Investigation of the Spectral Properties of a Non-Self-Adjoint Elliptic Differential Operator

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 H1 � L2(Ω) 1. (en, as the application of this new result, the resolvent of the considered operator in l-dimensional space Hilbert Hl � L2(Ω) l is obtained utilizing some analytic techniques and diagonalizable way.


Introduction
Let Ω be a bounded domain in R n with smooth boundary zΩ (i.e., zΩ ∈ C ∞ ). We introduce the weighted Sobolev space H 1 � W 2 2,β (Ω) ℓ as the space of complex value functions u (x) defined on Ω with the finite norm: We denote by � H 1 the closure of C ∞ 0 (Ω) ℓ in H with respect to the above norm, i.e., � H is the closure of C ∞ 0 (Ω) in H 1 � W 2 2,β (Ω) ℓ . e notion C ∞ 0 (Ω) 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 type acting on Hilbert space H ℓ � L 2 (Ω) ℓ with Dirichlet-type boundary conditions. Here, ρ(x) ∈ C 1 (0, 1) is a positive function that satisfies the following conditions: . . , n) and the functions α ij (x) satisfy the uniformly elliptic condition, i.e., there exists c > 0 such that Furthermore, suppose that q(x) ∈ C 2 (Ω; EndC ℓ ) such that for each x ∈ (Ω), the matrix function q (x) has nonzero simple eigenvalues μ j (x) ∈ C 2 (Ω)(1 ≤ j ≤ ℓ) arranged in the complex plane in the following way: where Φ � z ∈ C: |argz| ≤ φ , φ ∈ (0, π) is a closed angle with zero vertex (i.e., the eigenvalues μ j (x) of q(x) lie on the complex plane and outside of the closed angle Φ). For a closed extension of operator A with respect to space H � W 2 2,β (Ω) ℓ above, we need to extend its domain to the closed domain (see [1,2]), where the local space W 2 2,loe (Ω) ℓ is the functions Here, and in the sequel, the value of the function arg z ∈ (− π,π] and ‖A‖ denotes the norm of the bounded operator A: H ⟶ H.
To get a feeling for the history of the subject under study, refer to our earlier papers [3][4][5]. Indeed, this paper was written in continuing on our earlier papers. is 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.

The Resolvent Estimate of Degenerate Elliptic Differential Operators on H in Some Special Cases
arg q en, for sufficiently large modulus λ ∈ Φ, the inverse operator (A − λI) − 1 exists and is continuous in H, and the following estimates are valid: where M Φ , C Φ > 0 are sufficiently large numbers depending on S (Φ set is defined in the previous sections). e symbol ‖.‖ stands for the norm of a bounded arbitrary operator T in H.
Proof. Here, to establish eorem 1, we will first prove the assertion of eorem 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 set Let operator A now satisfy (7), (8). en, there exists a complex number Z ∈ C (notice that we can take Z � e iy , for a fix real Υ ∈ (− π, π]) such that |Z � e iΥ | � 1, and so In view of the uniformly elliptic condition, we have and take . From this, and according to c ′ ≤ Re Zq(x) in (10), we then multiply these two positive relations with each other, implying that Multiplying both sides of the latter relation by the positive term ρ 2α (x) and then integrating both sides, we will have Now by applying the integration by parts and using Dirichlet-type condition, then the right sides of the latter relation without multiple ReZ become Hence, International Journal of Mathematics and Mathematical Sciences 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.
ese operators are associated with the closed sectorial bilinear forms that are densely defined in H (for more explanation of the well-known eorem 1, see chapter 6 in [2]).
is is why we extend the domain of operator A to the closed domain in space H above. erefore, From this and the above inequality, we will have i.e., Since c 1 n i�1 Ω ρ 2β (x)| (zy/zx i )(x)| 2 dx is positive, we will have either is inequality ensures that the operator (A − λI) is one to one, which implies that ker (A − λI) � 0. erefore, the inverse operator (A − λI) − 1 exists, and its continuity follows from the proof of estimate (9) of eorem 1. To prove (9), we Since is estimate completes the proof of the assertion of eorem 1 together with estimate (9). Now, we start to prove estimate (10) of eorem 1. As in the above argument, we drop the positive term c ′ |λ| Ω |y(x)| 2 dx from It follows that Equivalently Set (A − λI) − 1 f, f ∈ Hin the latter relation, and proceeding by similar calculation as in the proof of estimate (9), we then obtain (29) Consequently, by (9), this implies that To this end, we will have us, here, the proof of estimate (10) is finished; i.e., this completes the proof of eorem 1. Now let condition (8) not hold. en we will have the following statement.

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 eorem 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 International Journal of Mathematics and Mathematical Sciences [3]), and let the complex function q (x) satisfy the following equations: q(x) ∈ CΦ; (∀x ∈ Ω ).
(33) en, for sufficiently large modulus λ ∈ Φ, the inverse operator (A − λI) − 1 exists and is continuous in H, and the following estimates hold: where M Φ ′ , C Φ > 0 are sufficiently large numbers depending on Φ .