On the domains of regularly solvable operators in the direct sum spaces

A general ordinary quasi-differential expressions τ_(1 ),τ_(2 ),…,τ_(n ) each of order n with complex coefficients and their formal adjoint are τ_1^+, τ_2^+, …, τ_n^+ can be defined on the interval [a, b) respectively, we give a characterization of all regularly solvable operators and their adjoints generated by a general ordinary quasi-differential expressions τ_(jp )in the direct sum of Hilbert spaces L_(w )^2 (a_p,b_p ),p=1,…,N. This characterization is an extension of those obtained in the case of one interval with one and two singular end-points, and is a generalization of those proved in the symmetric case.

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Introduction

In Jiangang, Zhaowen and H. Sun. Jiong 2011 [1] considered the problem of Sturm-Liouville differential equation -(𝑝𝑦 ′ ) ′ + 𝑞𝑦 = 𝜆𝑤𝑦 on (𝑎, 𝑏), -∞ ≤ ≤ 𝑎 < 𝑏 ≤ +∞, (1.1) where 𝑝, 𝑞 are complex functions, 𝑝(𝑥) ≠ 0 and 𝑤(𝑥) > 0 a. e. on (𝑎, 𝑏), 𝑝 -1 , 𝑞, 𝑤 are all locally integrable functions on (𝑎, 𝑏), 𝜆 is the so-called spectral parameter. They studied the classification of equation (1.1) according to the number of square integrable solutions of the equation (1.1) in suitable weighted integrable spaces. In [2] Amos considered the problem that all solutions of second-order ordinary differential equation 𝜏[𝑦] = 𝜆𝑤𝑦 (𝜆 ∈ ℂ) are in 𝐿 𝑤 2 (𝑎, ∞) when 𝜏 is a second-order symmetric ordinary differential expression of the form 𝜏[𝑦] = -(𝑝𝑦 ′ ) ′ + 𝑞𝑦 on [𝑎, ∞) under sufficient conditions on the coefficients 𝑝 and 𝑞. The case that not all solutions are in 𝐿 𝑤 2 (𝑎, ∞) was considered by Atkinson and Evans 1972 [3] [3, Theorem 1]. In Everitt and Zettl 1986 [4] considered the problem of characterizing all self-adjoint differential operators which can be generated by a formally symmetric Sturm-Liouville differential expression 𝜏 𝑝 defined on two intervals 𝐼 𝑝 , (𝑝 = 1,2) with boundary conditions at the end points. An interesting feature of their work is the possibility of generating self-adjoint operators in this way which are not expressible as the direct sum of self-adjoint differential operators defined in the separate intervals. In Jiong Sun 1986 [5] gives a characterization of the self-adjoint extensions of the minimal operator 𝑇 0 generated in 𝐿 𝑤 2 (0, 𝑏) by a formally-symmetric differential expression 𝜏 of arbitrary order 𝑛. If the minimal operator 𝑇 0 has deficiency indices (ℓ, ℓ), the domain of any self-adjoint extension of 𝑇 0 is described in terms of ℓ boundary conditions involving the square-integrable solutions of the differential equation 𝜏[𝑢] = 𝜆𝑢 for 𝜆 ∈ Ȼ\ℝ. In [6,7] J. Knowles and Zai-Jiu-Shang 1988 [7] gave a characterization of the boundary conditions which determine the domain of any 𝐽 -self-adjoint extension of the minimal operator 𝑇 0 with maximal deficiency index in the case when the field of regularity, П(𝑇 0 ), of 𝑇 0 was non-empty. This is achieved by using Sun Jiong , s results (see [5] ) with only one singular end-point. Evans and Sobhy El-sayed 1990 [8] and [9][10][11][12][13] extended their results for all regularly solvable operators and their adjoints in the case of one interval with one singular end-point and in the case of two singular end-points 𝑎 and 𝑏.

𝑎 and 𝑏. The domains of these operators are described in terms of boundary conditions featuring 𝐿 𝑤 2 (𝑎, 𝑏)-solutions of the equations 𝜏[𝑢] = 𝜆𝑤𝑢 and 𝜏 + [𝑣] = 𝜆 ̅ 𝑤𝑣 at both singular end-points 𝑎 and 𝑏. Their results include those of Jiong Sun 1986 [5] concerning self-adjoint realizations of a symmetric expressions 𝜏 when the minimal operator has equal deficiency indices, and Zai-Jiu Shang 1988 [7] concerning the 𝐽 -self-adjoint operators as special case. Our objective in this research is to generalize of Evans and Sobhy E. Ibrahim, 1990 Sun Jiong and Zai-Jiu-Shang , s 1988 results and Zettl 1986 in [4][5][6][7][8][9][10][11] and [15] for the general ordinary quasi-differential expressions 𝜏 1 , 𝜏 2 , … , 𝜏 𝑛 each of order 𝑛 with complex coefficients generated by a general Shin-Zettl matrices (see [8,16,17] ) in direct sum spaces such that the operators defined on each of the separate intervals 𝐼 𝑝 = (𝑎 𝑝 , 𝑏 𝑝 ), 𝑝 = 1,2, … , 𝑁. The left-hand end-point of 𝐼 𝑝 is assumed to be regular but the right-hand endpoint may be regular or singular.

Notation and Preliminaries

We begin with a brief survey of adjoin pairs of operators and their associated regularly solvable operators; a full treatment may be found in [5] , [8][9][10][11] , [12,13] , [15][16][17] , [18] , [Chapter III] and [19] . 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 𝑛𝑢𝑙(𝑇), is the dimension of 𝑁(𝑇) and the deficiency of 𝑇, written 𝑑𝑒𝑓(𝑇), is the co-dimension of 𝑅(𝑇) in 𝐻; thus if 𝑇 is densely defined and 𝑅(𝑇) is closed, then 𝑑𝑒𝑓(𝑇) = 𝑛𝑢𝑙(𝑇 * ). The Fredholm domain of T is (in the notation of [26,24,9]) the open subset ∆ 3 (𝑇) of Ȼ consisting of those values of λ ∈ Ȼ which are such that (𝑇-λ𝐼) is a Fredholm operator, where I is the identity operator in 𝐻. Thus λ ∈ ∆ 3 (𝑇) if and only if (𝑇-λ𝐼) has closed range and finite nullity and deficiency. The index of (𝑇-λ𝐼) is the number

This is defined for λ ∈ ∆ 3 (𝑇).

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

The field of regularity П(𝐴) of 𝐴 is the set of all λ 𝜖 Ȼ for which there exists a positive constant 𝐾(λ) such that

or, equivalently, on using the Closed Graph Theorem, and 𝑛𝑢𝑙(𝐴 -λ𝐼) = 0 and 𝑅(𝐴 -λ𝐼) is closed.

The joint field of regularity П(𝐴, 𝐵) of 𝐴 and 𝐵 is the set of λ 𝜖 Ȼ which are such that λ ∈ П(𝐴), 𝜆 ̅ ∈ П(𝐵) and both 𝑑𝑒𝑓(𝐴 -λ𝐼) and 𝑑𝑒𝑓(𝐵 -𝜆 ̅ 𝐼) may be finite. An adjoin pair of 𝐴 and 𝐵 is said to be compatible if П(𝐴, 𝐵) ≠ Ø.

We now turn to the quasi-differential expressions defined in terms of a Shin-Zettl matrix 𝐴 on an interval 𝐼.

Definition 2.1:

The set 𝑍 𝑛 (𝐼) of Shin-Zettl matrices on 𝐼 consists of 𝑛 × 𝑛-matrices 𝐴 = {𝑎 𝑟𝑠 }, 1 ≤ 𝑟, 𝑠 ≤ 𝑛, whose entries are complex-valued functions on 𝐼 which satisfy the following conditions: (i) a rs ∈ 𝐿 𝑙𝑜𝑐 1 (𝐼) (1 ≤ 𝑟, 𝑠 ≤ 𝑛, 𝑛 ≥ 2) (ii) a r,r+1 ≠ 0 a. e. on I (1 ≤ r ≤ n -1)

(2.3) (iii) a rs = 0, a. e. on I (2 ≤ 𝑟 + 1 < 𝑠 ≤ 𝑛).

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

The quasi-differential expression 𝜏 associated with the matrix 𝐴 is given by: 𝜏[𝑦] ≔ 𝑖 𝑛 𝑦 [𝑛] , (𝑛 ≥ 2 ), (2.5) this being defined on the set:

The formal adjoint 𝜏 + of 𝜏 defined by the matrix 𝐴 + ∈ 𝑍 𝑛 (𝐼) is given by: 𝜏 + [𝑧] ≔ 𝑖 𝑛 𝑧 + [𝑛] , for all 𝑧 ∈ 𝑉(𝜏 + ); (2.7) this being defined on the set:

where 𝑧 + [𝑟-1] , the quasi-derivatives associated with the matrix 𝐴 + ∈ 𝑍 𝑛 (𝐼),

Note that: (𝐴 + ) + = 𝐴 and so (𝜏 + ) + = 𝜏. We refer to [1,4,6,8] , [9][10][11][12][13] and [14][15][16][17][18][19][20] for a full account of the above and subsequent results on quasi-differential expressions.

Definition 2.2: For ∈ 𝑉(𝜏), 𝑣 ∈ 𝑉(𝜏 + ) and 𝛼, 𝛽 ∈ 𝐼, we have the Green , s formula:

𝑢 [𝑟] (𝑥) 𝑣̅ + [𝑛-𝑟-1] (𝑥))

See [5,8] , [9][10][11][12][13] , [15] , [Corollary 1] and [16,18] .

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

The equation, We shall be concerned with the case when 𝑎 is a regular end-point of (2.14), the end-point 𝑏 being allowed to be either regular or singular. Note that, in view of (2.9) an end-point of 𝐼 is regular for the equation (2.14), if and only if it is regular for the equation,

(2.17) see [4, 9], [10 -20] and [22] .

Note that, at a regular end-point 𝑎, say, 𝑦 [𝑟-1] (𝑎) (𝑧 The subspaces 𝐷(𝜏) and 𝐷(𝜏 + ) of 𝐿 𝑤 2 (𝑎, 𝑏) are the domains of the so-called maximal operators 𝑇(𝜏) and 𝑇(𝜏 + ) respectively, defined by:

For the regular problem the minimal operators 𝑇 0 (𝜏) and 𝑇 0 (𝜏 + ), are the restrictions of 𝑤 -1 𝜏[𝑦] and 𝑤 -1 𝜏[𝑧] to subspaces: 𝐷 0 (𝜏) ≔ {𝑦: 𝑦 ∈ 𝐷(𝜏), 𝑦 [𝑟-1] (𝑎) = 𝑦 [𝑟-1]

respectively. The subspaces 𝐷 0 (𝜏) and 𝐷 0 (𝜏 + ) are dense in 𝐿 𝑤 2 (𝑎, 𝑏), 𝑇 0 (𝜏) and 𝑇 0 (𝜏 + ) are closed operators (see [1,4,9] , [10][11][12][13][14] , [15, Corollary 3] and [16 -21] .

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

and with 𝑇 0 ′ (𝜏 + ) defined similarly. These operators are densely-defined and closable in 𝐿 𝑤 2 (𝑎, 𝑏); and we defined the minimal operators 𝑇 0 ′ (𝜏) and 𝑇 0 ′ (𝜏 + ) to be their respective closures (see [1,5,9] , [15,16,18] and [21] ). We denote the domains of 𝑇 0 (𝜏) and 𝑇 0 (𝜏 + ) by 𝐷 0 (𝜏) and 𝐷 0 (𝜏 + ) respectively. It can be shown that:

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

see [13, Section 5] in the case when 𝜏 = 𝜏 + and compare with treatment in [5,8] , [9][10][11][12][13] , [14] and [18, Section III.10.3] in general case. Note that 𝑇 0 (𝜏) and 𝑇(𝜏) are closed and densely-defined operators on 𝐻.

The Operators in Direct Sum Spaces

The operators here are no longer symmetric but direct sums

on any finite number of intervals 𝐼 𝑝 = (𝑎 𝑝 , 𝑏 𝑝 ), 𝑝 = 1,2, … , 𝑁, where 𝑇 0 (𝜏 𝑝 ) is the minimal operator generated by 𝜏 𝑝 in 𝐼 𝑝 and 𝜏 𝑝 + denotes the formal adjoint of 𝜏 𝑝 , which form an adjoint pair of closed operators in ⊕ 𝑝=1 𝑁 𝐿 𝑤 𝑝 2 ( 𝐼 𝑝 ). 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 . When 𝐼 𝑖 ∩ 𝐼 𝑗 = Ø, 𝑖 ≠ 𝑗, 𝑖, 𝑗 = 1, … , 𝑁, the direct sum space ⊕ 𝑝=1 𝑁 𝐿 𝑤 𝑝 2 ( 𝐼 𝑝 ), can be naturally identified with the space 𝐿 𝑤 𝑝 2 ( ∪ 𝑝=1 𝑁 𝐼 𝑝 ) where 𝑤 = 𝑤 𝑝 on 𝐼 𝑝 , 𝑝 = 1, … , 𝑁. This is of particular significance when ∪ 𝑝=1 𝑁 𝐼 𝑝 may be taken as a single interval; see [12] , [14] and [20] .

We now establish by [4] , [9] and [11] some further notation.

where 𝑓 ̃= {𝑓 1 , … , 𝑓 𝑁 }, 𝑔 ̃= {𝑔 1 , … , 𝑔 𝑁 } and (. , . ) 𝑝 the inner-product defined in (2.13). Note that 𝑇 0 (𝜏) is a closed densely-defined operator in 𝐻.

We summarize a few additional properties of 𝑇 0 (𝜏) in the form of a Lemma. In the case with one singular end-point

Proof: The proof is similar to that in [3, lemma 3.1] and [9-11], and therefore omitted.

For ∈ П[𝑇 0 (𝜏)], 𝑇 0 (𝜏 + )], we define 𝑟, 𝑠 and 𝑚 as follows:

Then 0 ≤ 𝑟, 𝑠 ≤ 𝑛𝑁 and by Lemma 3.4, 𝑚 is constant on П[𝑇 0 (𝜏), 𝑇 0 (𝜏 + )], and

For П[𝑇 0 (𝜏), 𝑇 0 (𝜏 + )] ≠ Ø, the operators which are regularly solvable with respect to 𝑇 0 (𝜏) and 𝑇 0 (𝜏 + ) are characterized by the following theorem: Proof: The proof is similar to that in [5] , [7][8][9][10] . [18, Theorem III.3.6] and [21,22] , For 𝜆 ∈ П[𝑇 0 (𝜏 𝑝 ), 𝑇 0 (𝜏 𝑝 + )], define 𝑟 𝑝 , 𝑠 𝑝 and 𝑚 𝑝 be defined by (3.8) We can therefore define the following:

Next, we state the following results, the proofs are similar to those in [4] , [9][10][11][12] , [14,15] and [18, Section 4].

A central role in the argument is played by the matrices; see also [9 -12] . The results in this section are extension of those in [1,4,5,7,8] , [9][10][11][12][13][14] and [19][20][21][22] . Then 𝑔 𝑗𝑝 ∈ 𝐷[𝑇(𝜏 𝑝 + )], by [4] , [9] and [11] we may choose Φ 𝑗𝑝 (𝑗 = 𝑟 𝑝 + 𝑥 𝑘𝑝 , (𝑗 = 1, … , 𝑟 𝑝 , 𝑝 = 1,2, … , 𝑁) (4.7) and by [4] , [9] and [11] we may choose 𝛹 𝑗𝑝 (𝑗 = 1, … . , 𝑟 𝑝 ) ∈ 𝐷[𝑇(𝜏 𝑝 )] such that for 𝑘 = 1,2, … , 𝑛 and some 𝑐 𝑝 ∈ [𝑎 𝑝 , 𝑏 𝑝 ),

On noting that

But 𝐽 𝑛×𝑛 has rank 𝑛 and so we infer that

Since 𝑢 = {𝑢 𝑝 } ∈ 𝐷 0 (𝑟 𝑝 ), we have that [𝑢, 𝑣](𝑏 𝑝 ) = 0 for all 𝑣 = {𝑣 𝑝 } ∈ 𝐷 0 (𝜏 𝑝 + ).

, on using the notation in (3.23). In view of (3.25), we conclude that

We obtain from (4.9) and (4. has rank 𝑟.

It follows similarly that, {Φ 𝑘 } = {Φ 𝑘𝑝 : 𝑘 = 𝑟 𝑝 + 1, … . , 𝑚 𝑝 } ⊂ 𝐷[𝑇(𝜏 𝑝 + )] is linearly independent modulo 𝐷 0 (𝜏 𝑝 + ).

Finally, we prove (𝑖𝑖) in Theorem 3.5, .

The proof is therefore complete.

Remark 4.3:

Assume that 𝜏 = ⊕ 𝑝=1 𝑁 𝜏 𝑝 is formally 𝐽-symmetric, that is 𝜏 + = 𝐽𝜏𝐽, where J is the complex conjugation. Then the operator 𝑇 0 (𝜏) is the J-symmetric and 𝑇 0 (𝜏) and 𝑇 0 (𝜏 + )] = 𝐽[𝑇 0 (𝜏)]𝐽 form an adjoint pair with П[𝑇 0 (𝜏), 𝑇 0 (𝜏 + )] = П[𝑇 0 (𝜏)].

Discussion

In Everitt and Zettl 1986 [5] discussed the possibility of generating self-adjoint operators which are not expressible as the direct sums of self-adjoint operators defined in the separate intervals. In this section we extend this case to the case of general ordinary differential operators, i.e., we discuss the possibility of the regularly solvable operators which are not expressible as the direct sums of regularly solvable operators defined in the separate intervals 𝐼 𝑝 = (𝑎 𝑝 , 𝑏 𝑝 ), 𝑝 = 1, 2, 3, 4. We will refer to these operators as "New regularly solvable operators" if 𝑎 𝑝 is a regular end point and 𝑏 𝑝 is singular, then by [18, Theorem III.10.13] .

We now consider some of the possibilities:

Example 1. 𝑑 = 0. This is the minimal case in (5.1) and can only occur when all four end-points are singular. In this case 𝑇 0 (𝜏) is itself regularly solvable and has no proper regularly solvable extensions, see Edmunds and Evans [18, Chapter III] and [9,12] .

Example 2. 𝑑 = 𝑛 with one of 𝑑 1 , 𝑑 2 , 𝑑 3 and 𝑑 4 is equal to 𝑛 and all the others are equal to zero. We assume that 𝑑 1 = 𝑛 and 𝑑 2 = 𝑑 3 = 𝑑 4 = 0. The other possibilities are entirely similar. In this case we must have seven singular end-points and one regular. There are no new regularly solvable extensions and we have that, 𝑆 = 𝑆 1 ⊕ 𝑝=2 4 𝑇 0 (𝜏 𝑝 ), where 𝑆 1 is regularly solvable extension of 𝑇 0 (𝜏 1 ), i. e., all regularly solvable extensions of 𝑇 0 (𝜏) can be obtained by forming sums of regularly solvable extensions of 𝑇 0 (𝜏 𝑝 ), 𝑝 = 1, 2, 3, 4. These are obtained as in the "one interval" case.

Example 3. Six singular end-points and 𝑑 = 2𝑛. We consider two cases: (𝑖) One interval has two regular end-points, say, 𝐼 1 , and each one of the others has two singular end-points. Then, 𝑆 = 𝑆 1 ⊕ 𝑝=2 4 𝑇 0 (𝜏 𝑝 ), where 𝑆 1 is regularly solvable extension of 𝑇 0 (𝜏 1 ), generates all regularly solvable extensions of 𝑇 0 (𝜏).

(𝑖𝑖) There are two intervals, say, 𝐼 1 rand 𝐼 2 each one has one regular and one singular end-point and each one of the others has two singular end-points. In this case 𝑆 = 𝑆 1 ⊕ 𝑆 2 ⊕ 𝑝=3 4 𝑇 0 (𝜏 𝑝 ), and 𝑆 1 ⊕ 𝑆 2 generates all regularly solvable extensions of 𝑇 0 (𝜏). The other possibilities in the cases (𝑖) and (𝑖𝑖) are entirely similar.

Example 4: Five singular end-points and 𝑑 = 3𝑛. We consider two cases:

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