Certain domains of a new matrix constructed by Euler totient and its summation function

Merve İlkhan Kara; Dilek Aydın; Merve İlkhan Kara; Dilek Aydın

doi:10.3934/math.2025329

AIMS Mathematics

2025, Volume 10, Issue 3: 7206-7222. doi: 10.3934/math.2025329

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Research article

Certain domains of a new matrix constructed by Euler totient and its summation function

Merve İlkhan Kara ^,,
Dilek Aydın

Department of Mathematics, Faculty of Science and Arts, Düzce University, Düzce, Türkiye

Received: 30 December 2024 Revised: 12 March 2025 Accepted: 20 March 2025 Published: 28 March 2025
MSC : 40C05, 46A45, 47B07, 47B37

With the aid of the Euler totient function $ \varphi $ and its summation function $ \top $, a new matrix $ \Delta(\varphi, \top) = (\delta(\varphi, \top)_{nk}) $, where
$ \delta(\varphi, \top)_{nk} = \left\{ \begin{array} [c]{ccl} \frac{(-1)^{n-k} \top(k)}{\varphi(n)}, & {n-1\leq k\leq n}, \\ 0, & \text{otherwise} \end{array} \right. $
is constructed to define the domains $ \ell_p(\Delta(\varphi, \top)) $, $ \ell_\infty(\Delta(\varphi, \top)) $, and $ \ell_1(\Delta(\varphi, \top)) $. After obtaining the norms on these domains, it is proved that these spaces are linearly isomorphic to classical ones. Also, their dual spaces are determined. Finally, characterizations of several matrix mappings are stated and proved.
- arithmetic divisor sum function,
- matrix domain,
- dual space,
- matrix mapping
Citation: Merve İlkhan Kara, Dilek Aydın. Certain domains of a new matrix constructed by Euler totient and its summation function[J]. AIMS Mathematics, 2025, 10(3): 7206-7222. doi: 10.3934/math.2025329

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Abstract

With the aid of the Euler totient function $ \varphi $ and its summation function $ \top $, a new matrix $ \Delta(\varphi, \top) = (\delta(\varphi, \top)_{nk}) $, where

$ \delta(\varphi, \top)_{nk} = \left\{ \begin{array} [c]{ccl} \frac{(-1)^{n-k} \top(k)}{\varphi(n)}, & {n-1\leq k\leq n}, \\ 0, & \text{otherwise} \end{array} \right. $

is constructed to define the domains $ \ell_p(\Delta(\varphi, \top)) $, $ \ell_\infty(\Delta(\varphi, \top)) $, and $ \ell_1(\Delta(\varphi, \top)) $. After obtaining the norms on these domains, it is proved that these spaces are linearly isomorphic to classical ones. Also, their dual spaces are determined. Finally, characterizations of several matrix mappings are stated and proved.

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