrary order n defined by a general Shin -Zettl Matrix given in [3][5][9]-[17] and [19][20], and the minimal operator T 0 ( τ j p ) generated by w p − 1 τ j p [ . ] in L w p 2 ( I p ) , p = 1 , ⋯ , N , where w p is a positive weight function on the underlying interval I p . The end-points of I p may be regular or singular.

The study of the point spectra of product differential operators in direct sum spaces is a cornerstone of functional analysis and mathematical physics. This study has led to an understanding of representational properties, the solution of complex differential equations by transforming them into simplified algebraic systems, and the analysis of physical electronics, particularly in quantum mechanics, by representing complements as direct sums.

The domain and range of a linear operator T acting in a Hilbert space H will be denoted by D ( T ) and R ( T ) respectively and N ( T ) will denote its null space. The nullity of T , written n u l l ( T ) , is the dimension of N ( T ) and the deficiency of T , written d e f ( T ) , is the co-dimension of R ( T ) in H ; thus if T is densely defined and R ( T ) is closed , then d e f ( T ) = n u l l ( T * ) . The Fredholm domain of T is (in the notation of [5]) the open subset Δ 3 ( T ) of ℂ consisting of those values of λ ∈ ℂ which are such that ( T − λ I ) is a Fredholm operator, where I is the identity operator in H . Thus λ ∈ Δ 3 ( T ) if and only if ( T − λ I ) has closed range and finite nullity and deficiency. The index of ( T − λ I ) is the number i n d ( T − λ I ) = n u l l ( T − λ I ) − d e f ( T − λ I ) , this being defined for λ ∈ Δ 3 ( T ) .

Two closed densely defined operators A and B acting in a Hilbert space H are said to form an adjoint pair if A ⊂ B * and, consequently, B ⊂ A * ; equivalently, ( A x , y ) = ( x , B y ) for all x ∈ D ( A ) and y ∈ D ( B ) , where ( . , . ) denotes the inner-product on H .

Definition 2.1: The field of regularity Π ( A ) of A is the set of all λ ∈ ℂ for which there exists a positive constant K ( λ ) such that

or, equivalently, on using the Closed Graph Theorem, n u l l ( A − λ I ) = 0 and R ( A − λ I ) is closed.

The joint field of regularity Π ( A , B ) of A and B is the set of λ ϵ ℂ which are such that λ ∈ Π ( A ) , λ ¯ ∈ Π ( B ) and both d e f ( A − λ I ) and d e f ( B − λ ¯ I ) are finite. An adjoint pair A and B is said to be compatible if Π ( A , B ) ≠ ϕ .

Definition 2.2: A closed operator S in H is said to be regularly solvable with respect to the compatible adjoint pair A and B if A ⊂ S ⊂ B * and Π ( A , B ) ∩ Δ 4 ( S ) ≠ ∅ , where Δ 4 ( S ) = { λ : λ ∈ Δ 3 ( S ) , i n d ( S − λ I ) = 0 } .

The terminology “regularly solvable” comes from Visik’s paper [25].

Definition 2.3: The resolvent set ρ ( S ) of a closed operator S in H consists of the complex numbers λ for which ( S − λ I ) − 1 exists, is defined on H and is bounded. The complement of ρ ( S ) in ℂ is called the spectrum of S and written σ ( S ) . The point spectrum σ p ( S ) , continuous spectrum σ c ( S ) and residual spectrum σ r ( S ) are the following subsets of σ ( S ) (see [2][3] and [12]-[15]):

σ p ( S ) = { λ ∈ σ ( S ) : ( S − λ I )   is not injective } , i.e., the set of eigenvalues of S ;

For a closed operator S we have,

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 ([3], Chapter II] to be the sets:

where Δ 3 ( S ) and Δ 4 ( S ) have been defined earlier. The sets σ e k ( S ) are closed and σ e k ( S ) ⊂ σ e j ( S ) if k < j . The inclusion being strict in general. We refer the reader to [1][2] and ([3], Chapter IX) for further information about the sets σ e k ( S ) .

Given two operators A and B, both acting in a Hilbert space H, we wish to consider the product operator AB. This is defined as follows:

It may happen in general that D ( A B ) contains only the null element of H . However, in the case of many differential operators the domains of the product will be dense in H.

The next result gives conditions under which the deficiency of a product is the sum of the deficiencies of the factors.

Lemma 2.4 (cf. ([13], Theorem A)). Let A and B be closed operators with dense domains in a Hilbert space H. Suppose that λ = 0 is a regular type point for both operators and d e f ( A ) and d e f ( B ) are finite. Then A B is a closed operator with dense domain, has λ = 0 as a regular type point and

We refer to [5][6][11][12][14]-[19] and [22] for more details.

Evidently Lemma 2.4 extends to the product of any finite number of operators A 1 , A 2 , ⋯ , A n .

The quasi-differential expressions are defined in terms of a Shin-Zettl matrix F p on an interval I p . The set Z n ( I p ) of Shin-Zettl matrices on I p consists of n × n -matrices F p = { f r s p } , p = 1 , 2 , ⋯ , N , whose entries are complex-valued functions on I p which satisfy the following conditions:

f r s p = 0 , a.e., on I p , ( 2 ≤ r + 1 < s ≤ n ) , p = 1 , 2 , ⋯ , N .

For F p ∈ Z n ( I p ) , the quasi-derivatives associated with F p are defined by:

where the prime ' denotes differentiation.

The quasi-differential expression τ p associated with F p is given by:

this being defined on the set:

where A C l o c ( I p ) denotes the set of functions which are absolutely continuous on every compact subinterval of I p .

The formal adjoint τ p + of τ p is defined by the matrix F p + given by:

where y + [ r − 1 ] , the quasi-derivatives associated with the matrix F p + in Z n ( I p ) ,

Note that: ( F p + ) + = F p and so ( τ p + ) + = τ p . We refer to [4][6][9]-[20][22] and [26] for a full account of the above and subsequent results on quasi-differential expressions.

For u ∈ V ( τ p ) , v ∈ V ( τ p + ) and α , β ∈ I p , we have Green’s formula,

where,

see [1][4][6] ([9], Corollary 1) [11][13][19][21] and [26].

Let the interval I p have end-points a p , b p ( − ∞ ≤ a p < b p ≤ ∞ ) , and let w p : I p → ℝ be a non-negative weight function with w p ∈ L l o c 1 ( I p ) and w p > 0 (for almost all x ∈ I p ). Then H p = L w p 2 ( I p ) denotes the Hilbert function space of equivalence classes of Lebesgue measurable functions such that ∫ I p w p | f | 2 < ∞ ; the inner-product is defined by:

The equation

is said to be regular at the left end-point a p ∈ ℝ , if for all X ∈ ( a p , b p ) ,

otherwise (3.9) is said to be singular at a p . If (3.9) is regular at both end-points, then it is said to be regular; in this case we have,

We shall be concerned with the case when a p is a regular end-point of (3.9), the end-point b p being allowed to be either regular or singular. Note that, in view of (3.5), an end-point of I p is regular for (3.9), if and only if it is regular for the equation:

Note that, at a regular end-point a p , say, u [ r − 1 ] ( a p ) ( v + [ r − 1 ] ( a p ) ) , r = 1 , 2 , ⋯ , n is defined for all u ∈ V ( τ p ) ( v ∈ V ( τ p + ) ) . Set:

The subspaces D ( τ p ) and D ( τ p + ) of L w p 2 ( a p , b p ) are domains of the so-called maximal operators T ( τ p ) and T ( τ p + ) respectively, defined by:

For the regular problem, the minimal operators T 0 ( τ p ) and T 0 ( τ p + ) , p = 1 , 2 , ⋯ , N are the restrictions of w p − 1 τ p [ u ] and w p − 1 τ p + [ v ] to the subspaces:

respectively. The subspaces D 0 ( τ p ) and D 0 ( τ p + ) are dense in L w p 2 ( a p , b p ) and T 0 ( τ p ) and T 0 ( τ p + ) are closed operators (see [4][6] ([9], Section 3), [11]-[14], [19]-[22] and [26]).

In the singular problem we first introduce the operators T ′ 0 ( τ p ) and T ′ 0 ( τ p + ) ; T ′ 0 ( τ p ) being the restriction of w p − 1 τ p [ . ] to the subspace:

and with T ′ 0 ( τ p + ) defined similarly. These operators are densely-defined and closable in L w p 2 ( a p , b p ) ; and we define the minimal operators T 0 ( τ p ) and T 0 ( τ p + ) to be their respective closures (see [4][9][11]-[14]). We denote the domains of T 0 ( τ p ) and T 0 ( τ p + ) by D 0 ( τ p ) and D 0 ( τ p + ) respectively. It can be shown that:

because we are assuming that a p is a regular end-point. Moreover, in both regular and singular problems, we have

see ([9], Section 5) in the case when τ p = τ p + and compare with treatment in [4][11] ([12], Section III 10.3) and [13]-[20] in general case.

In the case of two singular end-points, the problem on ( a p , b p ) . is effectively reduced to the problems with one singular end-point on the intervals ( a p , c p ] . and [ c p , b p ) , where c p ∈ ( a p , b p ) .We denote by T ( τ p ; a p ) and T ( τ p ; b p ) the maximal operators with domains D ( τ p ; a p ) and D ( τ p ; b p ) , and denote T 0 ( τ p ; a p ) and T 0 ( τ p ; b p ) the closures of the operators T ′ 0 ( τ p ; a p ) and T ′ 0 ( τ p ; b p ) defined in (3.13) on the intervals ( a p , c p ] and [ c p , b p ) , respectively, see [3][7][10]-[19] and [26].

Let T ˜ ′ 0 ( τ p ) , p = 1 , ⋯ , N , be the orthogonal sum as:

T ˜ ′ 0 ( τ p ) is densely-defined and closable in L w p 2 ( a p , b p ) and its closure is given by, T ˜ 0 ( τ p ) = T 0 ( τ p ; a p ) ⊕ T 0 ( τ p ; b p ) , p = 1 , ⋯ , N .

Also,

and R [ T ˜ 0 ( τ p ) − λ I ] is closed if and only if R [ T 0 ( τ p ; a p ) − λ I ] and R [ T 0 ( τ p ; b p ) − λ I ] are both closed. These results imply in particular that,

We refer to ([3], Section 3.10.14), [11][13][14] and [18] for more details.

Remark 3.1:If S p a is a regularly solvable extension of T 0 ( τ p ; a p ) and S p b is a regularly solvable extension of T 0 ( τ p ; b p ) , then S = S p a ⊕ S p b is a regularly solvable extension of T ˜ 0 ( τ p ) , p = 1 , ⋯ , N . We refer to ([3], Section 3.10.4) and [11]-[17] for more details.

Next, we state the following results; the proof is similar to that in ([3], Section 3.10.4), [11]-[13][16] and [17].

Theorem 3.2: T ˜ 0 ( τ p ) ⊂ T 0 ( τ p ) , T ( τ p ) ⊂ T 0 ( τ p ; a p ) ⊕ T 0 ( τ p ; b p ) and

i n d [ T 0 ( τ p ) − λ I ] = n − d e f [ T 0 ( τ p ; a p ) − λ I ] − d e f [ T 0 ( τ p ; b p ) − λ I ] , and in particular, if λ ∈ Π [ T 0 ( τ p ) ] ,

Remark 3.3:It can be shown that

see ([3], Section 3.10.4).

Let H be the direct sum,

The elements of H will be denoted by f ˜ = { f 1 , ⋯ , f N } with f 1 ∈ H 1 , ⋯ , f N ∈ H N .

Remark 3.4:When I i ∩ I j = ∅ , i ≠ j ; i , j = 1 , ⋯ , N , the direct sum space ⊕ p = 1 N L w p 2 ( a p , b p ) can be naturally identified with the space L w 2 ( ∪ p = 1 N I p ) , where w p = w on I p , p = 1 , ⋯ , N . This remark is of significance when ∪ p = 1 N I p may be taken as a single interval, see [8][10][13][16] and [17].

We now establish by [3][8][10] and [13] some further notations,

Also,

We summarize a few additional properties of T 0 ( τ ) in the form of a Lemma.

Lemma 3.5: We have,

(i) [ T 0 ( τ ) ] * = ⊕ p = 1 N [ T 0 ( τ p ) ] * = ⊕ p = 1 N [ T ( τ p + ) ] ,

In particular,

(ii) n u l l [ T 0 ( τ ) − λ I ] = ∑ p = 1 N n u l l [ T 0 ( τ p ) − λ I ] ,

(iii) The deficiency indices of T 0 ( τ ) are given by:

d e f [ T 0 ( τ ) − λ I ] = ∑ p = 1 N d e f [ T 0 ( τ p ) − λ I ] for all λ ∈ Π [ T 0 ( τ p ) ] ,

d e f [ T 0 ( τ + ) − λ ¯ I ] = ∑ p = 1 N d e f [ T 0 ( τ p + ) − λ ¯ I ] for all λ ∈ Π [ T 0 ( τ p + ) ] .

Proof: Part (i) follows immediately from the definition of T 0 ( τ ) and from the general definition of an adjoint operator. The other parts are either direct consequences of part (i) or follow immediately from the definitions.

Lemma 3.6: ([13], Lemma 2.4). For λ ∈ Π [ T 0 ( τ ) , T 0 ( τ + ) ] ,

d e f [ T 0 ( τ ) − λ I ] + d e f [ T 0 ( τ + ) − λ ¯ I ] is constant and

In the problem with one singular end-point,

for all λ ∈ Π [ T 0 ( τ ) , T 0 ( τ + ) ] .

In the regular problem,

d e f [ T 0 ( τ ) − λ I ] + d e f [ T 0 ( τ + ) − λ ¯ I ] = 2 n N , for all λ ∈ Π [ T 0 ( τ ) , T 0 ( τ + ) ] .

We refer to [3][5][11][12][16] and [17] for more details.

Lemma 3.7: Let T 0 ( τ ) = ⊕ p = 1 N T 0 ( τ p ) be a closed densely-defined operator on H . Then,

Proof:The proof follows from Lemma 3.1 and since R [ T 0 ( τ ) − λ I ] is closed if and only if R [ T 0 ( τ p ) − λ I ] , p = 1 , 2 , ⋯ , N are closed.

Lemma 3.8:If S p , p = 1 , ⋯ , N , are regularly solvable with respect to T 0 ( τ p ) and T 0 ( τ p + ) , then S = ⊕ p = 1 N S p is regularly solvable with respect to T 0 ( τ ) and T 0 ( τ p + ) .

Proof:The proof follows from Lemma 3.5 and Lemma 3.7.

Remark 3.9: Let S = ⊕ p = 1 N S p be an arbitrary closed operator on H , and since λ ∈ ρ ( S ) if, and only if, n u l ( S − λ I ) = d e f ( S − λ I ) = 0 (see ([2], Theorem 1.3.2)), we have ρ ( S ) = ∩ p = 1 N ρ ( S p ) . We therefore have,

Also,

We refer to ([3], Chapter 9) for more details.

Theorem 3.10:(cf. ([21], Part II, Theorem 16.2.2)). Suppose f ∈ L l o c 1 ( I p ) and suppose that the conditions (3.1) are satisfied. Then given any complex numbers c j ∈ ℂ , j = 0 , 1 , ⋯ , n − 1 and x 0 ∈ ( a p , b p ) there exists a unique solution of the equation τ p [ φ p ] = w f in the intervals ( a p , b p ) which satisfies,

We refer to [1] and [3] for more details.

Theorem 3.11:(cf. [3] and ([21], Theorem II.2.5)). Let τ p be a regular quasi-differential expression of order n on the closed interval [ a p , b p ] . For f ∈ L l o c 2 ( I p ) , the equation τ p [ φ p ] = w f has a solution φ p ∈ V ( τ p ) satisfying,

if, and only if, f is orthogonal in L l o c 2 ( a p , b p ) to the solution space of τ + [ φ p ] = 0 , i.e.,

Corollary 3.12:(cf. ([21], Corollary II.2.6)). As a result from Theorem 3.11, we have that