Fine Spectra of Upper Triangular Triple-Band Matrices over the Sequence Space ()

The fine spectra of lower triangular triple-band matrices have been examined by several authors (e.g., Akhmedov (2006), Başar (2007), and Furken et al. (2010)). Here we determine the fine spectra of upper triangular triple-band matrices over the sequence space . The operator on sequence space on is defined by , where , with . In this paper we have obtained the results on the spectrum and point spectrum for the operator on the sequence space . Further, the results on continuous spectrum, residual spectrum, and fine spectrum of the operator on the sequence space are also derived. Additionally, we give the approximate point spectrum, defect spectrum, and compression spectrum of the matrix operator over the space and we give some applications

On the Paranormed Nörlund Sequence Space of Nonabsolute Type

Maddox defined the space ℓ(p) of the sequences x=(xk) such that ∑k=0∞‍|xk|pk<∞, in Maddox, 1967. In the present paper, the Nörlund sequence space Nt(p) of nonabsolute type is introduced and proved that the spaces Nt(p) and ℓ(p) are linearly isomorphic. Besides this, the alpha-, beta-, and gamma-duals of the space Nt(p) are computed and the basis of the space Nt(p) is constructed. The classes (Nt(p):μ) and (μ:Nt(p)) of infinite matrices are characterized. Finally, some geometric properties of the space Nt(p) are investigated

The space Lq of double sequences

The spaces BS, BS(t), CSp, CSbp, CSr and BV of double sequences have recently been studied by Altay and Başar [J. Math. Anal. Appl. 309(1)(2005), 70–90]. In this work, following Altay and Başar [1], we introduce the Banach space Lq of double sequences corresponding to the well-known space ℓq of single sequences and examine some properties of the space Lq. Furthermore, we determine the β(υ)-dual of the space and establish that the α- and γ-duals of the space Lq coincide with the β(υ)-dual; where 1 ≤ q < ∞ and υ ∈ {p, bp, r}

On the fine spectrum of the upper triangle double band matrix Δ+ on the sequence space c0

In this study, we determine the fine spectrum of the matrix operator ∆+ defined by an upper triangle double band matrix acting on the sequence space c_0 with respect to the Goldberg’s classification. As a new development, we give the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator ∆+ on c_0

On the Domain of the Triangle on the Spaces of Null, Convergent, and Bounded Sequences

We introduce the spaces of -null, -convergent, and -bounded sequences. We examine some topological properties of the spaces and give some inclusion relations concerning these sequence spaces. Furthermore, we compute -, -, and -duals of these spaces. Finally, we characterize some classes of matrix transformations from the spaces of -bounded and -convergent sequences to the spaces of bounded, almost convergent, almost null, and convergent sequences and present a Steinhaus type theorem

On statistically convergent sequences of closed sets

In this paper, we give the definitions of statistical inner and outer limits for sequences of closed sets in metric spaces. We investigate some properties of statistical inner and outer limits. For sequences of closed sets if its statistical outer and statistical inner limits coincide, we say that the sequence is Kuratowski statistically convergent. We prove some proporties for Kuratowski statistically convergent sequences. Also, we examine the relationship between Kuratowski statistical convergence and Hausdorff statistical convergence

Certain Sequence Spaces over the Non-Newtonian Complex Field

It is known from functional analysis that in classical calculus, the sets , , , and of all bounded, convergent, null and -absolutely summable sequences are Banach spaces with their natural norms and they are complete according to the metric reduced from their norm, where . In this study, our main goal is to construct the spaces , , , and over the non-Newtonian complex field and to obtain the corresponding results for these spaces, where

Domain of the Double Sequential Band Matrix in the Sequence Space

The sequence space was introduced by Maddox (1967). Quite recently, the domain of the generalized difference matrix in the sequence space has been investigated by Kirişçi and Başar (2010). In the present paper, the sequence space of nonabsolute type has been studied which is the domain of the generalized difference matrix in the sequence space . Furthermore, the alpha-, beta-, and gamma-duals of the space have been determined, and the Schauder basis has been given. The classes of matrix transformations from the space to the spaces , c and c0 have been characterized. Additionally, the characterizations of some other matrix transformations from the space to the Euler, Riesz, difference, and so forth sequence spaces have been obtained by means of a given lemma. The last section of the paper has been devoted to conclusion