Erratum to “Some Geometric Properties of the Domain of the Double Sequential Band Matrix 𝐵(𝑟,̃𝑠)̃ in the Sequence Space ℓ(𝑝)”

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The purpose of this short note is to rectify the misprints in Part (iii) of Proposition 5 and its proof in the recent paper by Nergiz and Başar [1]. The reader may refer for relevant terminology to Nergiz and Başar [1].

Now, we give the corrected Part (iii) of Proposition 5 and its proof.

Proposition 5. The modular 𝜎_𝑝 on ℓ(𝐵,𝑝) satisfies the following properties with 𝑝_𝑘≥1 for all 𝑘∈ℕ:  (iii) if 𝛼≥1, then 𝜎_𝑝(𝑥)≥𝛼𝜎_𝑝(𝑥/𝛼).

Proof. Consider the modular 𝜎_𝑝 on ℓ(𝐵,𝑝).  (iii) Let 𝛼≥1. Then, 𝛼/𝛼^𝑝𝑘≤1 for all 𝑝_𝑘≥1. So, we have 𝜎_𝑝(𝑥)=_𝑘𝑠_𝑘−1𝑥_𝑘−1+𝑟_𝑘𝑥_𝑘^𝑝𝑘≥_𝑘𝛼𝛼^𝑝𝑘𝑠_𝑘−1𝑥_𝑘−1+𝑟_𝑘𝑥_𝑘^𝑝𝑘=𝛼𝜎_𝑝𝑥𝛼, (10)which completes the proof of Part (iii).

We also record that the relation (19) must be corrected as follows: 1𝑘‖≤1‖𝑘𝑥𝑘𝑘𝑥=‖𝑥‖,‖𝑘𝑥‖≤|𝑘|‖𝑥‖. (19)