Báo cáo hóa học: "MATRIX TRANSFORMATIONS AND GENERATORS OF ANALYTIC SEMIGROUPS" doc

MATRIX TRANSFORMATIONS AND GENERATORS OF ANALYTIC SEMIGROUPS BRUNO DE MALAFOSSE AND AHMED MEDEGHRI Received 3 May 2006; Revised 8 June 2006; Accepted 15 June 2006 We establish a relation between the notion of an operator of an a nalytic semigroup and matrix transformations mapping from a set of sequences into χ,whereχ is either of the sets l ∞ , c 0 ,orc. We get extensions of some results given by Labbas and de Malafosse concerning applications of the sum of operators in the nondifferential case. Copyright © 2006 B. de Malafosse and A. Medeghri. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. 1. Introduction In this paper, we are interested in the study of operators represented by infinite matrices. Note that in [1], Altay and Bas¸ar gave some results on the fine spectrum of the difference operator Δ acting on the sequence spaces c 0 and c. Then they dealt with the fine spectrum of the operator B(r,s) defined by a matrix band over the sequence spaces c 0 and c.In de Malafosse [3, 5], there are results on the spectrum of the Ces ` aro matrix C 1 and on the matrix Δ considered as operators from s r to itself. Spectral proper ties of unbounded operators are used in the theory of the sum of operators. The notion of generators of analytic semigroup was developed in this way. Recall that this theory was studied by many authors such as Da Prato and Grisvard [2, 12], Fuhrman [11], Labbas and Terreni [16, 17]. Some applications can also be found in Labbas and de Malafosse [15]ofthesumof operators in the theory of summability in the noncommutative case. Some results were obtained in de Malafosse [4] on the equation Ax + Bx − λx = y for λ ≥ 0 (1.1) in a reflexive Banach set of sequences E,wherey ∈ E, A and B are two closed linear operators represented by infinite matrices with domains D(A) and D( B) included in E. Here we are interested in some extensions of results given in [15] using similar matri- ces A and B. Recall that the choice of these matrices was motivated by the solvability of Hindaw i Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 67062, Pages 1–14 DOI 10.1155/JIA/2006/67062 2 Matrix and generators of analytic semigroups a class of infinite-tr idiagonal systems. Then we study some spectral properties of A and B considered as matrix transformations in the sets (s 1/a ,l ∞ )and(s 1/β ,l ∞ ), or (s 0 1/a ,c 0 )and (s 0 1/β ,c 0 ), or (s (c) 1/a ,c)and(s (c) 1/β ,c). Then we show that (−A)and(−B)aregenerators of ana- lytic semigroups,whereD(A)andD(B) are of the form χ(A)andχ(B)withχ = l ∞ , c 0 ,orc. Here the relative boundedness with respect to A or B is not satisfied, so we are not within the framework of the classical pertur bation theory given by Kato [14]orPazy[18]. In this paper, we establish a relation between results in summability and basic notions used in the theory of the sum of operators. For this, we need to recall the following. 2. Preliminary results 2.1. Recall of some results in summability. Let M = (a nm ) n,m≥1 be an infinite matrix and consider the sequence x = (x n ) n≥1 . We will define the product Mx = (M n (x)) n≥1 with M n (x) =  ∞ m=1 a nm x m whenever the series are convergent for all n ≥ 1. Let s denote the set of all complex sequences. We write ϕ, c 0 , c and l ∞ for the sets of finite, null, convergent, and bounded sequences, respectively. For any g iven subsets X, Y of s,wewillsaythat the operator represented by the infinite matrix M = (a nm ) n,m≥1 maps X into Y, that is, M ∈ (X,Y), if the series defined by M n (x) =  ∞ m=1 a nm x m are convergent for all n ≥ 1and for all x ∈ X and Mx ∈ Y for all x ∈ X. For any subset X of s,wewillwrite MX ={y ∈ s : y = Mx for some x ∈ X}. (2.1) If Y is a subset of s, we will denote the so-called matrix domain by Y(M) ={x ∈ s : y = Mx ∈ Y}. (2.2) Let X ⊂ s beaBanachspace,withnorm· X .ByᏮ(X), we will denote the setofall bounded linear operators, mapping X into itself. We will say that L ∈ Ꮾ(X)ifandonlyif L : X → X is a linear operator and L ∗ Ꮾ(X) = sup x=0   Lx X /x X  < ∞. (2.3) It is well known that Ꮾ(X)isaBanachalgebrawiththenorm L ∗ Ꮾ(X) .ABanachspace X ⊂ s is a BK space if the projection P n : x → x n from X into C is continuous for all n. ABKspaceX ⊃ ϕ is said to have AK if for every x ∈ X, x = lim p→∞  p k =1 x k e k ,where e k = (0, ,1, ), 1 being in the kth position. It is well known that if X has AK, then Ꮾ(X) = (X,X), see [9, 13, 19]. Put now U + ={x = (x n ) n≥1 ∈ s : x n > 0foralln}.Forξ = (ξ n ) n≥1 ∈ U + ,wewilldefine the diagonal matrix D ξ = (ξ n δ nm ) n,m≥1 ,(whereδ nm = 0foralln = m and δ nm = 1 other- wise). For α ∈ U + ,wewillwrites α = D α l ∞ ,(cf.[3–10, 15]. The set s α is a BK space with the norm x s α = sup n≥1 (|x n |/α n ). The set of all infinite matrices M = (a nm ) n,m≥1 with M S α = sup n≥1  1 α n ∞  m=1   a nm   α m  < ∞ (2.4) B. de Malafosse and A. Medeghri 3 is a Banach algebra with identity normed by · S α .RecallthatifM ∈ (s α ,s α ), then Mx s α ≤M S α x s α for all x ∈ s α . Thus we obtain the following result, (cf. [7]) where we put B(s α ) = Ꮾ(s α )  (s α ,s α ). Lemma 2.1. For any given α ∈ U + , B(s α ) = S α = (s α ,s α ). In the same way, we will define the sets s 0 α = D α c 0 and s (c) α = D α c,(cf.[7]). The sets s 0 α and s (c) α are BK spaces with the norm · s α and s 0 α has AK. It was shown in [9, 10]thatfor any matrix M ∈ (s α ,s α ), we get M ∗ Ꮾ(s α ) =M ∗ Ꮾ(s 0 α ) =M ∗ Ꮾ(s (c) α ) =M S α . (2.5) In all what follows, we will use the next lemma. Lemma 2.2. Let α, β ∈ U + and let X, Y be subsets of s. Then M ∈  D α X,D β Y  iff D 1/β MD α ∈ (X,Y). (2.6) 2.2. O perator generators of analytic semigroups. We recall here some results given in Da Prato and Grisvard [2] and Labbas and Terreni [16, 17]. Let E be a Banach space. We consider two closed linear operators A and B, whose domains are D(A)andD(B)in- cluded in E.Foreveryx ∈ D(A)  D( B), we then define their sum Sx = Ax + Bx. The spectral properties of A and B are the following: (H) there are C A ,C B > 0, and ε A ,ε B ∈ ]0,π[suchthat ρ(A) ⊃  A =  z ∈ C :   Arg(z)   <π− ε A  ,   (A − zI) −1   £(E) ≤ C A |z| ∀ z ∈  A −{0}, ρ(B) ⊃  B =  z ∈ C :   Arg(z)   <π− ε B  ,   (B − zI) −1   £(E) ≤ C B |z| ∀ z ∈  B −{0}, ε A + ε B <π. (2.7) It is said that A and B are generators of analytic semigroups not strongly continuous at t = 0andwehaveσ(A)  σ(−B) = ∅ and ρ(A)  ρ(−B) = C . Thefollowingiswellknowninthecommutativecase: (A − ξI) −1 (B − ηI) −1 − (B − ηI) −1 (A − ξI) −1 = 0 ∀ξ ∈ ρ(A), η ∈ ρ(B), (2.8) if D(A)andD(B) are densely defined in E,itiswellknown(cf.[2]) that the bounded operator defined by L λ =− 1 2iπ  Γ (B + zI) −1 (A − λI − zI) −1 dz ∀λ>0, (2.9) 4 Matrix and generators of analytic semigroups where Γ is an infinite-sectorial curve lying in ρ(A − λI)  ρ(−B) coincides with (A + B − λI) −1 . In the following, we will consider matrix transformations A and B mapping in a set of sequences and we will show that they satisfy hypothesis (H). 3. Definition of the operators A and B We will consider two infinite matrices and deal with the case when A and B map into l ∞ or c 0 and with the case when A and B map into c. In each case, we will study their spectral properties. For given sequences a = (a n ) n≥1 , b = (b n ) n≥1 , β = (β n ) n≥1 ,andγ = (γ n ) n≥1 ,letA and B be the follow ing infinite matr ices: A = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ a 1 b 1 O ·· Oa n b n · ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ , B = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ β 1 O ·· γ n β n O · ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ . (3.1) 3.1. The case when A and B are operators mapping from D(A) and D(B) into E,where E = l ∞ or c 0 . The next conditions are consequences of results given in [15]. When E is either of the sets l ∞ or c 0 , we assume that A satisfies the following properties: a ∈ U + , a n is strictly increasing, lim n→∞ a n =∞, (3.2a) there is M A > 0suchthat   b n   ≤ M A ∀n. (3.2b) Similarly, we assume that B satisfies the next conditions: β ∈ U + ,lim k→∞ β 2k = L = 0, (3.3a) lim k→∞ β 2k+1 a 2k+1 =∞, (3.3b) (α) there is M B > 0suchthat   γ 2k   ≤ M B ∀n,(β)γ 2k+1 = o(1) (n −→ ∞ ). (3.3c) 3.2. The case when A and B are operators mapping D(A) and D(B) into c. Here we need to recall the characterization of (c,c). Lemma 3.1. A ∈ (c,c) if and only if (i) A ∈ S 1 , (ii) lim n→∞  ∞ m=1 a nm = l for some l ∈ C, (iii) lim n→∞ a nm = l m for some l m ∈ C, m = 1,2, We will see that D(A) = c(A) = s (c) 1/a and D(B) = c(B) = s (c) 1/β . Here we will show that neither of the sets D(A) = s (c) 1/a and D(B) = s (c) 1/β is embedded in the other. B. de Malafosse and A. Medeghri 5 Proposition 3.2. Let a, β ∈ U + . Then s (c) 1/a  s (c) 1/β , s (c) 1/β  s (c) 1/a (3.4) if and only if β/a, a/β / ∈ c. Proof. The inclusion s (c) 1/a ⊂ s (c) 1/β means that I ∈ (s (c) 1/a ,s (c) 1/β )andbyLemma 2.2,wehave D β/a ∈ (c,c). From Lemma 3.1,weconcludethats (c) 1/a ⊂ s (c) 1/β if and only if β/a ∈ c.So s (c) 1/a  s (c) 1/β is equivalent to β/a /∈ c. Similarly, we have s (c) 1/β  s (c) 1/a if and only if a/β /∈ c. This completes the proof.  We assume that A and B satisfy the following hypotheses. The matrix A is defined in (3.1)with a ∈ U + , a n is strictly increasing, lim n→∞ a n =∞, (3.5a) b ∈ c. (3.5b) For B givenin(3.1), we do the following hypotheses: β ∈ U + ,lim n→∞ β n =∞, (3.6a) lim k→∞ β 2k+1 a 2k+1 =∞,lim k→∞ β 2k a 2k = l = 0, (3.6b) γ ∈ c. (3.6c) This lead to the next remark. Remark 3.3. The choice of β in (3.6b) is justified by Proposition 3.2 and so neither of the sets D(A) = s (c) 1/a and D(B) = s (c) 1/β is embedded in the other one. We will see in Proposition 5.7 and Theorem 5.8 that we need to have (3.5a)and(3.6a). Then we will see that A and B are closed operators when b, γ ∈ c. Finally, notice that D(B) ⊂ c means 1/β ∈ c which is trivial ly satisfied in (3.6a) and it is the same for A. 4. First properties of the operators A and B 4.1. The case when the operators A and B are considered as matrix maps from D(A) and D(B) into E,whereE is equal to l ∞ ,orc 0 . In this section, we will assume A and B satisfy (3.2)and(3.3). For the convenience of the reader, recall the following well-know n results. Lemma 4.1. (i) A ∈ (l ∞ ,l ∞ ) if and only if A ∈ S 1 . (ii) A ∈ (c 0 ,c 0 ) if and only if A ∈ S 1 and lim n→∞ a nm = 0 for each m = 1,2, Proposition 4.2. (i) A ∈ (s 1/a ,l ∞ ) and A ∈ (s 0 1/a ,c 0 ). (ii) B ∈ (s 1/β ,l ∞ ) and B ∈ (s 0 1/β ,c 0 ). 6 Matrix and generators of analytic semigroups Proof. (i) We have [AD 1/a ] nn = 1and[AD 1/a ] n,n+1 = b n /a n+1 for all n,and[AD 1/a ] nm = 0 otherwise. Then   AD 1/a   S 1 = sup n  1+   b n   a n+1  = O(1) (n −→ ∞ ). (4.1) So by Lemma 4.1,wehaveAD 1/a ∈ (l ∞ ,l ∞ )andbyLemma 2.2, A ∈ (s 1/a ,l ∞ ). The proof is similar for A ∈ (s 0 1/a ,c 0 ) note that in this case, [AD 1/a ] nm → 0(n →∞)foreachm ≥ 1. (ii) Now [BD 1/β ] nn = 1and[BD 1/β ] n,n−1 = γ n /β n−1 for all n,and[BD 1/β ] nm = 0 other- wise. By (3.3a), (3.3c)(β), we have γ 2n+1 β 2n = o(1) (n −→ ∞ ) (4.2) and by (3.2a), (3.3b)and(3.3c)(α), we get γ 2n β 2n−1 = γ 2n a 2n−1 a 2n−1 β 2n−1 = o(1) (n −→ ∞ ). (4.3) Then   BD 1/β   S 1 = sup n  1+   γ n   β n−1  = O(1) (n −→ ∞ ). (4.4) So BD 1/β ∈ (l ∞ ,l ∞ )andB ∈ (s 1/β ,l ∞ ). Finally, we obtain B ∈ (s 0 1/β ,c 0 ) reasoning as above.  We deduce that if E = l ∞ = s 1 , the matrix A is defined on D(A) = s 1/a and B is defined on D(B) = s 1/β . It can be shown that l ∞ (A) = s 1/a and l ∞ (B) = s 1/β .WhenE = c 0 ,wewill see in Theorem 5.6(i), (ii) that D(A) = c 0 (A) = s 0 1/a and D( B) = c 0 (B) = s 0 1/β .Wededuce from (3.3a), (3.3b) that in each case, neither of the sets D(A)andD(B)isembeddedin the other. 4.2. The case when the operators A and B are considered as matrix maps from D(A) and D(B) into c. We assume that A and B satisfy (3.5)and(3.6). From the preceding, we immediately get the following. Proposition 4.3. A ∈ (s (c) 1/a ,c) and B ∈ (s (c) 1/β ,c). Proof. It is enough to notice that by (3.5)wehave(1+b n /a n+1 ) n≥1 ∈ c.ThenfromLemma 3.1,weconcludethatA ∈ (s (c) 1/a ,c). We also have by (3.6), γ n β n−1 −→ 0(n −→ ∞ ) (4.5) so BD 1/β ∈ (c,c)andB ∈ (s (c) 1/β ,c).  We will see in Theorem 5.8(i), (ii) that D(A) = c(A) = s (c) 1/a and D(B) = c(B) = s (c) 1/β . B. de Malafosse and A. Medeghri 7 5. The matrices A and B as operator generators of an analytic semigroup In this section, we will show that A and B are generators of analytic semigroup in each case E = l ∞ , E = c 0 ,orE = c. 5.1. The case when A and B are considered as matrix maps from D(A) and D(B) into E, where E = l ∞ or c 0 . In this section, A and B satisfy (3.2)and(3.3).Thenextresultwas shown in [15] in the case when A ∈ (s 1/a ,l ∞ )andB ∈ (s 1/β ,l ∞ )witha n = a n , a>1, and β was defined by β 2n = 1andβ 2n+1 = (2n +1)!foralln,soweomittheproof. Proposition 5.1. In the space l ∞ , the two linear operators A and B are closed and satisfy the following: (i) D(A) = s 1/a , (ii) D(B) = s 1/β , (iii) D( A) = l ∞ , D(B) = l ∞ , (iv) there are ε A , ε B > 0 (with ε A + ε B <π) such that   (A − λI) −1   ∗ Ꮾ(l ∞ ) ≤ M |λ| ∀ λ = 0,   Arg(λ)   ≥ ε A ,   (B + μI) −1   ∗ Ꮾ(l ∞ ) ≤ M |μ| ∀ μ = 0,   Arg(μ)   ≤ π − ε B . (5.1) This result shows that −A and −B satisfy hypothesis (H) and σ(−A)  σ(B) = ∅.So −A and −B are generators of the analytic semigroups e (−At) and e (−Bt) not strongly con- tinuous at t = 0. We have similar results when A and B are matrix maps into c 0 .Werequire some elementary lemmas whose proofs are left to the reader. Lemma 5.2. Let ε ∈ ]0,π/2[ and let x 0 > 0 beareal.Then   x 0 − λ   ≥ x 0 sinε ∀λ ∈ C with   Arg(λ)   ≥ ε. (5.2) Lemma 5.3. Let x 0 > 0 beareal.Then   x 0 − λ   ≥| λ|sinθ ∀λ =|λ|e iθ /∈ R − ,   x 0 − λ   ≥| λ|∀λ ∈ R − . (5.3) We can state the following result where we will use the fact that for any α ∈ U + , since s 0 α is a BK space with AK, we have Ꮾ(s 0 α ) = (s 0 α ,s 0 α ). As we have seen in (2.5), for any matrix C ∈ (s 0 α ,s 0 α ), we have C ∗ Ꮾ(s 0 α ) =C ∗ (s 0 α ,s 0 α ) =C S α . Proposition 5.4. (i) Let ε A ∈ ]0,π/2[.Foreveryλ ∈ C with |Arg(λ)|≥ε A , the infinite matrix A − λI considered as an operator in s 0 1/a is invertible and (A − λI) −1 ∈  c 0 ,s 0 1/a  . (5.4) 8 Matrix and generators of analytic semigroups (ii) Let ε B ∈ ]0,π/2[.Foreveryμ ∈ C with |Arg(μ)|≤π − ε B , the infinite matrix B + μI considered as an operator in s 0 1/β is invertible and (B + μI) −1 ∈  c 0 ,s 0 1/β  . (5.5) Proof. (i) Fix ε A ∈ ]0,π/2[ and consider the infinite-sectorial set  ε A =  λ ∈ C :   Arg(λ)   <ε A  . (5.6) For any λ/ ∈  ε A ,put χ n = b n a n − λ (5.7) and D  λ = D (1/(a n −λ)) n .Then[D  λ (A − λI)] nn = 1, [D  λ (A − λI)] n,n+1 = χ n for all n and [D  λ (A − λI)] nm = 0 otherwise. By Lemma 5.2,wehave   χ n   ≤ M A a n sinε A ∀n and all λ/∈  ε A . (5.8) Since a n tends to infinity as n tends to infinit y, there is n 0 such that   χ n   ≤ 1 2 ∀n ≥ n 0 and all λ/∈  ε A . (5.9) Consider now the infinite matr ix  T λ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ · T −1 λ · 0 ··· 1 01 · · ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (5.10) where T λ is the matrix of order n 0 defined by [T λ ] nn = 1for1≤ n ≤ n 0 ;[T λ ] n,n+1 = χ n for 1 ≤ n ≤ n 0 − 1, and [T λ ] nm = 0 otherwise. Elementary calculations give T −1 λ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 1 −χ 1 χ 1 χ 2 −χ 1 χ 2 χ 3 ·· · 1 −χ 2 χ 2 χ 3 ·· · 1 −χ n χ n χ n+1 · · 0 1 χ n 0 −1 1 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ . (5.11) B. de Malafosse and A. Medeghri 9 Putting Ꮽ λ = D  λ (A − λI)  T λ we easily get [Ꮽ λ ] nn = 1foralln,[Ꮽ λ ] n,n+1 = χ n for n ≥ n 0 , and [Ꮽ λ ] nm = 0 otherwise. Then by Lemma 2.2 and since the sequence a is increasing , we get   I − Ꮽ λ   ∗ (s 0 1/a ,s 0 1/a ) =   I − Ꮽ λ   S 1/a = sup n≥n 0    χ n   a n a n+1  ≤ 1 2 ∀λ/∈  ε A . (5.12) Since Ꮽ λ = Ꮽ λ − I + I ∈ (s 0 1/a ,s 0 1/a )and(5.12)holds,Ꮽ λ is invertible in the Banach al- gebra of all bounded operators Ꮾ(s 0 1/a ) = (s 0 1/a ,s 0 1/a )mappings 0 1/a to itself and Ꮽ −1 λ ∈ (s 0 1/a ,s 0 1/a ). Then for any given y ∈ c 0 , we successively get y  = D  λ y = (y n /(a n − λ)) n≥1 ∈ s 0 1/a , Ꮽ −1 λ y  ∈ s 0 1/a ,  T λ (Ꮽ −1 λ y  ) ∈ s 0 1/a ,and(A − λI) −1 =  T λ Ꮽ −1 λ D  λ ∈ (c 0 ,s 0 1/a ). So we have shown (i). (ii) For ε B ∈ ]0,π/2[, let Σ B ={μ ∈ C : |Arg(μ)|≤π − ε B } and put χ  n = γ n β n + μ . (5.13) To deal with the inverse of B + μI, we need to study the sequences |γ 2k+1 |/β 2k and |χ  2k |β 2k / β 2k−1 .By(3.3a), (3.3b), we have γ 2k+1 β 2k −→ 0(k −→ ∞ ). (5.14) On the other hand, for every μ ∈ Σ B ,weget   χ  2k   β 2k β 2k−1 ≤ M B β 2k sinε B β 2k β 2k−1 = M B sinε B 1 β 2k−1 , (5.15) 1 β 2k−1 = a 2k−1 β 2k−1 1 a 2k−1 = o(1) (k −→ ∞ ). (5.16) From (5.14)and(5.16), we deduce that there is n 1 such that   γ 2k+1   1 β 2k ≤ 1 2 sinε B for 2k +1≥ n 1 ,   χ  2k   β 2k β 2k−1 ≤ 1 2 for 2k ≥ n 1 ∀μ ∈ Σ B . (5.17) As in (i), define the matrices D  μ = D (1/(β n +μ) n ) and  R μ = ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ · R −1 μ · 0 ··· 1 01 · · ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ , (5.18) 10 Matrix and generators of analytic semigroups where R μ is the matrix of order n 1 − 1definedby[R μ ] nn = 1foralln,by[R μ ] n,n−1 = χ  n for 2 ≤ n ≤ n 1 − 1, and by [R μ ] nm = 0 otherwise. Then elementary calculations show that the matrix Ꮾ μ =  R μ D  μ (B + μI)isdefinedby[Ꮾ μ ] nn = 1foralln,by[Ꮾ μ ] n,n−1 = χ  n for n ≥ n 1 , and by [Ꮾ μ ] nm = 0 otherwise. Then for any μ ∈ Σ B ,   I − Ꮾ μ   ∗ (s 0 1/β ,s 0 1/β ) = sup n≥n 1    χ  n   β n β n−1  = max  τ 1 ,τ 2  , (5.19) where τ 1 = sup k≥n 1 /2   χ  2k   β 2k β 2k−1 , τ 2 = sup k≥(n 1 −1)/2   χ  2k+1   β 2k+1 β 2k . (5.20) By Lemma 5.2,weget |β 2k + μ|≥β 2k sinε B for all μ ∈ Σ B and τ 1 ≤ 1/2. Then τ 2 ≤ 1 2 sinε B 1 β 2k+1 sinε B β 2k+1 = 1 2 for 2k ≥ n 1 . (5.21) This implies I − Ꮾ μ  ∗ (s 0 1/β ,s 0 1/β ) ≤ 1/2. Reasoning as in (i) with (B + μI) −1 = Ꮾ −1 μ  R μ D  μ ,we conclude that B + μI considered as an operator from s 0 1/β into c 0 is invertible and (B + μI) −1 ∈ (c 0 ,s 0 1/β )forallμ ∈ Σ B . This concludes the proof.  Remark 5.5. As a direct consequence of the preceding, it is trivial that c 0 (A − λI) = s 0 1/a ∀λ ∈ C,   Arg(λ)   ≥ ε A , c 0 (B + μI) = s 0 1/β ∀μ ∈ C,   Arg(μ)   ≤ π − ε B . (5.22) We immediately obtain the next result. Theorem 5.6. In the space c 0 , the two linear operators A and B areclosedandsatisfythe following: (i) D(A) = c 0 (A) = s 0 1/a , (ii) D(B) = c 0 (B) = s 0 1/β , (iii) D( A) = c 0 , D(B) = c 0 , (iv) there are ε A , ε B > 0 (with ε A + ε B <π) such that   (A − λI) −1   ∗ Ꮾ(c 0 ) ≤ M |λ| ∀ λ = 0,   Arg(λ)   ≥ ε A ,   (B + μI) −1   ∗ Ꮾ(c 0 ) ≤ M |μ| ∀ μ = 0,   Arg(μ)   ≤ π − ε B . (5.23) Proof. Show that A is a closed operator. For this, consider a sequence x  p = (x np ) n≥1 tend- ing to x = (x n ) n≥1 in c 0 ,asp tends to infinity, w here x  p ∈ s 0 1/a for all p.ThenAx  p → y (p → ∞ )inc 0 ,thatisforanyn,wehaveA n (x  p ) → A n (x) = y n (p →∞). It remains to show that x ∈ s 0 1/a . For this, note that since b ∈ l ∞ and x ∈ c 0 ,weconcludethata n x n = y n − b n x n+1 tends to a zero as n tends to infinity. The proof for B is similar. [...]... infinity The proof for B is similar The proof of statements (i) and (ii) comes from Proposition 4.3 and follows the same lines as that for Theorem 5.6(i), (ii) (iii) Follows exactly the same lines as that in the proof given in the case when E = l∞ in [15, Proposition 3, page 196] The proof of (iv) is a consequence of Proposition 5.7(i) and follows exactly the same lines as that in the proof of Theorem 5.6... contribution to improve the presentation of the paper The work of the second author is supported by the Agence Universitaire de la Francophonie (AUF) 14 Matrix and generators of analytic semigroups References [1] B Altay and F Basar, On the fine spectrum of the generalized difference operator B(r,s) over the ¸ sequence spaces c0 and c, International Journal of Mathematics and Mathematical Sciences 2005 (2005),... 1-2, 53–71 (2000) , Application of the sum of operators in the commutative case to the infinite matrix theory, [4] Soochow Journal of Mathematics 27 (2001), no 4, 405–421 , Properties of some sets of sequences and application to the spaces of bounded difference [5] sequences of order μ, Hokkaido Mathematical Journal 31 (2002), no 2, 283–299 , On matrix transformations and sequence spaces, Rendiconti... supn≥n0 (|χn |) ≤ 1/2 and then in the ∞ ≤ 2−m = 2 (5.28) m=0 Finally, by (5.9), (5.10), and (5.11), we have supλ∈ ε Tλ S1 < ∞ and we conclude that / A (A − λI)−1 ∗ 0 ) ≤ M/ |λ| for all λ ∈ εA We get similar results for B + μI This con/ Ꮾ(c cludes the proof of (iv) 5.2 The case when A and B are matrix maps from D(A) and D(B) into c In this section, A and B satisfy conditions given in (3.5) and (3.6) We will... 41–60 [10] B de Malafosse and V Rakocevic, Applications of measure of noncompactness in operators on the p spaces sα , s0 , s(c) and lα , to appear in Journal of Mathematical Analysis and Applications α α [11] M Fuhrman, Sums of operators of parabolic type in a Hilbert space: strict solutions and maximal regularity, Advances in Mathematical Sciences and Applications 4 (1994), no 1, 1–34 [12] P Grisvard,... Jarrah and E Malkowsky, Ordinary, absolute and strong summability and matrix transformations, Filomat 2003 (2003), no 17, 59–78 [14] T Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, vol 132, Springer, New York, 1966 [15] R Labbas and B de Malafosse, An application of the sum of linear operators in infinite matrix theory, Faculty of Sciences University of. .. Matrix and generators of analytic semigroups Proposition 5.7 (i) Let εA ∈]0,π/2[ For every λ with | Arg(λ)| ≥ ε, the infinite matrix A − λI considered as an operator in s(c) is invertible and 1/a (A − λI)−1 ∈ c,s(c) 1/a (5.29) (ii) Let εB ∈ ]0,π/2[ For every μ with | Arg(μ)| ≤ π − εB , the infinite matrix B + μI considered as an operator in s(c) is invertible and 1/β (B + μI)−1 ∈ c,s(c) 1/β (5.30) Proof... Journal of Mathematics and Mathematical Sciences [7] 2003 (2003), no 28, 1783–1801 , On the sets of sequences that are strongly α-bounded and α-convergent to naught with [8] index p, Seminario Matematico Universit` e Politecnico di Torino 61 (2003), no 1, 13–32 a , The Banach algebra Ꮾ(X), where X is a BK space and applications, Matematichki Vesnik [9] 57 (2005), no 1-2, 41–60 [10] B de Malafosse and V... Ꮽ−1 D1/a y ∈ s0 , and x = A−1 y = T0 Ꮽ0 1 D1/a y ∈ s0 This 1/a 1/a 1/a shows that c0 (A) ⊂ s0 and since s0 ⊂ c0 (A), we conclude that c0 (A) = s0 The proof is 1/a 1/a 1/a similar for B (iii) Let x = (1/an )n≥1 ∈ c0 and assume x p = (xnp )n≥1 tend to x in s0 , that is, 1/a xp − x s1/a = sup an xnp − n 1 an − 0 → (p − ∞) → (5.25) Since an tends to infinity, we should have xnp → 1/an and an xnp → 1 (p... 18, 3005–3013 [2] G Da Prato and P Grisvard, Sommes d’op´rateurs lin´aires et ´quations diff´rentielles op´ratione e e e e nelles, Journal de Math´ matiques Pures et Appliqu´ es Neuvi` me S´ rie 54 (1975), no 3, 305– e e e e 387 [3] B de Malafosse, Some properties of the Ces` ro operator in the space sr , Faculty of Sciences Unia versity of Ankara Series A1 Mathematics and Statistics 48 (1999), no 1-2, . 10.1155/JIA/2006/67062 2 Matrix and generators of analytic semigroups a class of infinite-tr idiagonal systems. Then we study some spectral properties of A and B considered as matrix transformations in the. (s 1/a ,l ∞ )and( s 1/β ,l ∞ ), or (s 0 1/a ,c 0 )and (s 0 1/β ,c 0 ), or (s (c) 1/a ,c )and( s (c) 1/β ,c). Then we show that (−A )and( −B)aregenerators of ana- lytic semigroups,whereD(A)andD(B) are of the. A ∈ (s 1/a ,l ∞ ) and A ∈ (s 0 1/a ,c 0 ). (ii) B ∈ (s 1/β ,l ∞ ) and B ∈ (s 0 1/β ,c 0 ). 6 Matrix and generators of analytic semigroups Proof. (i) We have [AD 1/a ] nn = 1and[ AD 1/a ] n,n+1 =