Abstract
Non-self-adjoint operators have many applications, including quantum and heat equations. On the other hand, the study of these types of operators is more difficult than that of self-adjoint operators. In this paper, our aim is to study the resolvent and the spectral properties of a class of non-self-adjoint differential operators. So we consider a special non-self-adjoint elliptic differential operator (Au)(x) acting on Hilbert space and first investigate the spectral properties of space . Then, as the application of this new result, the resolvent of the considered operator in -dimensional space Hilbert is obtained utilizing some analytic techniques and diagonalizable way.
1. Introduction
Let Ω be a bounded domain in Rn with smooth boundary . We introduce the weighted Sobolev space as the space of complex value functions u (x) defined on Ω with the finite norm:
We denote by the closure of in H with respect to the above norm, i.e., is the closure of in . The notion stands for the space of infinitely differentiable functions with compact support in Ω. In this paper, we investigate the spectral properties. In particular, we estimate the resolvent of a non-self-adjoint elliptic differential operator of typeacting on Hilbert space with Dirichlet-type boundary conditions. Here, is a positive function that satisfies the following conditions:where if and if and if and if , and the functions satisfy the uniformly elliptic condition, i.e., there exists c > 0 such that
Furthermore, suppose that such that for each , the matrix function q (x) has nonzero simple eigenvalues arranged in the complex plane in the following way:where is a closed angle with zero vertex (i.e., the eigenvalues of lie on the complex plane and outside of the closed angle Φ). For a closed extension of operator A with respect to space above, we need to extend its domain to the closed domain(see [1, 2]), where the local space is the functions in this form Here, and in the sequel, the value of the function arg and denotes the norm of the bounded operator A: .
To get a feeling for the history of the subject under study, refer to our earlier papers [3–5]. Indeed, this paper was written in continuing on our earlier papers. This study is sufficiently more general than our earlier papers; here, we obtain the resolvent estimate of operator A, which satisfies the special and general conditions.
2. The Resolvent Estimate of Degenerate Elliptic Differential Operators on H in Some Special Cases
Theorem 1. Let A in (2), i.e., assume that operator A is acting on Hilbert space H = L2 (Ω) with Dirichlet-type boundary conditions, and the sector Ω be defined as in Section 1. Let the complex function q (x) satisfy the following conditions:
Then, for sufficiently large modulus , the inverse operator exists and is continuous in H, and the following estimates are valid:where are sufficiently large numbers depending on S ( set is defined in the previous sections). The symbol stands for the norm of a bounded arbitrary operator T in H.
Proof. Here, to establish Theorem 1, we will first prove the assertion of Theorem 1 together with estimate (9). So, as in Section 1 for a closed extension of operator A (for more explanation, see chapter 6 in [3]), we need to extend its domain to the closed setLet operator A now satisfy (7), (8). Then, there exists a complex number (notice that we can take Z = eiy, for a fix real ) such that , and so In view of the uniformly elliptic condition, we haveand take which implies that . From this, and according to in (10), we then multiply these two positive relations with each other, implying thatMultiplying both sides of the latter relation by the positive term and then integrating both sides, we will haveNow by applying the integration by parts and using Dirichlet-type condition, then the right sides of the latter relation without multiple ReZ becomeHence,
Here, the symbol (,) denotes the inner product in H.
Notice that the above equality in (16) is obtained by the well-known theorem of the m-sectorial operators which are closed by extending its domain to the closed domain in H. These operators are associated with the closed sectorial bilinear forms that are densely defined in H (for more explanation of the well-known Theorem 1, see chapter 6 in [2]). This is why we extend the domain of operator A to the closed domain in space H above. Therefore,From (10), we have Multiply this inequality by . It follows thatFrom this and the above inequality, we will havei.e.,Since is positive, we will have eitherorThis inequality ensures that the operator is one to one, which implies that ker . Therefore, the inverse operator exists, and its continuity follows from the proof of estimate (9) of Theorem 1. To prove (9), we set in (19), implying thatSince , thensowhich implies that Since , then . This estimate completes the proof of the assertion of Theorem 1 together with estimate (9). Now, we start to prove estimate (10) of Theorem 1. As in the above argument, we drop the positive term dx fromIt follows thatEquivalentlySet in the latter relation, and proceeding by similar calculation as in the proof of estimate (9), we then obtainSince , thenConsequently, by (9), this implies thatTo this end, we will haveThus, here, the proof of estimate (10) is finished; i.e., this completes the proof of Theorem 1.
Now let condition (8) not hold. Then we will have the following statement.
3. The Resolvent Estimate of Some Classes of Degenerate Elliptic Differential Operators on H
In this section, we will derive a new general theorem by dropping the assumption (8) from Theorem 1 in Section 2.
Theorem 2. As in Section 1, let Φ be some closed sector with vertex at 0 in the complex plane (for more explanation, see [3]), and let the complex function q (x) satisfy the following equations:
Then, for sufficiently large modulus , the inverse operator exists and is continuous in H, and the following estimates hold:where are sufficiently large numbers depending on .
Proof. Let us (9) not satisfy. To prove the assertion of Theorem 2 together with (34), we construct the functions so that each one of the functions as the function q (x) in Theorem 1 satisfies (8).
Therefore, letsatisfyIn view of Theorem 1 and by (9) and (10), set Ar = A in the definition of the differential operator, which implies thatis acting on H whereDue to the assertion of Theorem 1, for , the inverse operator exists and is continuous in space and satisfiesLet us introduceHere is the multiplication operator in H by the function . Consequently, it is easily verified thatwhere ; supp and supp are contained in supp . Let us take the right side of (41) equal to I + T (λ). Thus, we will haveNow according to Section 2, if we put A = Ar for r = 1,…,m in (8), we will haveOwing to the definition of T (λ) in (41) easily, it follows thatSince is a sufficiently large number, it easily implies that . From this and using the well-known theorem in the operator theory, we conclude that I + T (λ) and so (A − λI) G (λ) are invertible. Hence, ((A − λI) G (λ))−1 exists and is equal toBy adding + I and –I to the right side of (44), it follows thatWe now setThenIn view of kT (λ) < 1 and (44), we now estimate F (λ) by the following geometric series:i.e., ., for we will havei.e., . Now from (45), we haveThereforei.e., here the assertion of Theorem 2 is proved. Therefore, to complete the proof Theorem 2, we must prove the estimate (34). To the end, according to the latter inequality, we have and since , it follows thatThis completes the proof of Theorem 2.
4. On the Resolvent Estimate of the Differential Operator in
As in Section 1, let the differential operatoract on Hilbert space with Dirichlet-type boundary conditions, and suppose that such that for each , the matrix function q (x) has nonzero simple eigenvalues arranged in the complex plane in the following way:where
Furthermore, suppose that for , we have
Now, according to Theorem 1, but here instead of operator A which acts on the space H = L2 (Ω), let operator A act on the space . Now by the assumption of Section 1, we will have the following theorem in the general case.
Theorem 3. Let (58) and (59) and the assumptions of Section 1 hold for operator A as in (2), then for sufficiently large modulus , the inverse operator exists and is continuous in the space and the following estimate holds: where are sufficiently large numbers depending on and .
Proof. Now by applying the eigenvalues of the matrix function q (x), we define the operators such thatwhere its extension domains arewhich, as operator A in Theorem 1, the operators Aj, , acts on space H = L2 (Ω) (notice that here the operators Aj are the same operator A in Section 2, i.e., to define the operators Aj, we just change the function in operator A by the eigenvalues functions μj (x), of matrix ). The conditions which we consider on the eigenvalues μj (x) of the matrix function in Section 1 guarantee that one can convert the matrix to the diagonal form , where and . Consider space ( times). Put where the operatoracts on the direct sum ( times) in which and (Uu)(x) = U (x) u (x); (). Consequently, it follows thatwhereUsing (9) and (10), we have where and . Now by the Hardy-type inequality, we estimate the operator as follows:Since by (3), we have the following inequality:Now by (3) and estimate (9), it followsThen, for sufficiently large in modulus of ; consequently,Proceeding as at the end of Section 2 (e.g., see (43)) from , it easily follows that is inversible and then that is inversible, that is,Then by adding , the last relation we have is Since , in a calculation as in Section 2, take . Then, satisfiesConsequently, sincePut as in (39). By (72) and (73), we have and it follows that , so Now we prove estimate (39). Since for , we can get the corresponding estimate , and this impliesSince , we havewhich implies so that the proof of the fundamental Theorem 3 in the general case is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
The authors contributed equally to this work. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to thank the Department of Mathematics and Statistics at McGill University for its hospitality during their sabbatical leave period and the Department of Mathematics, Lorestan University, Khorramabad, Iran.