On the solvability of the (SSIE) (s (c) R ) B(r;s;t) ⸦ s (c) x , involving the infinite triple band matrix B (r; s; t)

Abstract

In this article, we consider the infinite triple band matrix B(r,s,t), with r, s, t ≠ 0. Then, under the condition ∆ = s² −4rt, t, −s and r > 0, we state an interesting characterization of the set ℐ^(c) _R (r,s,t) of all positive sequences x = (x_n)_n∈N, such that (s^(c)_R) _B(r,s,t) ⊂ s ^(c) _x for R >0. Then, we obtain some numerical applications, and results associated with the fine spectrum theory. 
Finally, we consider the triple band matrix B (1,2s,as²) and we solve the (SSIE) (s^(c)_R) _B(1,2s,as²) ⊂ s ^(c) _x and we state some tauberian results, using the Silverman- Toeplitz theorem. These results extend those stated in [37, 8, 9].

Keywords: Matrix transformations; Silverman-Toeplitz theorem; Tauberian theorem; (SSIE); band matrix B(r,s,t).

2010 Mathematics Subject Classification. 40H05, 46A45