Fine spectrum of the upper triangular matrix U(r, 0, 0,s) over the sequence spaces c 0 and c

Transcription

1 Proyeccione Journal of Mathematic Vol 37, N o, pp 85-0, March 208 Univeridad Católica del Norte Antofagata - Chile Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the equence pace c 0 and c Binod Chandra Tripathy Tripura Univerity, India and Rituparna Da Sikkim Manipal Intitute of Technology, India Received : March 207 Accepted : March 207 Abtract Fine pectra of variou matrice have been examined by everal author In thi article we have determined the fine pectrum of the upper triangular matrix U(r, 0, 0,) on the equence pace c 0 and c Key Word : Spectrum of an operator; matrix mapping; equence pace; upper triangular matrix; fine pectrum AMS Claification : 47A0; 47B37; 40C05; 40C5; 40D20; 40H05

2 86 Binod Chandra Tripathy and Rituparna Da Introduction The tudy of pectrum and fine pectrum for variou operator are made by variou author Okutoyi [5] determined the pectrum of the Ceàro operator C on the equence pace bv 0 The fine pectra of the Ceàro operator C over the equence pace bv p,( p< ) wadeterminedby Akhmedov and Başar [2] Altay and Başar [3, 4] determined the fine pectrum of the difference operator and the generalized difference operator B(r, ) on the equence pace c 0 and c The pectrum and fine pectrum of the Zweier Matrix on the equence pace and bv were tudied by Altay and Karakuş [5] Altun [6, 7] determined the fine pectra of triangular Toeplitz operator and tridiagonal ymmetric matrice over ome equence pace Fine pectra of operator B(r,, t) over the equence pace and bv and generalized difference operator B(r, ) over the equence pace p and bv p,( p< ) were tudied by Bilgiç and Furkan [9, 0] Akhmedov and El-Shabrawy [] determined the fine pectrum of the operator a,b on the equence pace c Panigrahi and Srivatava [6, 7] tudied the pectrum and fine pectrum of the econd order difference operator 2 uv on the equence pace c 0 and generalized econd order forward difference operator 2 uvw on the equence pace Fine pectrum of the generalized difference operator v on the equence pace wa invetigated by Srivatava and Kumar [9] Fine pectra of upper triangular double-band matrix U(r, ) over the equence pace c 0 and c were tudied by Karakaya and Altun [4] The pectra of ome matrix clae ha been invetigated recently by Rhoade [8], Tripathy and Da [20, 2, 22], Tripathy and Pal [23, 24, 25, 26, 27] and Tripathy and Saikia [28] In thi paper, we hall determine the pectrum and fine pectrum of the upper triangular matrix U(r, 0, 0,) over the equence pace c 0 and c, where U(r, 0, 0,)= r, if n = k u nk uch that u n k =, if n +3=k for all n, k N 0 and 6= 0 0, otherwie 2 Preliminarie and Background Let X and Y be Banach pace and T : X Y be a bounded linear operator By R(T ), we denote the range of T,ie R(T )={y Y : y = Tx,x X}

3 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 87 By B(X),we denote the et of all bounded linear operator on X into itelf If T B(X), then the adjoint T of T i a bounded linear operator on the dual X of X defined by (T f)(x) =f(tx), for all f X and x X Let X 6= {θ} be a complex normed linear pace, where θ i the zero element and T : D(T ) X be a linear operator with domain D(T ) X WithT, we aociate the operator T λ = T λi, where λ i a complex number and I i the identity operator on D(T ) If T λ ha an invere which i linear, we denote it by Tλ,thati Tλ =(T λi), and call it the reolvent operator of T A regular value λ of T i a complex number uch that (R): Tλ (R2): Tλ (R3): Tλ exit, i bounded i defined on a et which i dene in X ie R(T λ )=X The reolvent et of T, denoted by ρ(t,x), i the et of all regular value λ of T It complement σ(t,x)=c ρ(t,x) in the complex plane C i called the pectrum of T Furthermore, the pectrum σ(t,x) i partitioned into three dijoint et a follow: The point(dicrete) pectrum σ p (T,X) i the et uch that Tλ doe not exit Any uch λ σ p (T,X) i called an eigenvalue of T The continuou pectrum σ c (T,X) i the et uch that Tλ exit and atifie (R3), but not (R2), that i, Tλ i unbounded The reidual pectrum σ r (T,X) i the et uch that Tλ exit (and may be bounded or not), but doe not atify (R3), that i, the domain of Tλ i not dene in X From Goldberg [3], if X i a Banach pace and T B(X), then there are three poibilitie for R(T )andt : (I) R(T )=X, (II) R(T ) 6= R(T )=X (III) R(T ) 6= X and

4 88 Binod Chandra Tripathy and Rituparna Da () T exit and i continuou, (2) T exit but i dicontinuou, (3) T doe not exit Applying Goldberg [3] claification to T λ,wehavethreepoibilitie for T λ and Tλ ; (I) T λ i urjective, (II) R(T λ ) 6= R(T λ )=X, (III) R(T λ ) 6= X, and () T λ i injective and Tλ (2) T λ i injective but Tλ (3) T λ i not injective i continuou, i dicontinuou, If thee poibilitie are combined in all poible way, nine different tate are created which may be hown a in the Table 2

5 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 89 Thee are labeled by: I,I 2,I 3,II, II 2, II 3, III, III 2 and III 3 Ifλ i a complex number uch that T λ I or T λ I 2,then λ i in the reolvent et ρ(t,x)oft The further claification give rie to the fine pectrum of T If an operator i in tate II 2 for example, then R(T ) 6= R(T )=X and T exit but i dicontinuou and we write λ II 2 σ(t,x) By w, we denote the pace of all real or complex valued equence Any vector ubpace of w i called a equence pace Throughout the paper c, c 0,, repreent the pace of all convergent, null, abolutely ummable and bounded equence repectively Let λ and µ be two equence pace and A =(a nk )beaninfinite matrix of real or complex number a nk,wheren, k N 0 = {0,, 2,} Then, we ay that A define a matrix mapping from λ into µ, and we denote it by A : λ µ, if for every equence x =(x k ) λ, the equence Ax = {(Ax) n }, the A-tranform of x, i in µ, where (2) (Ax) n = X a nk x k,n N 0 k=0 By (λ : µ), we denote the cla of all matrice uch that A : λ µ Thu, A (λ : µ) if and only if the erie on the right hand ide of equation (2) converge for each n N 0 and every x λ and we have Ax = {(Ax) n } n N0 µ for all x λ The upper triangular matrix U(r, 0, 0,)ianinfinite matrix of the form U(r, 0, 0,)= r r r r 0 0 where 6= 0 The following reult will be ued in order to etablih the reult of thi article Lemma 2 [Wilanky [29] Theorem 36, Page 6] The matrix A = (a nk ) give rie to a bounded linear operator T B(c) from c to itelf if and only if: (i) the row of A are in and their norm are bounded,

6 90 Binod Chandra Tripathy and Rituparna Da (ii) the column of A are in c, (iii) the equence of row um of A i in c The operator norm T i the upremum of norm of the row Corollary 2 U(r, 0, 0,):c c i a bounded linear operator and U(r, 0, 0,) (c:c) = r + Lemma 22 [Wilanky [29] Example 845 A, Page 29] The matrix A =(a nk ) give rie to a bounded linear operator T B(c 0 ) from c 0 to itelf if and only if: (i) the row of A are in and their norm are bounded, (ii) the column of A are in c The operator norm T i the upremum of norm of the row Corollary 22 U(r, 0, 0,):c 0 c 0 i a bounded linear operator and U(r, 0, 0,) (c0 :c 0 ) = r + Lemma 23 [ Goldberg [3], Page 59] T ha a dene range if and only if T i one to one Lemma 24 [ Goldberg [3], Page 60] T ha a bounded invere if and only if T i onto 3 Fine pectrum of the operator U(r, 0, 0,) on the equence pace c 0 From now onward we denote the matrix U(r, 0, 0,)byU The following reult will be ued for etablihing ome reult of thi ection Lemma 3 [Akhmedov and El-Shabrawy [], Lemma 2] Let (c n ) and (d n ) be two equence of complex number uch that lim c n = c and n c < Define the equence (z n ) of complex number uch that z n+ = c n+ z n + d n+ for all n N 0 Then (i) if (d n ) i bounded, then (z n ) i bounded (ii) if (d n ) i convergent then (z n ) i convergent

7 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 9 (iii) if (d n ) i a a null equence, then (z n ) i a null equence In view of Lemma 3 we can formulate the following reult: Lemma 32 Let (c n ) and (d n ) be two equence of complex number uch that lim c n = c and c < Define the equence (z n ) of complex number n uch that z n+k = c n z n + d n for all n N 0 Then (i) if (d n ) i bounded, then (z n ) i bounded (ii) if (d n ) i convergent then (z n ) i convergent (iii) if (d n ) i a a null equence, then (z n ) i a null equence Theorem 3 The point pectrum of the operator U over c 0 i given by σ p (U, c 0 )={λ C : λ r < } Proof Let λ be an eigenvalue of the operator U Then there exit x 6= θ =(0, 0, 0, 0,)inc 0 uch that Ux = λx Then, we have rx 0 + x 3 = λx 0 rx + x 4 = λx rx 2 + x 5 = λx 2 rx n + x n+3 = λx n, n 0 Without lo of any generality we may aume that x 0 6= 0 Then, by uing the recurrence relation rx n + x n+3 = λx n, n 0, we have x 3 = λ r x 0 x 6 = λ r x 3 = x 9 = λ r x 6 = x 3n = λ r 2 x 0 λ r 3 x 0 λ r n x 0, n 0

8 92 Binod Chandra Tripathy and Rituparna Da Since, x =(x n ) c 0, o the ubequence (x 3n ) alo converge to 0 But (x 3n ) converge to 0 if and only if λ r < Hence, σ p (U, c 0 )={λ C : λ r < } 2 If T : c 0 c 0 i a bounded linear operator repreented by a matrix A, then it i known that the adjoint operator T : c 0 c 0 i defined by the tranpoe A t of the matrix A It hould be noted that the dual pace c 0 of c 0 i iometrically iomorphic to the Banach pace of all abolutely P ummable equence normed by x = x n Theorem 32 The point pectrum of the operator U over c 0 i given by n=0 σ p (U,c 0 = )= Proof Let λ be an eigenvalue of the operator U Then there exit x 6= θ =(0, 0, 0,)in uch that U x = λx Then, we have rx 0 = λx 0 rx = λx rx 2 = λx 2 x 0 + rx 3 = λx 3 x + rx 4 = λx 4 x 2 + rx 5 = λx 5 x n 3 + rx n = λx n, n 3 If x k i the firt non-zero entry of the equence (x n ),thenλ = r Then from the relation x k + rx k+3 = λx k+3,wehavex k =0 But 6= 0and hence, x k = 0, a contradiction Hence, σ p (U,c 0 = )= 2 Theorem 33 For any λ C, U λi ha a dene range Proof By Theorem 32, σ p (U,c 0 = )= Hence, U λi ie (U λi) i one to one for all λ C So, by applying Lemma 23, we get the required reult 2 Corollary 3 The reidual pectrum of the operator U over c 0 i given by σ r (U, c 0 )=

9 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 93 Proof 2 Since, U λi ha a dene range for all λ C, oσ r (U, c 0 )= Theorem 34 The continuou pectrum and the pectrum of the operator U over c 0 are repectively given by σ c (U, c 0 )={λ C : λ r = } and σ(u, c 0 )={λ C : λ r } Proof Let y =(y n ) be uch that Uλ x = y for ome x =(x n) Then we have following ytem of linear equation: (r λ)x 0 = y 0 (r λ)x = y (r λ)x 2 = y 2 x 0 +(r λ)x 3 = y 3 x +(r λ)x 4 = y 4 x n 3 +(r λ)x n = y n, n 3 Solving thee equation we get, x 0 = x = x 2 = x 3 = x 4 = x 5 = x 6 = x 7 = x 8 = r λ y 0 r λ y r λ y 2 r λ (y 3 x 0 )= r λ y 3 r λ (y 4 x )= r λ y 4 r λ (y 5 x 2 )= r λ y 5 r λ (y 6 x 3 )= r λ y 6 r λ (y 7 x 4 )= r λ y 7 r λ (y 8 x 5 )= r λ y 8 (r λ) 2 y 0 (r λ) 2 y (r λ) 2 y 2 (r λ) 2 y 3 + (r λ) 2 y 4 + (r λ) 2 y (r λ) 3 y 0 2 (r λ) 3 y 2 (r λ) 3 y 2

10 94 Binod Chandra Tripathy and Rituparna Da Now, X x n n=0 = r λ r λ X n=0 Ã y n + r λ 2 + r λ + 2 r λ 2 + X y n + 2 X r λ n=0 3 y n + n=0! y a y =(y n ) Let λ C be uch that r λ > Then x n r λ y n=0 Therefore, x =(x n ) and hence, for r λ > the operator Uλ = (U λi) i onto So, by Lemma 24, U λ = U λi ha a bounded invere for all λ C be uch that r λ > Therefore, λ/ {λ C : r λ } λ/ σ c (U, c 0 )ando, σ c (U, c 0 ) {λ C : r λ } Now, σ(u, c 0 )=σ p (U, c 0 ) σ r (U, c 0 ) σ c (U, c 0 ) {λ C : r λ } By Theorem 3, we have {λ C : r λ < } = σ p (U, c 0 ) σ(u, c 0 ) Since σ(u, c 0 )iacompactet,oiticloedandhence, {λ C : r λ < } σ(u, c 0 )=σ(u, c 0 ) and therefore,{λ C : r λ } σ(u, c 0 ) Hence, σ(u, c 0 )={λ C : λ r } Since σ(u, c 0 ) i dijoint union of σ p (U, c 0 ),σ r (U, c 0 )andσ c (U, c 0 ), therefore σ c (U, c 0 )={λ C : λ r = } 2 Theorem 35 If λ r <, thenλ I 3 σ(u, c 0 ) P

11 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 95 Proof Let λ r < Then by Theorem 3, λ σ p (U, c 0 )So λ atifie Goldberg condition 3 To get the reult we need to how that U λi i urjective when λ r < Let y =(y n ) c 0 be uch that U λ x = y for ome x =(x n ) Then (r λ)x 3n + x 3n+3 = y 3n ; n 0 (r λ)x 3n+ + x 3n+4 = y 3n+ ; n 0 (r λ)x 3n + x 3n+3 = y 3n+2 ; n 0 Now, (r λ)x 3n + x 3n+3 = y 3n (r λ) x 3n+3 = x 3n + y 3n Let x 3n = z n, x 3n+3 = x 3(n+) = z n+, c n = (r λ), d n+ = y 3n Then z n+ = c n+ z n + d n+ Now, lim c n = (r λ) n = c and c = r λ < Alo, a y =(y n ) c 0,o(d n+ ) c 0 Hence, by Lemma 3(iii), (z n )=(x 3n ) c 0 Similarly we can how that (x 3n+ ), (x 3n+2 ) c 0 Therefore, (x n ) c 0 if and only if λ r < Therefore,U λi i onto if λ r < So,λ atifie Goldberg condition I Hence the reult 2 4 Fine pectrum of the operator U(r, 0, 0,) on the equence pace c Theorem 4 The point pectrum of the operator U over c i given by σ p (U, c) ={λ C : λ r < } {r + }

12 96 Binod Chandra Tripathy and Rituparna Da Proof Let λ be an eigenvalue of the operator U Then there exit x 6= θ =(0, 0, 0, 0,)inc uch that Ux = λx Then, we have rx 0 + x 3 = λx 0 rx + x 4 = λx rx 2 + x 5 = λx 2 rx n + x n+3 = λx n, n 0 If x =(x n ) c i a contant equence, then λ = r + Thenλ = r + i an eigen value of the operator U If x =(x n ) c i not a contant equence, then proceeding exactly a Theorem 3 we get a ubequence of x =(x n ) which i alo convergent if and only if λ r < Hence, σ p (U, c) ={λ C : λ r < } {r + } 2 If T : c c i a bounded linear operator repreented by a matrix A, then à T : c! c acting on C ha a matrix repreentation of the form χ 0 b A t, where i χ the limit of the equence of row um of A minu the um of the limit of the column of A, andb i the column vector whoe k th entry i the limit of the k th column of A for each n N 0 For U = U(r, 0, 0,):c c, the matrix of the operator U = U(r, 0, 0,) B( )ioftheform U = U(r, 0, 0,) = à r U(r, 0, 0,) t! = r r r r r r Theorem 42 The point pectrum of the operator U over c i given by σ p (U,c = C )={r + }

13 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 97 Proof Let λ be an eigenvalue of the opertaor U Then there exit x 6= θ =(0, 0, 0, )in uch that U x = λx Then, we have (r + )x 0 = λx 0 rx = λx rx 2 = λx 2 rx 3 = λx 3 x + rx 4 = λx 4 x 2 + rx 5 = λx 5 x n 3 + rx n = λx n, n 4 If x 0 6= 0, then λ = r + Thu, λ = r + i an eigenvalue of U correponding to the eigenvector (x 0, 0, 0, )withx 0 6=0 Let λ 6= r + Then, x 0 =0 Let x k be the firt non-zero entry of the equence x =(x n ) Thenfrom the relation x k 3 + rx k = λx k,wegetλ = r Now from the relation x k + rx k+3 = λx k+3,wegetx k = 0 Since, 6= 0owehavex k =0, a contradiction So, U doe not have any other eigenvalue other than λ = r + Hence, σ p (U,c = C )={r + } 2 Theorem 43 U λ : c c ha a dene range if and only if λ 6= r + Proof By Theorem 42, σ p (U,c = C )={r + } Hence, U λi ie Uλ =(U λi) i one to one for all λ C \{r + } So, by applying Lemma 23, we get the reult 2 Corollary 4 The reidual pectrum of the operator U over c i given by σ r (U, c) = Proof Let λ σ r (U, c) Then, U λ doe not have a dene range By Theorem 43, we get λ = r + Therefore,σ r (U, c) ={r +} and hence, r + σ p (U, c) σ r (U, c), a contradiction Hence, the reult 2

14 98 Binod Chandra Tripathy and Rituparna Da Theorem 44 The continuou pectrum and the pectrum of the operator U over c are repectively given by σ c (U, c) ={λ C : λ r = }\{r +} and σ(u, c) ={λ C : λ r } Proof Proceeding exactly a in Theorem 34, we can how that σ(u, c) ={λ C : λ r } Since,σ(U, c) i dijoint union of σ p (U, c), σ r (U, c) andσ c (U, c), thefore uing Theorem 4 and Corollary 4, we get σ c (U, c) ={λ C : λ r = } \ {r + } 2 It i known from Cartlidge [] that, if a matrix operator A i bounded in c, thenσ(a, c) =σ(a, ) Therefore, we have the following reult: Corollary 42 The pectrum of the operator U over i given by σ(u, )={λ C : λ r } Theorem 45 If λ r <, thenλ I 3 σ(u, c) Proof The proof of the theorem i analogou to the proof of the Theorem 35 2 Reference [] A M Akhmedov and S R El-Shabrawy: On the fine pectrum of the operator a,b over the equence pace c, Comput Math Appl, 6(0), pp , (20) [2] A M Akhmedov and F Başar: The fine pectra of the Ceàro operator C over the equence pace bv p ( p< ), Math J Okayama Univ, 50, pp 35-47, (2008) [3] BAltayandFBaşar: On the fine pectrum of the difference operator on c 0 and c, Inf Sci, 68, pp , (2004) [4] B Altay and F Başar: On the fine pectrum of the generalized difference operator B(r, ) over the equence pace c 0 and c, Int J Math Math Sci, 2005:8, pp , (2005) [5] B Altay and M Karakuş: On the pectrum and the fine pectrum of the Zweier matrix a an operator on ome equence pace, Thai J Math, 3(2), pp 53-62, (2005)

15 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the 99 [6] M Altun: On the fine pectra of triangular Toeplitz operator, Appl Math Comput, 27, pp , (20) [7] M Altun: Fine pectra of tridiagonal ymmetric matrice, Int J Math Math Sci, 20, Article ID 6209, 0 page, (20) [8] F Başar: Summability Theory and It Application, Bentham Science Publiher,e-book, Monograph, Itanbul, 202, ISBN: [9] H Bilgiç and HFurkan: On the fine pectrum of operator B(r,, t) over the equence pace and bv, Math Comput Model, 45, pp , (2007) [0] H Bilgiç and HFurkan: On the fine pectrum of the generalized difference operator B(r, ) over the equence pace and bv p ( <p< ), Nonlinear Anal, 68, pp , (2008) [] J P Cartlitdge,Weighted mean matrice a operator on p,phddiertation,indiana Univerity,Indiana,(978) [2] H Furkan, H Bilgiç and B Altay: On the fine pectrum of operator B(r,, t) overc 0 and c, Comput Math Appl, 53, pp , (2007) [3] S Goldberg: Unbounded Linear Operator-Theory and Application, Dover Publication, Inc, New York, (985) [4] V Karakaya and M Altun: Fine pectra of upper triangular doubleband matrice, J Comp Appl Math, 234, pp , (200) [5] J I Okutoyi: On the pectrum of C a an operator on bv 0, J Autral MathSoc(SerieA)48, pp 79-86, (990) [6] B L Panigrahi and P D Srivatava: Spectrum and the fine pectrum of the generalied econd order difference operator 2 uv on equence pace c 0, Thai J Math, 9(), pp 57-74, (20) [7] B L Panigrahi and P D Srivatava: Spectrum and fine pectrum of the generalized econd order forward difference operator 2 uvw on the equence pace, Demontration Mathematica, XLV(3), pp , (202) [8] B E Rhoade: The fine pectra for weighted mean operator, Pac J Math, 04(), pp , (983)

16 00 Binod Chandra Tripathy and Rituparna Da [9] P D Srivatava and S Kumar: Fine pectrum of the generalized difference operator v on the equence pace, Thai J Math, 8(2), pp , (200) [20] B C Tripathy and R Da: Spectra of the Rhaly operator on the equence pace bv 0, Bol Soc Parana Mat, 32(), pp , (204) [2] B C Tripathy and R Da: Spectrum and fine pectrum of the lower triangular matrix B(r, 0, ) over the equence pace c, Appl Math Inf Sci, 9(4), pp , (205) [22] B C Tripathy and R Da: Spectrum and fine pectrum of the upper triangular matrix U(r, ) over the equence pace c, Proyeccione J Math, 34(2), pp 07-25, (205) [23] B C Tripathy and A Paul: Spectra of the operator B(f,g) onthe vector valued equence pace c 0 (X), Bol Soc Parana Mat, 3(), pp 05-, (203) [24] B C Tripathy and A Paul: Spectrum of the operator D(r, 0, 0,) over the equence pace p and bv p, Hacet J Math Stat, 43(3), pp , (204) [25] B C Tripathy and A Paul: The pectrum of the operator D(r, 0, 0,) over the equence pace bv 0, Georgian J Math, 22(3), pp , (205) [26] BC Tripathy and A Paul: The pectrum of the operator D(r, 0,,0,t) over the equence pace p and bv p, Afrika Mat, 26(5-6), pp 37-5, (205) [27] B CTripathy and A Paul: The pectrum of the operator D(r, 0, 0,) over the Sequence Space c 0 and c, Kyungpook Math J, 53(2),pp , (203) [28] B C Tripathy and P Saikia: On the pectrum of the Ceàro operator C on bv, Math Slovaca, 63(3), pp , (203) [29] A Wilanky: Summability Through Functional Analyi, NorthHolland, 984

17 Fine pectrum of the upper triangular matrix U(r, 0, 0,) over the0 [30] M Yeşilkayagil and F Başar: On the fine pectrum of the operator defined by the lambda matrix over the pace of null and convergent equence, Abtr Appl Anal, 203, Art ID , 3page, (203) Binod Chandra Tripathy Department of Mathematic Tripura Univerity Suryamaninagar, Agartala Tripura, India tripathybc@rediffmailcom and Rituparna Da Department of Mathematic Sikkim Manipal Intitute of Technology Sikkim , India rituparnada ghy@rediffmailcom