Spectra of the rhaly operator on the sequence space $\bar{bv}_0 \cap \ell_\infty$

Bol. Soc. Paran. Mat. c SPM –ISSN-2175-1188 on line SPM: www.spm.uem.br/bspm (3s.) v. 32 1 (2014): 265–277. ISSN-00378712 in press doi:10.5269/bspm.v32i1.19490 Spectra of the Rhaly Operator on the Sequence Space bv0 ∩ ℓ∞ Binod Chandra Tripathy and Rituparna Das abstract: In this article we have determined the spectra of the Rhaly operator on the class of bounded statistically null bounded variation sequence space. We have also determined the dual of the bounded statistically null bounded variation sequence space. Key Words: Spectra; Dual space; Rhaly operator; bounded variation. Contents 1 Preliminaries and background 265 2 The sequence space bv0 ∩ ℓ∞ 267 3 Some important results 269 4 Matrix operators on bv0 ∩ ℓ∞ 269 5 Dual space of bv0 ∩ ℓ∞ 272 6 Spectra of the Rhaly operator on the sequence space bv0 ∩ ℓ∞ 274 1. Preliminaries and background Let X and Y be Banach spaces and T : X → Y be a bounded linear operator. By R(T ), we denote the range of T , i.e. R(T ) = {y ∈ Y : y = T x, x ∈ X}. Throughout B(X) denotes the set of all bounded linear operators on X into itself. If T ∈ B(X), then the adjoint T ∗ of T is a bounded linear operator on the dual X ∗ of X defined by (T ∗ f )(x) = f (T x), for all f ∈ X ∗ and x ∈ X. Let X 6= {θ} be a complex normed space and T : D(T ) → X be a linear operator with domain D(T ) ⊆ X. With T , we associate the operator Tλ = T − λI, where λ is a complex number and I is the identity operator on D(T ). If Tλ has an inverse which is linear, we denote it by Tλ−1 , that is Tλ−1 = (T − λI)−1 , 2000 Mathematics Subject Classification: 40H05, 40C99, 46A35, 47A10. 265 Typeset by BSP style. M c Soc. Paran. de Mat. 266 Binod Chandra Tripathy and Rituparna Das and call it the resolvent operator of T . Let X 6= {θ} be a complex normed space and T : D(T ) → X be a linear operator with domain D(T ) ⊆ X. A regular value λ of T is a complex number such that (R1) Tλ−1 exists, (R2) Tλ−1 is bounded (R3) Tλ−1 is defined on a set which is dense in X i.e. R(Tλ ) = X. The resolvent set of T , denoted by ρ(T, X), is the set of all regular values λ of T . Its complement σ(T, X) = C − ρ(T, X) in the complex plane C is called the spectrum of T . Furthermore, the spectrum σ(T, X) is partitioned into three disjoint sets as follows: The point(discrete) spectrum σ p (T, X) is the set such that Tλ−1 does not exist. Any such λ ∈ σ p (T, X) is called an eigenvalue of T . The continuous spectrum σ c (T, X) is the set such that Tλ−1 exists and satisfies (R3), but not (R2), that is, Tλ−1 is unbounded. The residual spectrum σ r (T, X) is the set such that Tλ−1 exists (and may be bounded or not), but does not satisfy (R3), that is, the domain of Tλ−1 is not dense in X. By w, we denote the space of all real or complex valued sequences. Throughout the paper c, c0 , bv, c, c0 , bv, bs, ℓ1 , ℓ∞ represent the spaces of all convergent, null, bounded variation, statistically convergent, statistically null, statistically bounded variation, bounded series, absolutely summable and bounded sequences respectively. Also bv0 = bv ∩ c0 and bv0 = bv ∩ c0 . Let λ and µ be two sequence spaces and A = (ank ) be an infinite matrix of real or complex numbers ank , where n, k ∈ N0 = {0, 1, 2, ...}. Then, we say that A defines a matrix mapping from λ into µ, and we denote it by A : λ → µ , if for every sequence x = (xk ) ∈ λ, the sequence A = {(Ax)n }n∈N0 , the A-transform of x, is in µ, where (Ax)n = ∞ X ank xk , n ∈ N0 . (1) k=0 For simplicity in notation, throughout the summation without limits runs from 0 to ∞. By (λ, µ), we denote the class of all matrices such that A : λ → µ. Thus, A ∈ (λ, µ) if and only if the series on the right hand side of (1) converges for each Spectra of the Rhaly Operator on the Sequence Space bv0 ∩ ℓ∞ 267 n ∈ N0 and every x ∈ λ and we have Ax = {(Ax)n }n∈N0 ∈ µ for all x ∈ λ. Our main focus in this paper is on the Rhaly matrix A = Ra , where  a0 0 0  a1 a1 0  Ra =  a2 a2 a2 . . . where all the entries are real or complex. 0 0 0 .  ... ... , ... ... lim On taking L = n→∞ (n + 1)an , Rhaly [9] determined the spectrum of Ra on the Hilbert space ℓ2 of square summable sequences. Mustafa Yildirim [18] determined the spectrum of Ra on the sequence spaces c0 and c with the assumptions lim (a) L = n→∞ (n + 1)an exists, finite and nonzero, (b) an > 0 for all n, (c) ai 6= aj for i 6= j. Yildirim [20] under the same assumptions has determined the fine spectrum of Ra on the sequence space c0 . Yildirim [16] determined the spectrum of Ra on the sequence space bv0 with the assumptions lim (a) L = n→∞ (n + 1)an exists, finite and nonzero, (b) an > 0 for all n, (c) (an ) is a monotone decreasing sequence. The purpose of this paper is to determine the spectrum of Ra on the sequence space bv0 ∩ ℓ∞ under the same conditions used by Yildirim in [16]. Recently the spectra of some matrix operators have been investigated by Tripathy and Paul ( [12,13]), Tripathy and Saikia [14] and others. 2. The sequence space bv0 ∩ ℓ∞ A sequence (xn ) is said to be bounded variation sequence if (∆xn ) ∈ ℓ1 , where ∆xn = xn − xn+1 , for all n ∈ N0 . 268 Binod Chandra Tripathy and Rituparna Das lim 1 A subset E of N is said to have natural density δ(E) if δ(E) = n→∞ n n P χE (k) k=1 exists, where χE is the characteristic function of E. Clearly, δ(E) = 0 for all finite subset E of N and δ(E c ) = δ(N − E) = 1 − δ(E). A sequence (xn ) is said to be statistically convergent to L if for every ε > 0, δ({k ∈ N : |xk − L| ≥ ε}). We write xn stat → L or stat − lim xn = L. Alternatively, a sequence (xn ) is said to be statistically convergent to L if and only if there exists a subset K = {ki : i ∈ N } of N such that δ(K) = 1 and lim i→∞ xki = L. A sequence (xn ) is said to be a sequence of statistically bounded variation if (∆xni ) ∈ ℓ1 such that δ({ni : i ∈ N }) = 1, where ∆xni = xni − xni +1 for all i ∈ N and we denote (xn ) ∈ bv. Let us consider the sequence (xn ) defined by xn = ( n, if n = k 2 , k ∈ N, n−1 , otherwise. Clearly (xn ) ∈ bv. The above example shows that bv contains some unbounded sequence too. Now, let us consider the sequence (xn ) given by xn = ( 1, if n = i2 , i ∈ N, 0, otherwise. Clearly (xn ) is bounded. Now, let K1 = {n ∈ N : n = i2 , i ∈ N }. Then δ(K1 ) = 0. If K = N − K1 , then lim δ(K) = 1. If K = {ki : i ∈ N }, then i→∞ xki = 0. That is, xn stat → 0 and hence, P xn ∈ c0 . Also |xki − xki +1 |(= 0) < ∞. So (xn ) ∈ bv. Then (xn ) ∈ bv ∩ c0 ∩ ℓ∞ . i Let us denote bv0 = bv ∩ c0 . In this paper we will mainly deal with this type of sequence spaces. Clearly, P bv0 ∩ℓ∞ is a Banach space with respect to the norm ||x|| = ||(xn )|| = |xn −xn+1 |. n Spectra of the Rhaly Operator on the Sequence Space bv0 ∩ ℓ∞ 269 3. Some important results We procure the following results those will be used in establishing the results of this article. Lemma 3.1 (Tripathy [11], Theorem 5). x = (xn ) ∈ bv if and only if there exists sequences (un ) an (...truncated)