A class of strongly stable approximation for unbounded operators

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1. Introduction Let (H;D(H)) be a self-adjoint unbounded operator on a Hilbert space H. With the purpose of finding the spectrum set sp(H) of the operator H by using numerical approach, the conventional methods used are the projection methods (see e.g. [1] and [2]). Precisely, let (Pk)k2N be a sequence of orthogonal projections Pk : H ! Lk , where the closed set Lk is a subspace of D(H) . In the theory of spectral approximation, we seek whether or not lim k!1 sp(Pk H Pk) = sp(H) . Generally, the result is negative, where for k large enough, the set sp(Hk) may contain points that do not belong to the set sp(H) . The weakness of projection method is well known in numerical analysis as the spectral pollution problem, this is an important problem in several areas in the field of applied mathematics (see e.g. [3], [4] and [5]). In this paper, we use an alternative method, the generalized spectral method, which has been introduced in [6]. This new method is based on the concept of the generalized spectrum (see [7] and [8]). Let T and S be two bounded linear operators defined on a Banach space X , we define the generalized resolvent, re(T; S) = fz 2 C : (T zS) : X ! X is beijective g: The complementary set of the generalized resolvent set is the generalized spectrum, denoted sp(T; S) . We say that is a generalized eigenvalue of the couple (T; S) if there exists u 2 X n f0g such that Tu = Su: 220 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela The subspace Ker(T S) is called the generalized spectral subspace corresponding to . The space of all bounded linear operator defined on the Banach space X is denoted by BL(X) . We consider now an unbounded operator (A;D(A)) defined on X , we recall that the resolvent set of A is given by re(A) = fz 2 C : (A zI) : D(A) ! X; is beijective and (A zI) 1 2 BL(X)g; and the spectrum set of A is sp(A) = C n re(A) . In this work, under the assumption re(A) 6= ; , we prove that each spectral problem associated to A has an equivalent generalized spectral problem which means that there exist two bounded operators T and S defined on X , satisfying sp(T; S) = sp(A) . Furthermore, if is an eigenvalue of A, then is a generalized eigenvalue of the couple (T; S) and Ker(A I) = Ker(T S): (1.1) Through the numerical approximation of the bounded operators T and S by sequences of bounded operators (Tk)k2N and (Sk)k2N defined on X; where they converge in an appropriate sense to T and S; we prove that lim k!1 sp(Tk; Sk) = sp(T; S): The limit here is understood as a combination of the following Property U and Property L, where they are naturally extended from the classical case with S = I (see [9]). Property U: if k 2 sp(Tk; Sk) and k ! , then 2 sp(T; S) . Property L: if 2 sp(T; S) , then there exists ( k)k2N such that k 2 sp(Tk; Sk) and k ! . We organize this paper as follows: throughout section 2, we construct the theoretical foundations of the generalized spectral method. This theory is a generalization of the classical case when S = I (see [9]). In section 3, we prove that the Property U and Property L hold under appropriate convergence of (Tk)k2N and (Sk)k2N to T and S respectively. Finally, a numerical application is given for the case of Schr¨odinger’s operator, where our numerical results show the coherence and the effectiveness of the generalized spectrum method (see [11]). 2. Generalized spectrum Let (X; k k) be a Banach space. The space BL(X) is the set of all bounded linear operators on X equipped with the subordinated operator norm, kAk = supfkAxk : x 2 X; kxk = 1g; A 2 BL(X): Let T and S be two operators in BL(X) , for z 2 re(T; S) , we set R(z; T; S) = (T zS) 1 A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 221 as the generalized resolvent operator. Let 2 sp(T; S) be a generalized eigenvalue, we say that has a finite algebraic multiplicity if dim Ker(T S) < 1: We remark that, if the operator S is invertible, then sp(T; S) = sp(S 1T); but if S 1 does not exist, the generalized spectrum set can be a bounded set, or the whole C; or an empty set. The next three results are a generalization of a classical case when S = I . The proofs are provided in [6]. Theorem 2.1. Let 2 re(T; S) and 2 C; where j j < jjR( ; T; S)Sjj 1 . Then 2 re(T; S) . Corollary 2.1. The set sp(T; S) is closed in C. Theorem 2.2. The function R( ; T; S) : re(T; S) ! BL(X) is analytic, and its derivative is given by R( ; T; S)SR( ; T; S) . We consider now an unbounded operator A with domain D(A) X . The following theorem shows that every unbounded operator allows a pair of two bounded operators in BL(X) which expresses it in the terms of the generalized spectrum. Theorem 2.3. If re(A) 6= ; , then there exist T; S 2 BL(X) such that sp(A) = sp(T; S): In particulary, is an eigenvalue for A if and only if is a generalized eigenvalue for the couple (T; S) . In addition, the equality (1.1) is satisfied. P r o o f. Let 2 re(A) . We define S; T : X ! D(A) as follows: S = (A I) 1; T = A(A I) 1: It is clear that T; S 2 BL(X) . To show that sp(A) = sp(T; S); we prove that re(A) = re(T; S): Let 2 re(A); i. e. there exists operator (A I) 1 2 BL(X): Then (A I)(A I) 1 = I + ( )(A I) 1 2 BL(X): So as [(A I)(A I) 1] 1 = (A I)(A I) 1 = I + ( )(A I) 1 2 BL(X); 222 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela we get A(A I) 1 (A I) 1 2 BL(X) ) (T S) 1 2 BL(X): Thus, it is proved that 2 re(T; S): Inversely, let 2 re(T; S) . To show that (A I) 1 2 BL(X) , we prove that (A I) is bijective. Firstly, check the injectivity. Let u 2 D(A) , using the fact that A commutes with (A I) 1 we have (A I) 1Au = A(A I) 1u = u + (A I) 1u: (2.2) Taking into consideration the equality (2.2), we find (A I)u = 0 ) (A I) 1(A I)u = 0 ) (A I)(A I) 1u = 0 ) [A(A I) 1 (A I) 1]u = 0 ) (T S)u = 0 ) u = 0: Secondly, prove the surjectivity. For all y 2 X we show that (A I)x = y has a solution x 2 D(A) . Put x = (A I) 1(T S) 1y ; it is clear that x 2 D(A) (the fact that (A I) 1 : X ! D(A) ), moreover we have (A I)(A I) 1(T S) 1y = [A(A I) 1 (A I) 1](T S) 1y = (T S)(T S) 1y = y: Furthermore, we can see, upon the choice of the vector x; that kxk k(A I) 1k k(T S) 1k kyk; so k(A I) 1k k(A I) 1k k(T S) 1k; which implies that (A I) 1 2 BL(X) and therefore 2 re(A) . Now, we show that the equality (1.1) holds. Let be a generalized eigenvalue of the couple (T; S) , then there exists u 2 Xnf0g such that Tu = Su , thus Tu = Su ) A(A I) 1u = (A I) 1u ) u = ( )(A I) 1 ) u 2 D(A): By applying (A I) on Tu = Su , we find that Au = u . Inversely, let be an eigenvalue of A, then Au = u . So, by applying (A I) 1 on Au = u and using the fact that (A I) 1Au = A(A I) 1u for all u 2 D(A) , we find that Tu Su = 0 . We note that the choice of the couple (T; S) as a function of the resolvent operator of A is not unique (see the numerical application below). The next results represent the theoretical background of the generalized spectrum approach. A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 223 Theorem 2.4. Let T; S 2 BL(X) , and let be a generalized eigenvalue with finite algebraic multiplicity, isolated in sp(T; S) . We denote by the Cauchy contour separating from sp(T; S) . Then the operator P = 1 2i Z (T zS) 1S dz (2.3) defines a projection from X to X , and we have PX = Ker(T S): (2.4) P r o o f. To show that the operator P given by (2.3) is a projection form X to X , see the book [8, p. 50]. Now to prove the equality (2.4), firstly we fix 2 re(T; S); where for any Cauchy contour associated with we assume that 62 . For 2 ; we have S T = ( S T)[( ) 1I ( S T) 1S]( ) which gives ( S T) 1 = [( ) 1I ( S T) 1S] 1( ) 1( S T) 1: Thus, we can see that ( ) 1 is an eigenvalue of the operator ( S T) 1S . Indeed u 2 Ker(T S) ) (T S)u = 0 ) ( S T) 1( S T + T S)u = u ) ( S T) 1Su = ( ) 1u ) u 2 Ker(( S T) 1S ( ) 1I): We reverse the last process and get Ker(T S) = Ker(( S T) 1S ( ) 1I): Now, under the choice of , we can see that for all Cauchy contour , -( ) is also a Cauchy contour of the eigenvalue ( ) 1 where -( ) = ( ) 1: We put B = ( S T) 1S and z = ( ) 1 for any 2 . Following this notation we have ( S T) 1S = z[ I + z(zI B) 1]: Thus, integrating over the ; we get 1 2 i Z ( S T) 1S d = 1 2 i Z -( ) z[ I + z(zI B) 1] dz z2 = 1 2 i Z -( ) [ z 1I + (zI B) 1] dz = 1 2 i Z -( ) 1 z dz I + 1 2 i Z -( ) (zI B) 1 dz = Pf( ) 1g; 224 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela where Pf( ) 1g is the spectral projection associated with the operator ( S T) 1S around ( ) 1 . Hence, according to the spectral decomposition theory, PX = Pf( ) 1gX = Ker(( S T) 1S ( ) 1I) = Ker(T S): Now, we show some results obtained in the qualitative aspect for the generalized spectrum theory. We denote by B(0; k) C the ball with center 0 and radius k > 0: Theorem 2.5. Let T; S 2 BL(X) , then there exists k > 0 such that sp(T; S) B(0; k) if and only if 0 62 sp(S) . P r o o f. We assume that sp(T; S) B(0; k) , then for 2 re(T; S); we get S T = ( S T)[( S T) 1S ( ) 1]( ): (2.5) As 2 re(T; S) , we have that 2 sp(T; S) () ( ) 1 2 sp(( S T) 1S): So, the inclusion sp(T; S) B(0; k) implies the relation 0 62 sp(( S T) 1S) ; otherwise, 0 2 sp(( S T) 1S) implies 1 2 sp(T; S) . Thus 0 62 sp(( S T) 1S) gives 0 62 sp(S) . We denote by spp(T; S) the set of all generalized eigenvalues. It is clear that when X is a finite-dimensional space, the generalized spectrum consists only of the generalized eigenvalues, except f1g: Theorem 2.6. Let T; S 2 BL(X) , if S is compact, then sp(T; S) = spp(T; S) [ f1g: P r o o f. We use the expression (2.5). Since the operator ( S T) 1S is compact, sp(T; S) is a set of isolated points. Let 2 sp(T; S) , then there is 2 sp(( S T) 1S); where = ( ) 1 . Hence there exists u 2 X n f0g such that ( S T) 1Su = u ) ( S T) 1( S S)u = u ) u + ( S T) 1(T S)u = u ) Tu = Su: A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 225 3. Generalized spectrum approximation Let T; S 2 BL(X); where re(T; S) 6= ; , and let (Tk)k2N and (Sk)k2N be two sequences in BL(X) . We will use the following conditions: (H1) S is a compact operator in BL(X) , (H2) k(Tk T)xk ! 0; k(Sk S)xk ! 0 for all x 2 X , (H3) k(Tk T)Tk ! 0 , (H4) k(Sk S)Tk ! 0 . In the sequel, we write p! to express the pointwise convergence, while the norm convergence is denoted by n! . P r o p o s i t i o n 3.1. (see [6]) Let T , ~ T , S , ~ S 2 BL(X) . For all z 2 re(T; S) , if kR(z; T; S) h (T ~ T) z(S ~ S) i k < 1; then z 2 re( ~ T; ~ S) , and the next inequality is satisfied kR(z; ~ T; ~ S)k kR(z; T; S)k 1 kR(z; T; S) h (T ~ T) + z(S ~ S) i k : R e m a r k 3.1. According to our assumptions in (H1) (H4) we can easily conclude that [(Tk T) (Sk S)] (T zS) n! 0; for all z 2 re(T; S) . P r o p o s i t i o n 3.2. Let A; B and C be three bounded operators such that 0 =2sp(B) and AB n! C , then B 1A n! B 1CB 1 . P r o o f. We note that jjB 1A B 1CB 1jj jjB 1jj jjAB Cjj jjB 1jj: Theorem 3.7. Property U. For k 2 N, under (H1) (H4) , if k 2 sp(Tk; Sk) and k ! ; then 2 sp(T; S): P r o o f. We assume that 2 re(T; S) . Since the set re(T; S) is open in C; as stated in Corollary 2.1, there exists r > 0 such that E := f 2 C : j j < rg re(T; S): On the other side, for all z 2 E and for all k 2 N, we find that Tk zSk = (T zS) [I + R(z; T; S)[(T Tk) z(S Sk)]] : Using Remark 3.1 and Proposition 3.2 with A = [(T Tk) (S Sk)]; B = (T S); so, there exists k0 2 N such that kR(z; T; S) [(T Tk) (S Sk)] k 1 2 ; 226 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela for all k k0 . Then, by Proposition 3.1, we find z 2 re(Tk; Sk) such that kR(z; Tk; Sk)k 2kR(z; T; S)k; 8k k0; but k ! , thus there exists k1 2 N such that k 2 E re(Tk; Sk) for k k1 , which form a contradiction. In numerical test, we calculate the quantity sup dist( ; sp(T; S)) : 2 sp(Tk; Sk) ; its convergence to 0 implies the Property U. We mention that dist( ; sp(T; S)) = inf y2sp(T;S) j yj: Lemma 3.1. (see [10]) Let P1 and P2 be two projections on X such that jj(P1 P2)P1jj < 1; then dim P1X dim P2X . Lemma 3.2. Let z 2 re(T; S) , under (H1) (H4) there exists a positive integer k0 2 N such that for k k0 , z 2 re(Tk; Sk) and R(z; Tk; Sk) n! R(z; T; S): P r o o f. Let z 2 re(T; S) , we have Tk zSk = (T zS) [I + R(z; T; S)[(T Tk) z(S Sk)]] ; for all k 2 N. As stated above in the demonstration of Theorem 3.7, we find z 2 re(Tk; Sk) for all k k0; and R(z; Tk; Sk) is uniformly bounded for all k 2 N. On the other side, for z 2 re(T; S) re(Tk; Sk); R(z; Tk; Sk) R(z; T; S) = R(z; T; S) [(T Tk) z(S Sk)]R(z; Tk; Sk): Since R(z; T; S) [(T Tk) z(S Sk)] n! 0 (according Remark 3.1 and to Proposition 3.2) and R(z; Tk; Sk) is uniformly bounded for all k 2 N, we have that R(z; T; S) [(T Tk) z(S Sk)] (Tk zSk) 1 n! 0: A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 227 Theorem 3.8. Let be a generalized eigenvalue of finite type, isolated in sp(T; S) . We denote by the Cauchy contour separating from sp(T; S) . Under (H1) (H4); there exists k0 2 N such that for each k k0; we have dim PX = dim PkX; where P = 1 2i Z R(z; T; S)S dz; Pk = 1 2 i Z R(z; Tk; Sk) 1Sk dz: P r o o f. For z 2 and k k0; we see that R(z; Tk; Sk)Sk R(z; T; S)S = [R(z; Tk; Sk) R(z; T; S)] S [R(z; Tk; Sk) R(z; T; S)] (S Sk) R(z; T; S)(S Sk): From (H1) (H4) we easily find that (S Sk)(T zS) n! 0 , thus according to Proposition 3.2 we have R(z; T; S)(S Sk) n! 0 . Now by using Lemma 3.2, we have R(z; Tk; Sk)Sk R(z; T; S)S n! 0: Finally, we apply Lemma 3.1 and find that dim PX = dim PkX for k k0 . Theorem 3.9. Property L. Let be a generalized eigenvalue of finite type, isolated in sp(T; S) . Under (H1) (H4) there exists a sequence k 2 sp(Tk; Sk) such that k ! . P r o o f. Let be the Cauchy contour separating from sp(T; S) . We set k 2 int( ) sp(Tk; Sk): Since re(T; S) 3 z 7! R(z; T; S)S and re(Tk; Sk) 3 z 7! R(z; Tk; Sk)Sk are analytic functions, and Pk n! P , we find ( k)k2N = ; () int( ) sp(T; S) = ;: We fix > 0 such that the sequence ( k)k2N belongs to B; where B = fz 2 C : jz j g: On the other hand, it is enough to show that every convergent subsequence of ( k)k2N converges to itself. Indeed, let a subsequence ( k0)k02N converge to ~ where ~ 6= . By Property U proved in Theorem 3.7, we see that ~ 2 sp(T; S); but ~ 2 B and sp(T; S) B = f g , hence = ~ , thus k ! . The last theorem shows that for every generalized eigenvalue of finite type isolated in sp(T; S) , there exists a sequence ( k)k2N converging to such that k 2 sp(Tk; Sk) . The next result shows that the generalized eigenvectors associated to k converge to the generalized eigenvector associated with . 228 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela We define the notion of gap between two closed subspaces Z and Y of X as gap(Z; Y ) = max (Z; Y ); (Y;Z) ; where (Z; Y ) = sup dist(z; Y ) : z 2 Z; kzk = 1 : Theorem 3.10. Let M = PX and Mk = PkX for k 2 N. Then gap(M;Mk) ! 0: P r o o f. Let u 2 M = PX such that kuk = 1 . For k 2 N large enough we have dist(u;Mk) ku Pkuk = kPu Pkuk kP Pkk: Let u 2 Mk = PkX such that kuk = 1: For k 2 N large enough dist(u;M) ku Puk = kPku Puk kP Pkk; which implies gap(M;Mk) kPk Pk: 4. Numerical application As an example for which the numerical results are available by other approaches, we consider the following problem from [11]; it is also studied in [13]. We consider the unbounded operator A defined on L2(0;+1) by the differential equation Au := u00 + x2u; x 2 [0;+1); u(0) = 0: This is the harmonic oscillator problem with domain D(A) = H2(0;+1) n u 2 L2(0;1) : Z 1 0 x2 juj2dx < 1 o : First, according to the theory of pseudo spectrum for self-adjoint operators (see [6], [11] and [14]) we can find sp(A) = [ a>0 sp(Aa); (4.6) where Aa is the Schr¨odinger operator which has the same formula as A in L2(0; a) , but with the Dirichlet condition at the point a . The domain of Aa is given by D(Aa) = H2(0; a) H1 0 (0; a): Let a > 0 , we denote by La the Laplacien operator defined on L2(0; a) by Lau = u00; D(L) = H2(0; a) H1 0 (0; a): A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 229 P r o p o s i t i o n 4.3. (see [12]) La is invertible and its inverse is the bounded operator Sa defined by Sau(x) = Z a 0 Gf0;ag(x; y)u(y)dy; u 2 L2(0; a); where Gf0;ag(x; y) = ( x(a y) a 0 x y a; y(a x) a 0 y x a: Let Ta be a bounded operator defined on L2(0; a) to itself by Tau(x) = u(x) + Z a 0 Gf0;ag(x; y)y2u(y)dy: Theorem 4.11. sp(A) = [ a>0 sp(Ta; Sa): P r o o f. According to (4.6), we need only to show that sp(Aa) = sp(Ta; Sa) for all a > 0 . Let be an eigenvalue of Aa with the eigenvector u 2 D(Aa) n f0g: By applying Sa to Aau = u; we get Tau = Sau; which implies that is a generalized eigenvalue of the couple (Ta; Sa) with the eigenvector u 2 L2(0; a) n f0g . Inversely, let be a generalized eigenvalue of the couple (Ta; Sa) with the eigenvector u 2 L2(0; a) n f0g , i. e. Tau = Sau , so u = Sau Sa(v u) ) u = Sa( u v u); where v(x) = x2 . Since u + vu 2 L2(0; a) , we have u 2 D(La) = D(Aa) , then u + Sa(v u) = Sau ) Lau + v u = u: Now, for a > 0 we use Kantorovich’s projection method to approach the operators Ta and Sa . We define a subdivision of [0; a] for n 2 by hn = a n 1 ; xi = (i 1)hn; 1 i n: Let Ta;n and Sa;n be the approximation operators of Ta and Sa by means of Kantorovich’s projection methods (see [9]), given for all x 2 [0; a] by Ta;nun(x) un(x) + Xn i=1 Z a 0 Gf0;ag(xi; y)y2un(y)dy - ei(x); Sa;nun(x) Xn i=1 Z a 0 Gf0;ag(xi; y)un(y)dy - ei(x); 230 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela where, for 2 i n 1 , ei(x) = 8< : 1 jx xij hn ; xi 1 x xi+1 0; otherwise; e1(x) = 8< : x2 x hn ; x1 x x2 0; otherwise; en(x) = 8< : x xn 1 hn ; xn 1 x xn 0; otherwise: By applying Kantorovich’s projection method [9] to the equation Tau = Sau; we get the approximate equation un(x) + Xn i=1 Z a 0 Gf0;ag(xi; y)y2un(y)dy - ei(x) = n Xn i=1 Z a 0 Gf0;ag(xi; y)un(y)dy - ei(x); x 2 [0; a]: Denote by 1 and 2 the two vectors 1(i) = Z a 0 Gf0;ag(xj ; y)y2un(y)dy; 2(i) = Z a 0 Gf0;ag(xj ; y)un(y)dy; 1 i n; then we can rewrite the previous approximate equation as un(x) + Xn i=1 1(i)ei(x) = n Xn i=1 2(i)ei(x): (4.7) Multiplying first equation (4.7) by Gf0;ag(xj ; x)x2 for 1 j n and integrating over [0; a] , we obtain n Xn i=1 2(i) Z a 0 Gf0;ag(xj ; x)x2ei(x)dx - = Z a 0 Gf0;ag(xj ; x)x2un(x)dx + Xn i=1 1(i) Z a 0 Gf0;ag(xj ; x)x2ei(x)dx - : The latter equation is equivalent to the matrix equation 1 + A 1 = nA 2; (4.8) where A is a matrix defined by A(i; j) = Z a 0 Gf0;ag(xj ; x)x2ei(x)dx; 1 i; j n: A CLASS OF STRONGLY STABLE APPROXIMATION FOR UNBOUNDED OPERATORS 231 In the same way, multiplying equation (4.7) by Gf0;ag(xj ; x) for 1 j n and integrating over [0; a] , we also obtain n Xn i=1 2(i) Z a 0 Gf0;ag(xj ; x)ei(x)dx - = Z a 0 Gf0;ag(xj ; x)un(x)dx + Xn i=1 1(i) Z a 0 Gf0;ag(xj ; x)ei(x)dx - ; the latter equation is equivalent to the matrix equation 2 + B 1 = nB 2; (4.9) where B is a matrix defined by B(i; j) = Z a 0 Gf0;ag(xj ; x)ei(x)dx; 1 i; j n: So, by using this process, we have transformed the equation (4.7) into the system of two matrix equations (4.8) and (4.9), namely 1 + A 1 = nA 2; 2 + B 1 = nB 2: We also can write this system as (In-n + A) 1 + On-n 2 = nOn-n 1 + nA 2; B 1 + 2 = nOn-n 1 + nB 2; where In-n is the identity matrix with dimension n - n and On-n is the null matrix with dimension n - n . This leads to the matrix generalized eigenvalue problem A + In-n On-n B In-n 1 2 = n On-n A On-n B 1 2 : Finally, we use the command "eig" in Matlab to calculate the generalized eigenvalue of A + In-n On-n B In-n ; On-n A On-n B : We mention that Kantorovich’s projection method gives the norm convergence (see [9]) which satisfies our assumption in (H1) (H4): We fix n = 200 to approach the eigenvalues in our example. The following table 1 shows that the Kantorovich’s method converges perfectly compared with the exact eigenvalue. 232 A. Khellaf, S. Benarab, H. Guebbai, W. Merchela Table 1: The numerical results for a=5 Exact eigenvalue Kantorovich’s method 3 3.0001972 7 7.0009887 11 11.0026039 15 15.0103317 19 19.0806050 5. Conclusion Our study shows the efficiency of the generalized spectrum method, theoretically and numerically. This technique appears to be a computationally attractive tool for resolving the spectral pollution. We resolved this spectral pollution by treating the analytical question: to find the bounded operators T and S representing the spectrum proprieties of an unbounded operator A in the theory of generalized spectrum.

About the authors

Ammar Khellaf

Universit´e 8 Mai 1945

Email: amarlasix@gmail.com; khellaf.ammar@univ-guelma.dz
Post-Graduate Student, Mathematics Department B.P. 401, Guelma 24000, Algeria

Sarra Benarab

Derzhavin Tambov State University

Email: benarab.sarraa@gmail.com
Post-Graduate Student, Functional Analysis Department 33 Internatsionalnaya St., Tambov 392000, Russian Federation

Hamza Guebbai

Universit´e 8 Mai 1945

Email: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz
Associate Professor of the Mathematics Department B.P. 401, Guelma 24000, Algeria

Wassim Merchela

Derzhavin Tambov State University

Email: merchela.wassim@gmail.com
Post-Graduate Student, Functional Analysis Department 33 Internatsionalnaya St., Tambov 392000, Russian Federation

References

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