On <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si16.svg" display="inline" id="d1e25"> <mml:mi>q</mml:mi> </mml:math> -Pell sequence spaces: A study of operator ideals and geometric properties

The matrix \( \mathcal{P}(q) = \{\Psi_{\lambda \mu}(q)\}_{\lambda, \mu \in \mathbb{N}} \), called the \( q \)-Pell matrix, with elements determined by \[ \mathcal{P}(q) = \begin{cases} \dfrac{2~q^{\mu-1}~\Psi{_\mu(q)}}{\Psi{_{\lambda+1}(q)}+\Psi{_{\lambda}(q)}-1} , & \text{if } 1 \leq \mu \leq \lambda, \\ 0, & \text{if } \mu > \lambda. \end{cases} \] with \( \{\Psi_{\lambda}{(q)}\} \) representing the \( q \)-Pell sequence. Using \( \mathcal{P}(q) \) is defined by \[\Psi{_{\lambda}(q)}=(1+q^{\lambda+1})\Psi{_{\lambda-1}(q)}+\Psi{_{\lambda-2}(q)}\] for \(\lambda \geq 2\),~\(\Psi{_0(q)}=0,\Psi{_1(q)}=1\). In this context, the Schauder basis is derived for the space \( \ell_p(\mathcal{P}(q)) \) for \( 1 \leq p \leq \infty \). The key geometric properties of \( \ell_p(\mathcal{P}(q)) \) and \( \ell_\infty(\mathcal{P}(q)) \), such as the approximation property, the Dunford–Pettis property, the Hahn-Banach extension property, and rotundity, are examined. Finally, the solidity properties of these sequence spaces are studied.