On the Fine Spectrum of the Operator Defined by the Lambda Matrix over the Spaces of Null and Convergent Sequences

Abstract

The main purpose of this paper is to determine the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces 0 and c. As a new development, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator Λ on the sequence spaces 0 and c. Finally, we present a Mercerian theorem. Since the matrix Λ is reduced to a regular matrix depending on the choice of the sequence ( ) having certain properties and its spectrum is firstly investigated, our work is new and the results are comprehensive.

Some Topological and Geometric Properties of Some New Spaces ofλ-Convergent and Bounded Series

“…Proof. Since (12) holds and , 0 , and are -spaces with the respect to their natural norms and the matrix Λ is a triangle, Theorem 4.3.12 of Wilansky [13, page 63] gives the fact that , 0 , and are -spaces with the given norms. This completes the proof.…”

“…The same authors have studied the spaces of -convergent sequences and almost convergence [11]. Also, the fine spectrum of the operator defined by lambda matrix over the spaces of null and convergent sequences has been studied by Yeşilkayagil and Başar [12].…”

On the fine spectrum of the generalized difference operator defined by a double sequential band matrix over the sequence space

“…Karaisa and Başar [13,14,27] have determined the fine spectrum of the upper triangular triple band matrix A(r, s, t) over some sequence spaces. Yeşilkayagil and Başar [40] have computed the fine spectrum with respect to Goldberg's classification of the operator defined by the lambda matrix over the sequence spaces c0 and c. Finally, Dündar and Başar [18] have studied the fine spectrum of the matrix operator ∆ + defined by an upper triangle double band matrix acting on the sequence space c0 with respect to the Goldberg's classification. At this stage, the following table may be useful: σ(A, λ) σp(A, λ) σc(A, λ) σr(A, λ) refer to: 1) , c) σc(∆ (1) , c) σr(∆ (1) , c)…”

On the spaces of $λ-$ convergent and bounded series

“…Same authors has been studied on spaces of λ− convergent sequences and almost convergence [11]. Also, on the fine spectrum of the operator defined by lambda matrix over the spaces of null and convergent sequences has been studied by Yeşilkayagil and Başar [12].…”