On the fine spectrum of the operator Δa, b over the sequence space c

The main aim of this paper is to complete and improve the former results for the spectrum and …ne spectrum of the generalized di¤erence operator a;b over the sequence space c which were proved by the authors in [A.M. Akhmedov, S.R. El-Shabrawy, On the …ne spectrum of the operator a;b over the sequence space c, Comput. Math. Appl. 61 (2011) 2994-3002]. The improved results cover a wider class of linear operators which are represented by in…nite lower triangular double-band matrices. Illustrative examples showing the advantage of the present results are also given. 1. Introduction and preliminaries Several authors have studied the spectrum and …ne spectrum of linear operators de…ned by lower and upper triangular matrices over some sequence spaces [1-22]. Throughout this paper, letX be a Banach space. ByR(T ); T ; X ; B(X); (T;X), p(T;X), r(T;X) and c(T;X), we denote the range of T , the adjoint operator of T , the space of all continuous linear functionals on X, the set of all bounded linear operators on X into itself, the spectrum of T on X, the point spectrum of T on X, the residual spectrum of T on X and the continuous spectrum of T on X, respectively. We shall write c and c0 for the spaces of all convergent and null sequences, respectively. Also by l1 we denote the space of all absolutely summable sequences. We assume here some familiarity with basic concepts of spectral theory and we refer to Kreyszig [23, pp. 370-372] for basic de…nitions such as resolvent operator, resolvent set, spectrum, point spectrum, residual spectrum and continuous spectrum of a linear operator. Also, we refer to Goldberg [24, pp. 58–71] for Goldberg’s classi…cation of spectrum. In [6], we have de…ned the operator a;b on the sequence space c as follows: a;bx = a;b(xk) = (akxk + bk 1xk 1) 1 k=0 with x 1 = b 1 = 0; (1) where (ak) and (bk) are convergent sequences of nonzero real numbers such that limk!1 ak = a, limk!1 bk = b 6= 0 and the following condition is satis…ed ja akj 6 = jbj , for all k 2 N: (2) 1991 Mathematics Subject Classi…cation. primary 47A10; secondary 47B37. Key words and phrases. Spectrum of an operator, In…nite matrices, Sequence space c. 1 2 ALI M. AKHMEDOV, SAAD R. EL-SHABRAWY JFCA-2012/3(S) It is easy to verify that the operator a;b can be represented by a lower triangular double-band matrix of the form a;b = BBB@ a0 0 0 b0 a1 0 0 b1 a2 .. .. .. . . . CCCA : In [6], the following results are obtained: Result 1: [6, Corollary 1.2] . The operator a;b : c ! c is a bounded linear operator with the norm k a;bkc = sup k (jakj+ jbk 1j). Result 2: [6, Theorem 2.2] . ( a;b; c) = D[E, whereD = f 2 C : j aj jbjg and E = fak : k 2 N; jak aj > jbjg. Result 3: [6, Theorem 2.3] . p ( a;b; c) = E; if there exists m 2 N : ai 6= aj 8i 6= j m; ?; otherwise. Result 4: [6, Theorem 2.6] . (i) f 2 C : j aj < jbjg [ fa+ bg r ( a;b; c) ; (ii) fak : k 2 Ng n p ( a;b; c) r ( a;b; c) ; (iii) 2 C : sup k ak bk < 1 r ( a;b; c) ; (iv) r ( a;b; c) 2 C : inf k ak bk < 1 [ fa+ bg ; (v) r ( a;b; c) ((D [ E)nG) [ fa+ bg ; where the set G is de…ned as 2 G if and only if there exists k0 2 N such that j akj = jbkj ; for all k k0: Result 5: [6, Theorem 2.8] . (i) c ( a;b; c) (f 2 C : j aj = jbjg [ E) n ( p ( a;b; c) [ fa+ bg) ; (ii) c ( a;b; c) (D [ E) \ 2 C : sup k ak bk 1 n ( p ( a;b; c) [ fa+ bg) ; (iii) Gn fa+ bg c ( a;b; c) ; (iv) 2 C : inf k ak bk 1 \ f 2 C : j aj jbjg n fa+ bg c ( a;b; c) : Result 6: [6, Theorem 2.12] . If 2 (f 2 C : j aj < jbjg n fak : k 2 Ng)[ fa+ bg ; then 2 III2 ( a;b; c). Result 7: [6, Theorem 2.13] . If there exists m 2 N such that ai 6= aj for all i; j m; then 2 E if and only if 2 III3 ( a;b; c). In this paper, we weaken the conditions on the sequences (ak) and (bk), assuming only that (ak) and (bk) are convergent sequences of real numbers, bk 6= 0 for all k 2 N; and that the limit of the sequence (bk) does not equal zero. We continue to get some new results even from these weaker conditions. Our new theorems give better results while conditions imposed are much weaker than in [6]. Moreover, JFCA-2012/3(S) NOTES ON THE FINE SPECTRUM OF THE OPERATOR a;b ... 3 some examples are given to show the ability and simplicity of applying the new results. 2. Main results and proofs Throughout this section, (ak) and (bk) are assumed to be two convergent sequences of real numbers with lim k!1 ak = a, lim k!1 bk = b 6= 0 and bk 6= 0 for all k 2 N: (3) Note that, the condition (2) is not necessarly satis…ed. However, the following two results are still valid. Theorem 1. The operator a;b : c ! c is a bounded linear operator with the norm k a;bkc = sup k (jakj+ jbk 1j) : Theorem 2. ( a;b; c) = D [ E, where D = f 2 C : j aj jbjg and E = fak : k 2 N; jak aj > jbjg. The following theorem characterizes the set p( a;b; c) completely. Theorem 3. p( a;b; c) = E [K; where K = ( aj : j 2 N; jaj aj = jbj ; k Y i=m bi 1 aj ai ! is convergent sequence for some m 2 N )