The Fine Spectra of the Difference Operator Δ over the Sequence Spaces l 1 and bv

Transcription

1 International Mathematical Forum, 1, 2006, no 24, The Fine Spectra of the Difference Operator Δ over the Sequence Spaces l 1 and bv Kuddusi Kayaduman Gaziantep Üniversitesi Fen Edebiyat Fakültesi,Matematik Bölümü Gaziantep, Turkey kayaduman@gantepedutr Hasan Furkan Kahramanmaraş Sütçü İmam Üniversitesi Fen Edebiyat Fakültesi,Matematik Bölümü Kahramanmaraş, Turkey hasanfurkan@hotmailcom Abstract In the present paper, the fine spectrum of the difference operator on the sequence spaces l 1 and bv have been examined Mathematics Subject Classification: Primary: 47A10, Secondary: 47B37 Keywords: Spectrum of an operator, the sequence spaces l 1 and bv 1 Preliminaries, Background and Notation Let X and Y be the Banach spaces and T : X Y also be a bounded linear operator By R(T ), we denote the range of T, ie, R(T )={y Y : y = Tx, x X} By B(X), we also denote the set of all bounded linear operators on X into itself If X is any Banach space and T B(X) then the adjoint T of T is a bounded linear operator on the dual X of X defined by (T φ)(x) =φ(tx) for all φ X and x X Let X {θ} be a complex normed space and T : D(T ) X also be a linear operator with domain D(T ) X With T, we associate the operator T α = T αi, (1)

2 1154 Kuddusi Kayaduman and Hasan Furkan where α is a complex number and I is the identity operator on D(T ) If T α has an inverse, which is linear, we denote it by Tα 1, that is T 1 α =(T αi) 1 (2) and call it the resolvent operator of T The name resolvent is appropriate, since Tα 1 helps to solve the equation T α x = y Thus, x = Tα 1 y provided Tα 1 exists More important, the investigation of properties of Tα 1 will be basic for an understanding of the operator T itself Naturally, many properties of T α and Tα 1 depend on α, and spectral theory is concerned with those properties For instance, we shall be interested in the set of all α in the complex plane such that Tα 1 exists Boundedness of Tα 1 is another property that will be essential We shall also ask for what α s the domain of Tα 1 is dense in X, to name just a few aspects For our investigation of T, T α and Tα 1, we need some basic concepts in spectral theory which are given as follows (see [6, pp ]): Definition 11 Let X {θ} be a complex normed space and T : D(T ) X also be a linear operator with domain D(T ) X A regular value α of T is a complex number such that (R1) Tα 1 exists, (R2) Tα 1 is bounded, (R3) Tα 1 is defined on a set which is dense in X The resolvent set ρ(t ) of T is the set of all regular values α of T Its complement σ(t )=C\ρ(T ) in the complex plane C is called the spectrum of T Furthermore, the spectrum σ(t ) is partitioned into three disjoint sets as follows: The point (discrete) spectrum σ p (T ) is the set such that Tα 1 does not exist A α σ p (T ) is called an eigenvalue of T The continuous spectrum σ c (T ) is the set such that Tα 1 (R3) but not (R2), that is, Tα 1 is unbounded The residual spectrum σ r (T ) is the set such that T 1 α bounded or not) but not satisfy (R3), that is, the domain of Tα 1 in X exists and satisfies exists (and may be is not dense To avoid trivial misunderstandings, let us say that some of the sets defined above, may be empty This is an existence problem which we shall have to discuss Indeed, it is well-known that σ c (T )=σ r (T )= and the spectrum σ(t ) consists of only the set σ p (T ) in the finite dimensional case By w, we shall denote the space of all real valued sequences Any vector subspace of w is called as a sequence space We shall write l, c, c 0 and bv for the spaces of all bounded, convergent, null and bounded variation sequences, respectively Also by l 1 and bs, we denote the space of all absolutely convergent and bounded series, respectively

3 The Fine Spectra of the Difference Operator Δ Over l 1 and bv 1155 Let λ and μ be two sequence spaces and A =(a nk ) be an infinite matrix of real or complex numbers a nk, where n, k N = {0, 1, 2,} Then, we say that A defines a matrix mapping from λ into μ, and we denote it by writing A : λ μ, if for every sequence x =(x k ) λ the sequence Ax = {(Ax) n }, the A-transform of x, is in μ; where (Ax) n = k a nk x k, (n N) (3) For simplicity in notation, here and in what follows, the summation without limits runs from 0 to By (λ : μ), we denote the class of all matrices A such that A : λ μ Thus, A (λ : μ) if and only if the series on the right side of (3) converges for each n N and every x λ, and we have Ax = {(Ax) n } n N μ for all x λ 2 Main Results In this section, the fine spectrum of the difference operator Δ on the sequence spaces l 1 and bv has been examined The difference operator Δ is represented by the matrix Δ= Lemma 21 The matrix A =(a nk ) gives rise to a bounded linear operator T B(l 1 ) from l 1 to itself if and only if the supremum of l 1 norms of the columns of A is bounded Corollary 22 Δ:l 1 l 1 is a bounded linear operator and Δ (l1,l 1 ) =2 Lemma 23 The matrix A =(a nk ) gives rise to a bounded linear operator T B(bv) from bv to itself if and only if sup (a nk a n 1,k ) l < n k=l Corollary 24 Δ:bv bv is a bounded linear operator and Δ (l1,l 1 ) = Δ (bv,bv) The fine spectrum of the Cesàro operator on the sequence space l p has been studied by Gonzàlez [5], where 1 <p< Also, weighted mean matrices of operators on l p have been investigated by Cartlidge [3] The spectrum of the

4 1156 Kuddusi Kayaduman and Hasan Furkan Cesàro operator on the sequence spaces bv 0 and bv have also been investigated by Okutoyi [8] and Okutoyi [7], respectively More recently, the fine spectra of the difference operator Δ over the sequence space l p by Akhmedov and Başar [1] In this work, our purpose is to determine the fine spectrum of the difference operator Δ on the sequence spaces l 1 and bv Theorem 25 σ(δ,l 1 )={α C : α 1 1} Proof: Since Δ αi is triangle, hence (Δ αi) 1 exists Solving the equation (Δ αi)x = y for y we can formally derive that (Δ αi) 1 = (1 α) 1 0 (1 α) (1 α) 1 (1 α) (1 α) 3 (1 α) 2 Thus, we observe that (Δ αi) 1 (l 1 : l 1 ) if and only if sup k n=1 1 1 α n < One can see that (Δ αi) 1 (l 1 : l 1 ) whenever α 1 1 This completes the proof Theorem 26 σ p (Δ,l 1 )= Proof: This is immediate from the inclusion l 1 c 0, and by Theorem 22 in [2] If T : l 1 l 1 is a bounded linear operator with matrix A, then it is known that the adjoint operator T : l 1 l 1 is defined by the transpose of the matrix A Theorem 27 σ p (Δ,l 1 )={α C : α 1 1} Proof: Suppose Δ f = αf for f θ l 1 = l Then by solving the system of linear equations f 0 f 1 = αf 0 f 1 f 2 = αf 1 f 2 f 3 = αf 2 f k f k 1 = αf k (4)

5 The Fine Spectra of the Difference Operator Δ Over l 1 and bv 1157 we obtain that f k =(1 α) k f 0, (k N) This shows that f =(f k ) l 1 if and only if α 1 1 This means that f l 1 if f 0 0 and α 1 1, as asserted This completes the proof Now we give the following lemma required in the proof of next theorem Lemma 28 [4, p 59] T has a dense range if and only if T is one to one, where T denotes the adjoint operator of the operator T Theorem 29 σ r (Δ,l 1 )={α C : α 1 1} Proof: For α 1 1, the operator Δ αi is one to one in l 1 hence has an inverse But Δ αi is not one to one by Theorem 27 Now Lemma 28 yields the fact that the range of the operator Δ αi is not dense in l 1 and this step completes the proof Theorem 210 σ c (Δ,l 1 )= Proof: Since σ(t,x) is the disjoint union of the parts σ p (T,X), σ r (T,X) and σ c (T,X) we must have σ c (Δ,l 1 )= Theorem 211 σ(δ,bv)={α C : α 1 1} Proof: Solving the equation (Δ αi)x = y (5) for y bv, we can formally derive that 1/(1 α) 0 0 (Δ αi) 1 1/(1 α) 2 1/(1 α) 0 = 1/(1 α) 3 1/(1 α) 2 1/(1 α) Therefore, one can obtain by (5) that x =(Δ αi) 1 y At this situation, it is easy to show that x k x k 1 = k j=0 y j y j 1, (k N) (1 α) k j+1 and (Δ αi) 1 is bounded if and only if α 1 > 1 This completes the proof

6 1158 Kuddusi Kayaduman and Hasan Furkan Theorem 212 Δ B(bv) with the norm Δ (bv,bv) =2 Proof: The linearity of the operator Δ is trivial and so is omitted Let us take any x =(x k ) bv Then since ( ) Δx bv = x k x k 1 (x k 1 x k 2 ) k k x k x k 1 + k x k 1 x k 2 =2 x bv we derive from here that Further, by Theorem 211 we have that Δ (bv,bv) 2 (6) 2 Δ (bv,bv) (7) Therefore, we obtain by combining the inequalities (6) and (7) that Δ (bv,bv) =2 as desired This step concludes the proof Theorem 213 σ p (Δ,bv)= Proof: Suppose that Δx = αx for x θ =(0, 0, 0,)inbv Then, by solving the systems of linear equations x 0 = αx 0 x 1 x 0 = αx 1 x 2 x 1 = αx 2 we find that if x n0 is the first non-zero entry of the sequence x =(x n ) the α = 1 and x n0 +1 x n0 = x n0 +1 which implies that x n0 = 0 This is a contradiction Therefore σ p (Δ,bv)= If T : bv bv is a bounded matrix operator with matrix A, then T : bv bv acting on C bs has matrix of the form [ ] χ 0 b A t ;

7 The Fine Spectra of the Difference Operator Δ Over l 1 and bv 1159 where χ is the limit of the sequence of row sums of A minus the sum of the limit of the columns of A, and b is the column vector whose k th entry is the limit of the k th column of A for each k N [7] For Δ : bv bv, the matrix Δ B(bs) is of the form [ ] 0 0 Δ = 0 Δ t Theorem 214 σ p (Δ,bv )={α C : α 1 1} Proof: Suppose Δ f = αf for f θ bv = C bs Then by solving the system of linear equations 0 = αf 0 f 1 f 2 = αf 1 f 2 f 3 = αf 2 f k f k 1 = αf k (8) we obtain that f k =(1 α) k f 1, (k N) If α = 0 then we can choose f 0 0 and so f =(f 0, 0, 0, 0, ) is an eigenvector corresponding to α Ifα 0 then f 0 = 0, and by solving the system we obtain that f n =(1 α) n 1 f 1 (n 2) This means that f C bs iff n sup (1 α) k < n or sup n k=0 if and only if α 1 1, as asserted 1 (1 α) n+1 1 (1 α) < Theorem 215 σ r (Δ,bv)={α C : α 1 1} Proof: For α 1 1, the operator Δ αi is one to one hence has an inverse However, Δ αi is not one to one by Theorem 214 Now, Lemma 28 yields the fact that R(Δ αi) bv and this step concludes the proof Theorem 216 σ c (Δ,bv)=

8 1160 Kuddusi Kayaduman and Hasan Furkan Proof: The result immediately follows from the fact that σ(δ,bv) is the disjoint union of σ p (Δ,bv), σ r (Δ,bv) and σ c (Δ,bv) ACKNOWLEDGEMENTS The authors would like to express their indeptedness to Dr Bilâl Altay, İnönü Ün ivers ites i, Eğit im Fakültes i, Matematik Eğitimi Bölümü, Malatya/Türkiye, for his careful reading and corrections on the first draft of the present paper References [1] A M Akhmedov and F Başar, On the spectra of the difference operator in Δ over the sequence space l p, Demonstratio Math (to appear) [2] B Altay and F Başar, On the fine spectrum of the difference operator Δ on c 0 and c, Information Sciences, 168(2004), [3] J P Cartlidge, Weighted Mean Matrices as Operators on l p, Ph D Dissertation, Indiana University, 1978 [4] S Goldberg, Unbounded Lineer Operators, Dover Publications, Inc New York, 1985 [5] M Gonzàlez, The fine spectrum of the Cesàro operator in l p (1 <p< ), Arch Math 44(1985), [6] E Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons Inc New York-Chichester-Brisbane-Toronto, 1978 [7] J T Okutoyi, On the spectrum of C 1 as an operator on bv, Commun Fac Sci Univ Ank Ser A 1, 41(1992), [8] J I Okutoyi, On the spectrum of C 1 as an operator on bv 0, J Austral Math Soc Ser A,48 (1990), Received: October 16, 2005