The Products of Regularly Solvable Operators with Their Spectra in Direct Sum Spaces

In this paper, we consider the general quasi-differential expressions 1 2 , , , n     each of order n with complex coefficients and their formal adjoints on the interval   , . a b It is shown in direct sum spaces   2 , 1,2, , w p L I p N   of functions defined on each of the separate intervals with the cases of one and two singular end-points and when all solutions of the equation 1 0 n j j w u           and its adjoint 1 0 n j j w v            are in   2 , w L a b (the limit circle case) that all well-posed extensions of the minimal operator   0 1 2 , , , n T     have resolvents which are HilbertSchmidt 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. These results extend those of formally symmetric expression  studied in [1-10] and those of general quasi-differential expressions  in [11-19].


Introduction
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   0 T  and   0 T   generated by a general ordinary quasi-differential expression  and its formal adjoint   respectively, the minimal operators have the same finite dimension. This notion was originally due to Visik [20].
Akhiezer and Glazman [1] and Naimark [2] are showed that the self-adjoint extension S of the minimal operator   0 T  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 [15,16,18,19] Ibrahim extend their results for general ordinary quasi-differential expression  of n-th order with complex coefficients in the singular case.
In [3,8] Everitt and Zettl considered the problem of integrable square solutions of products of differential expressions 1 2 , , , n     and investigate the relationship between the deficiency indices of general symmetric differential expressions 1 2 , , , n     and those of the product expression 1 n j j    and in [17] Ibrahim considered the problem of the point spectra and regularity fields for products of a general quasi-differential operators. Our objective in this paper is a generalization of the results in [6,7,15,16,18,19] for the product quasi-differ- (and hence all C   . The end-points of p I assumed to be regular or may be singular.
We deal throughout this paper with a quasi-differential expression  of arbitrary order n defined by Shin-Zettl matrices [14], and the minimal operator I The endpoints a and b of I may be regular or singular endpoints.

Notation and Preliminaries
We begin with a brief survey of adjoint pairs of operators and their associated regularly solvable operators; a full treatment may be found in [2,7,11,Chapter III], [12,15,16,18]. The domain and range of a linear operator T acting in a Hilbert space H will be denoted by  (see [11,15], and [16]).
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 [11,Chapter 9] to be the sets: B The terminology "regularly solvable" comes from Visik's paper [20], while the notion of "well posed" was introduced by Zhikhar in his work on J -self adjoint operators in [21].
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 Evidently Lemma 2.5 extends to the product of any finite number of operators 1 2 , , , n A A A  .

Quasi-Differential Expressions in Direct Sum Spaces
The   . We refer to [3,9] and [17][18][19][20] for a full account of the above and subsequent results on quasi-differential expressions. For where, see [4,9] and [14][15][16][17][18] We shall be concerned with the case when p a is a regular end-point of (3.9), the end-point p b being allowed to be either regular or singular. Note that, in view of (3.5), an end-point of p I is regular for (3.9), if and only if it is regular for the equation, Note that, at a regular end-point p a , say, The subspaces For the regular problem the minimal operators to the subspaces: respectively. The subspaces closed operators (see [2,5,9,Section 3], [11,13,16]).
In the singular problem we first introduce the operators T   to be their respective closures (see [12,13,16,19] because we are assuming that p a is a regular end-point. Moreover, in both regular and singular problems, we have see [8,Section 5] . respectively, see ( [2,5,9,11,13] and [16]). Let the orthogonal sum as: and its closure is given by:  are both closed. These results imply in particular that, We refer to [11,Section 3.10.14], [16] and [18] for more details. S is a regularly solvable extension of . We refer to [11, Section 3.10.4], [16] and [18] for more details.

Theorem 3.2:
Let H be the direct sum, The elements of H will be denoted by , may be taken as a single interval, see [15] and [17]. We now establish by [8,10,11,13,15] and [18] and some further notations, We summarize a few additional properties of In the problem with one singular end-point, . We refer to [10,11,16] and [19] for more details.

The Product Operators
The proof of general theorems will be based on the results in this section. We start by listing some properties and results of quasi-differential expressions 1 2 , , , n     . For proofs the reader is referred to [3,8,10,17] and [19], and , for a complex n ber.
for P any polynomial with complex coefficients. Also we note that the leading coefficients of a product are the product of the leading coefficients. Hence the product of regular differential expressions is regular.
if and only if the following partial separation conditions are satisfied: where s is the order of product expression   We will say that the product 1 then from definition of the field of regularity we have  The special case of Corollary 4.4 when j    for 1, 2, , j n   and  is symmetric was established in [9]. In this case it is easy to see that the converse also holds.   is a constant which is independent of t .
Proof: The proof is similar to that in [2,9,13,15,17]. Lemma 4.5 contains the following lemma as a special case. . We refer to [20] for more details.

The Product Operators in Direct Sum Spaces
Next, we consider our interval is   In the regular problem, The proof is similar to that in [10, Lemma 2.4], [17] and [19] and therefore omitted.

The Case of Two Singular End-Points
For the case of two singular end-points, we consider our interval to be  