On The Fine Spectrum of Generalized Upper Triangular Triple-Band Matrices ( 2 uvw) t

Transcription

1 Filomat 30:5 ( DOI 098/FIL605363A Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: On The Fine Spectrum of Generalized Upper Triangular Triple-Band Matrices ( t Over the Sequence Space l Selma Altundağ ab Merve Abay a a Sakarya University Department of Mathematics 5487 Sakarya Turkey b Faculty of Engineering and Natural Sciences International University of Sarajevo Sarajevo Bosnia and Herzegovina Abstract In this work we determine the fine spectrum of the matrix operator ( t which is defined generalized upper triangular triple band matrix on l Also we give the approximate point spectrum defect spectrum and compression spectrum of the matrix operator ( t on l Introduction In functional analysis the spectrum of an operator generalizes the notion of eigenvalues for matrices The spectrum of an operator over a Banach space is partitioned into three parts which are point spectrum the continuous spectrum and residual spectrum The calculation of these three parts of the spectrum of an operator is called calculating the fine spectrum of the operator Several authors studied the spectrum and fine spectrum of linear operators defined by some triangle matrices over some sequence spaces We introduce knowledge in the existing literature concerning the spectrum ( and the fine spectrum The fine spectrum of the Cesàro operator on the sequence space l p for < p < was studied by Gonzalez [8] Reade [5] studied the spectrum of the Cesàro operator over the sequence space c 0 The fine spectrum of the difference operator over the sequence spaces c o and c has been studied by Altay and Başar [] The same authors have studied the fine spectrum of the generalized difference operator B(r s over c o and c in [] The fine spectra of over l and bv have been studied by Kayaduman and Furkan [] The fine spectrum of generalized difference operator B(r s over the sequence spaces l and bv has been studied by Furkan Bilgiç and Kayaduman [5] Recently the fine spectrum of B(r s t over the sequence spaces c o and c has been studied by Furkan et al [6] Vatan Karakaya and Muhammed Altun have studied U(r s which is upper triangular double-band matrices over the sequences c o and c [] In 0 Srivastava and Kumar have studied fine spectrum of generalized difference operator v on l [6] Fine spectrum of the generalized difference operator uv on sequence space l has been studied by Srivastava and Kumar [7] Ali Karaisa has studied fine spectrum of upper triangular doubleband matrices over sequence space l p ( < p < [9] Fathi and Lashkaripour have studied on the fine 00 Mathematics Subject Classification 47A0 47B37 47A39 Keywords Spectrum of an operator fine spectrum Goldberg s classification approximate point spectrum defect spectrumcompression spectrum Received: 5 April 04; Accepted: 7 September 04 Communicated by Eberhard Malkowsky The corresponding author: Selma Altundağ addresses: scaylan@sakaryaedutr (Selma Altundağ abaymerve@hotmailcomtr (Merve Abay

2 S Altundağ M Abay / Filomat 30:5 ( spectrum of generalized upper double-band matrices uv which is transpose of the uv over the sequence space on l [4] Quite recently Karaisa studied fine spectra of upper triangular triple-band matrices over the sequence space l p (0 < p < [0] Panigrahi and Srivastava have studied spectrum and fine spectrum of generalized second order forward difference operator on sequence space l [4] In this paper we study the fine spectrum of the transpose of matrix operator on the sequence space l Additionally we give the approximate point spectrum defect spectrum and compression spectrum of the matrix operator ( t on l Preinaries and Notations 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 with T = T Let X {θ} be a complex normed space and T : D (T X be a linear operator with domain D (T X By T associate the operator T α = T αi 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 T α that is T α = (T αi and it is called to be the resolvent operator of T The name is appropriate since T α helps to solve the equation T α x = y Thus x = T α y provided T α exist More important the investigation of properties of T α will be basic for an understanding of the operator T itself Naturally many properties of T α and T α 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 T α exist Boundedness of T α is another property that will be essential We shall also ask for what α s the domain of T α is dense in X to name just a few aspects For our investigaton of T T α and T α we need some basic concepts in spectral theory which are given as follows (see[0 pp ] Definition Let X {θ} be a complex normed space and T : D (T X also be a linear operator with domain D (T X A reguler value of α of T is a complex number such that (R T α exist (R T α is bounded (R3 T α is defined on a set which is dense in X The resolvent set ρ (T of T is the set of all reguler 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 spectrum σ p (T is the set such that T α 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 T α exist and satisfies (R3 but not (R

3 S Altundağ M Abay / Filomat 30:5 ( The residual spectrum σ r (T is the set such that T α exist (and may be bounded or not but not satisfy (R3 Therefore these three subspectras from disjoint subdivisions σ(t X = σ p (T X σ c (T X σ r (T X ( In this section following Appell et al [3] we call the three more subdivisions of the spectrum called the approximate point spectrum defect spectrum and compression spectrum Given a bounded linear operator T in a Banach space X we call a sequence (x k in X as a Wely sequence for T if x k = and Tx k 0 as k In what follows we call the set σ ap (T X := { α C : there exists a Wely sequence for αi T } ( the approximate point spectrum of T Moreover the subspectrum σ δ (T X := { α C : αi T is not surjective } (3 is called defect spectrum of T The two subspectra given by ( and (3 from a (not necessarily disjoint subdivisions σ (T X = σ ap (T X σ δ (T X of the spectrum There is another subspectrum σ co (T X := { α : R(αI T X } which is often called compression spectrum in the literature another (not necessarily disjoint decomposition The compression spectrum gives rise to σ (T X = σ ap (T X σ co (T X of spectrum Clearly σ p (T X σ ap (T X and σ co (T X σ δ (T X Moreover comparing these subspectra with those in ( we note that σ r (T X = σ co (T X /σ p (T X σ c (T X = σ (T X / [ σ p (T X σ co (T X ] Proposition Spectra and subspectra of an operator T B(X and its adjoint T B(X are related by following relations: (aσ(t X = σ(t X (bσ c (T X σ ap (T X (cσ ap (T X = σ δ (T X (dσ δ (T X = σ ap (T X (eσ p (T X = σ co (T X (fσ co (T X σ p (T X (gσ(t X = σ ap (T X σ p (T X = σ p (T X σ ap (T X

4 S Altundağ M Abay / Filomat 30:5 ( Relation (c-(f show that the approximate point spectrum is in a certain sense dual to the defect spectrum and the point spectrum dual to compression spectrum The equality (g implies in particular that σ(t X = σ ap (T X if T is a Hilbert space and T is normal Roughly speaking this shows that normal (in particular self-adjoint operator on Hilbert space are most similar to matrices in finite dimensional spaces (see [3] From Goldberg [7] If T B(X X a Banach space then there are three possibilities for R(T the range of T (AR(T = X (BR(T R(T = X (CR(T X and (T exists an is continuous ( T exists but is discontinuous (3T does not exist If these possibilities are combined in all ways nine different states are created These are labelled by: A A A 3 B B B 3 C C C 3 If an operator is in state C for example then R(T X and T exists but is discontinuous (see [7] Let X and Y be two sequence spaces and A = (a nk be an infinite matrix of real or complex numbers a nk where n k N = { 3 } Then we say that A defines a matrix mapping from X into Y and we denote it by writing A : X Y if for every sequence x = (x k X the sequence Ax = { (Ax n }n N the A-transform of x is in Y where (Ax n = a nk x k (n N (4 k By (X : Y we denote the class of all matrices A such that A : X Y Thus A (X : Y if and only if the series on the right side of (4 converges for each n N and every x X and we have Ax = { (Ax n }n N Y for all x X Lemma 3 The adjoint operator T of T is onto if and only if T has a bounded inverse Lemma 4 T has a dense range if and only if T is one to one Lemma 5 The matrix A = (a nk gives rise to a bounded linear operator T B (l from l to itself if and only if the supremum of l norms of the columns of A is bounded Corollary 6 σ r (T X σ p (T X σ r (T X σ p (T X 3 Main Results In this section we prove that operator ( t : l l is bounded linear operator and we compute spectrum point spectrum continuous spectrum residual spectrum approximate point spectrum defect spectrum and compression spectrum of the operator ( t over space l Let u = (u k is either a constant sequence or sequence of distinct positive real numbers with U = u k so that u k 0 for each k N 0 v = (v k is a sequence of positive real numbers such that v k 0 for each k N 0 with V = v k and w = (w k is a sequence of positive real numbers such that w k 0 for each k 0 with

5 S Altundağ M Abay / Filomat 30:5 ( W = w k and sup u k < U + V We define the operator ( t on sequence space l as k ( tx = (uk x k + v k x k+ + w k x k+ k=0 with x = 0 x = 0 where x = (x n l It is easy to verify that the operator ( t can be represented by the matrix u 0 v 0 w ( t 0 u v w 0 = 0 0 u v w If we take u = (r v = (s w = (t then the operator ( t reduces to A (r s t which is determined in [0] Thus the results of this paper is generalized condition results of many operator whose matrix respresentation is a upper trianguler triple-band matrix In this work if z is a complex number than by z we always mean the square root of z with non-negative real part If Re ( z = 0 then z represents the square root of z with Im ( z 0 The same results are obtained if z represents the other square root Theorem 3 The operator ( t : l l is a bounded linear operator and t ( (l = sup ( w k + v k+ + u k+ l k Proof Proof is simple Hence we omit ( t Theorem 3 σ p l = t Proof Let f = α f for θ f l Then by solving system of linear equation u 0 f 0 = α f 0 v 0 f 0 + u f = α f w 0 f 0 + v f + u f = α f w f + v f + u 3 f 3 = α f 3 w k f k + v k f k + u k f k = α f k (5 Part Suppose (u k is a constant sequence say u k = U for each k N 0 We consider that (5 Let f m be the first non-zero entry of the sequence ( f n So we get fm = 0 which is a contradiction to our assumption For this reason ( t σ p l = Part Assume that (u k is a sequence of distinct positive real numbers Consider θ f l which gives (5 system of equations For all α {u 0 u u } we have f k = 0 for all k N 0 which is a contradiction Assume that α = u m for some m Then f 0 = f = = f m = 0 t f = α f for

6 S Altundağ M Abay / Filomat 30:5 ( w m f m + v m f m + (u m+ α f m+ = 0 If f m 0 then If f m = 0 then f k = 0 for all k N which is a contradiction and so f k+ = f k+ f k v k u k+ u m f k f or all k m = V u m U > f or all k m Because u m < V + U So f l Consequently ( t σ p l = ( t { Theorem 33 σ r l = α C : V+ Proof αi is one to one by Theorem 3 Suppose ( y = αy for θ y l This gives (U α V 4W(U α } < = S where t = u o y 0 + v o y + w 0 y = αy 0 u y + v y + w y 3 = αy (6 If y 0 = y = 0 then y k = 0 for all k N 0 So y 0 0 y 0 and solving the system of linear equations (6 in terms of y 0 and y we get y k = ( b k 0 y b k y 0 (u0 α(u α(u α (u k α w 0 w w k where b k 0 and b k are defined as in [4] Let y 0 = and y = r y k+ u y k = k α b k0 y b k y 0 w k b k 0 y b k y 0 = r < V+ V provided 4W(U α (U α = r > So if r > then y = ( y k l which shows that ( αi is not one to one Lemma 4 gives that αi has not dense range t { } (U α Theorem 34 σ p l = α C : < = S V+ Proof This proof is elementary by Corollary 6 t { Remark 35 σ p l = α C : V V 4W(U α (U α V 4W(U α } < = S

7 Proof This proof is made similarly to Theorem 34 Theorem 36 σ r t l = S Altundağ M Abay / Filomat 30:5 ( Proof t αi is one to one for all α by Theorem 3 Hence ( t αi is a dense range for all α by Lemma 4 Accordingly σ r t l = Theorem 37 Assume { V = V and define set S 3 by α C : S 3 V+ Proof Let α S 3 and y = ( y k l Then by solving the equation (u 0 α x 0 = y 0 v 0 x 0 + (u α x = y w 0 x 0 + v x + (u α x = y w k x k + v k+ x k+ + (u k+ α x k+ = y k+ and in this way we can get x 0 = y 0 u 0 α x = u α y v + 0 (u α(u 0 α y 0 (U α V 4W(U α } t = S 3 Then σ l = [ t ] αi x = y We obtain x = u α y v + (u α(u α y + ( v 0 v (u 0 α(u α(u α w 0 (u α(u 0 α y0 ( t α u k for all k N 0 and α U by α S 3 Thus αi = (b nk exist and u 0 α 0 0 b nk = v 0 (u 0 α(u α v 0 v (u 0 α(u α(u α w 0 (u 0 α(u α u α 0 u α v (u α(u α Let b kk = u k α b v k+k = k (u k α(u k+ α for all k N 0 We can see x n = b n0 y n + b n y n + + b nn y 0 = n b nn k y k We can observe u k α = U α = a x Clearly a n = k=0 v k (u k α(u k+ α = V (U α = a (r n (r n V 4W(U α for n = 3 where r = V+ V 4W(U α (U α We may suppose that V 4W(U α Since α S 3 r < and thus we have + 4W(U α < V (U α V and r = V V 4W(U α (U α

8 S Altundağ M Abay / Filomat 30:5 ( Since z + z for any z C we must have 4W(U α < V (U α V which leads us to the fact that r < and r < r We can see that x n n yk b nn k k=0 for all n N 0 Taking it on the inequality (7 as n we get x y b nn k k=0 We can see that b k+k a b kk = k+ x a k+ = (r k+ (r k+ = r x (r k+ (r k+ < ( t This shows that αi is onto for r < and (7 t αi has a bounded inverse by Lemma 4 If V = 4W(U α then a n = ( ( n V n V (U α for all n Similarly x n n yk b nn k for all n N0 and taking it on the inequality (7 as n we get k=0 x y b nn k and b k+k b kk k=0 This means that a = k+ x a k+ = V (U α ( t αi Thus α σ c t l In that case < is onto for r < and t αi has a bounded inverse by Lemma 4 t σ c l σ α C : (U α V + V 4W (U α = S 3 (8 By Theorem 34 we get σ p t l = α C : (U α ( V + V 4W (U α < ( σ t l (9 Since the spectrum of any bounded operator is closed we have α C : (U α ( V + V 4W (U α ( σ t l (0

9 and from Theorem and (38 t σ l α C : S Altundağ M Abay / Filomat 30:5 ( Combining (0 and ( this completes the proof Remark 38 If V t { = V then σ l = α C : t { Theorem 39 σ c l = α C : (U α V + V 4W (U α ( V+ (U α V 4W(U α V } = (U α V 4W(U α } Proof This is clear by Theorem Theorem 30 Assume that V = V If (U α < V V 4W (U α t then α A 3 σ l t Proof By Remark 35 αi dosen t exist Let y = (y 0 y l Solving the linear equation t αi x = y (u 0 α x 0 + v o x + w 0 x = y 0 (u α x + v x + w x 3 = y (u α x + v x 3 + w x 4 = y (u k α x k + v k x k+ + w k x k+ = y k Let x 0 = 0 and x = 0 So x = w 0 y 0 x 3 = w y + v w o w y 0 x 4 = w y + v w w y + ( v v w 0 w w (u α w 0 w y0 Letc kk+ = w k c kk+3 = v k+ w k w k+ c kk+4 = v k+v k+ w k w k+ w k+ (u k+ α w k w k+ Hence we say that x k = c 0k y 0 + c k y + + c k k y k = k k x k sup(r k k k y k where n=0 c nk y n Then R k = w k + v k+ w k w k+ + v k+ v k+ w k w k+ w k+ u k+ α w k+ w k + for all k N0 Now by letting r = V+ V 4W(U α (U α we can observe and r = V V 4W(U α (U α c kk+ = W = t = V 4W(U α ( ( r r

10 ( c kk+3 = V = t W = V 4W(U α ( c kk+4 = V = t W 3 3 = c kk+5 = V3 = t W 4 4 = where t n = V 4W(U α ( n ( Since r > we have r > Let V 4W(U α S Altundağ M Abay / Filomat 30:5 ( V 4W(U α ( 3 ( V 4W(U α ( 4 ( r n r n r r r 3 r 3 r 4 r 4 n = 3 c Since kk+ t c kk+ = k+ t k = r < then Rk is convergent for all k N That is R k = t n n= V 4W(U α ( n r + n r < n= (R k is a convergent sequence of positive real numbers and R k < hence sup R k < k This shows x = (x k l If V = 4W(U α then t n = W n( (U α n ( n V Consequently c kk+ t c kk+ = k+ t k = U α V < So R k is convergent for all k N and R k = t n = n W (U α n V < n= n= n= (R k is a convergent sequence of positive real numbers and R k < hence sup R k < This shows ( k ( t x = (x k l Thus αi t is onto So we have α A 3 σ l Theorem 3 Let V = V The following statements hold: i σ ap t l = S 3 ii σ co t l = Proof ( i Since from Table determined [0] ( t t t σ ap l = σ l /C σ l We have ( by Theorem 36 ( t t C σ l = C σ l So σ ap t l = S 3 = ii From Table we have

11 S Altundağ M Abay / Filomat 30:5 ( t t t t σ co l = C σ l C σ l C 3 σ l By Theorem 3 σ co t l = ( t { Theorem 3 σ c l = α C : V+ U α V 4W(U α Proof The proof is obvious so is ommitted Theorem 33 Let V = V If (U α < V + ( V 4W (U α t then α C σ l = } ( t Proof By Theorem 3 αi is exist By Theorem 33 and 3 proof is completed t { Theorem 34 σ δ l = α C : V+ (U α Proof By Proposition (c it can be seen readily V 4W(U α } = 4 Acknowledgement This work was supported by the Sakarya University Research Fund with Project Number This article was studied partly in International University of Sarajevo Faculty of Engineering and Natural Sciences Sarajevo BIH The authors also thank Prof Dr Fuat GURCAN (Faculty Dean International University of Sarajevo Faculty of Engineering and Natural Sciences Sarajevo BIH for his contributions to this study References [] B Altay F Başar On the fine spectrum of the difference operator on c 0 and c Inform Sci68 ( [] B Altay F BaşarOn the fine spectrum of the generalized difference operator B (r s over the sequence spaces c 0 and c Int J Math Math Sci 8 ( [3] JAppell E Pascale A Vingoli Nonlinear Spectral Theory de Gruyter Series in Nonlinear Analysis and Applications0 Walter de Gruyter Berlin Germany (004 [4] J Fathi R Lashkaripour On the fine spectrum of generalized upper double-band matrices uv over the sequence space l Mat Vesnik 65 ( ( [5] H Furkan H Bilgiç K Kayaduman On the fine spectrum of the generalized difference operator B(r s over the sequence spaces l and bv Hokkaido Math J35 ( [6] H Furkan H Bilgiç B Altay On the fine spectrum of the operator B (r s t over c 0 and c Comput Math Appl 53 ( [7] S Goldberg Unbounded Linear Operator Dover publications Inc New York (985 [8] M Gonzalez The fine spectrum of the Cesàro operator in l p ( < p < Arch Math44 ( [9] A Karaisa Fine spectrum of upper triangular double-band matrices over the sequence space l p ( < p < Discrete Dyn Nat Soc vol 0 Article ID (0 9 pages [0] A Karaisa Fine spectrum of upper triangular triple-band matrices over the sequence space l p ( 0 < p < Abstr Appl Anal vol 03 Article ID 348(03 0 pages [] V Karakaya M Altun Fine spectra of upper triangular double-band matrices J Comput Appl Math34 ( [] K Kayaduman H Furkan The fine spectra of difference operator over the sequence spaces l and bv Int Math Forum 4 ( [3] E Kreyszig Introductory Functional Analysis with Applications John Wiley and Sons Inc New York (978 [4] B L Panigrahi P D Sirvastava Spectrum and fine spectrum of generalized second order forward difference operator on sequence UVW space l Demonstratio Math 45 (3 ( [5] J B Reade On the spectrum of the Cesàro operator Bull Lond Math Soc 7 ( [6] PD Srivastava and S Kumar Fine spectrum of generalized difference operator v on sequence space l Thai J Math 8 ( (00 33 [7] PD Srivastava S Kumar Fine spectrum of generalized difference operator uv on sequence space l Appl Math Comput 8 ( [8] A Wilansky Summability through Functional Analysis North-Holland Mathematics Studies Amsterdam-New York-Oxford (984