Double Band Matrix Research Articles

Orlicz Extension of New Sequence Spaces Engendered by the Composition of Binomial Matrix and Double Band Matrix

The present paper is emphasis on introducing Orlicz extension of new sequence spaces (i.e br,s0(M,G), br,sc(M,G) and br,s∞(M,G)) by way of the composition of binomial matrix and double band matrix, which are BK-spaces, moreover we prove that these spaces are linearly isomorphic to the spaces l∞, c0 and c. We also derive some inclusion relations. Additionally, we find the Schauder basis for these spaces and finally we also determine the α−, β− and γ− duals of these spaces.

On the spectrum of the upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence $c$

The upper triangular double band matrix $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ is defined on a Banach sequence space by $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})(x_{n})=(a_{n}x_{n}+b_{n}x_{n+1})_{n=0}^{\infty}$where $a_{x}=a_{y},~b_{x}=b_{y}$ for $x\equiv y~(mod3)$. The class of the operator$U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$includes, in particular, the operator $U(r,s)$ when $a_{k}=r$ and $b_{k}=s$ for all $k\in\mathbb{N}$, with $r,s\in\mathbb{R}$ and $s\neq 0$. Also, it includes the upper difference operator; $a_{k}=1$ and $b_{k}=-1$ for all $k\in\mathbb{N}$. In this paper, we completely determine the spectrum, the fine spectrum, the approximate point spectrum, the defect spectrum, and the compression spectrum of the operator $U(a_{0},a_{1},a_{2};b_{0},b_{1},b_{2})$ over the sequence space $c$.

Solvability of the Sequence Spaces Equations With Operators C<sub>a</sub>+C<sub>x</sub> = (C<sub>b</sub>)<sub>B</sub><sub>(r,s)</sub> And C<sub>a</sub>+C<sub>x</sub> = (C<sub>b</sub>)<sub>G</sub><sub>(α,α)</sub>

Given any sequence a = (an)n?1 of positive real numbers and any set E of complex sequences, we write Ea for the set of all sequences y = (yn)n?1 such that y/a = (yn/an)n?1 ? E; in particular, ca, (or s(c) a ) denotes the set of all sequences y such that y/a converges. In this paper, we solve sequence spaces equations of the form (Ex)B(r,s) = Ea, where E 2 {c0, c, `1}. Then we apply these results to the solvability of each of the (SSE) with operators ca + cx = (cb)B(r,s) and ca + cx = (cb)G(,), where B (r, s) is a double band matrix, and G(, ) is the factorable matrix with positive sequences and , that is, the triangle whose the nonzero entries are defined by [G(, )]nk = nk.

On topological properties of spaces obtained by the double band matrix

Abstract Let λ denote any one of the spaces ℓ∞ and ℓp and λ(Ť) be the domain of the band matrix Ť. We study ℓp(Ť) for 1 ≤ p ≤ ∞ and give some inclusions and its topological properties. Also, we define the alpha−, beta− and gamma− duals of the space ℓp(Ť). Finally, we give some matrix mappings.