Solvability of new (SSIE) involving the continuous and residual spectra of the generalized difference operator B (r, s) on c0

Let U+ be the set of all positive sequences. Then, given any sequence z = (zn)n≥1 ∈ U+ and any set E of complex sequences, we write Ez for the set of all sequences y = (yn)n≥1 such that y/z = (yn/zn)n≥1 ∈ E. We use the notation sz = (ℓ∞)z. In this paper, for given r, s≠ 0 and for every λ ∈ ℂ, we determine the set of all positive sequences x = (xn)n≥1 that satisfy the (SSIE) with an operator (c0)B(r,s)−λI ⊂ Ɛ + sx, where Ɛ ⊂ sθ for some θ ∈ U+ is a linear space of sequences, in each of the cases, (1) |λ - r| > |s|, or λ = r, (2) |λ - r| = |s| and (3) |λ - r| < |s| and λ ≠ r. These cases are associated with the continuous and residual spectra σc (B (r, s), c0) and σr (B (r, s), c0), of B (r, s) on c0, determined by Altay and Başar in [2]. We apply these results to the solvability of the (SSIE) (c0)B(r,s)−λI ⊂ s(c)R +sx for all λ ∈ ℂ and R > 0. Then we deal with the (SSIE) (c0)Δ−λI ⊂ bvp + sx and (c0)B(r,s)−λI ⊂ ERɑ + sx, for E = c0, c, or ℓ∞, where Rɑ, ɑ ∈ U+, is the Rhaly matrix. These results extend those stated in [21].

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023).