Compact matrix operators on the sequence spaces involving Riordan numbers

Recently, the sequence spaces $\ell _{p}(\mathcal {R})=(\ell _{p})_{\mathcal {R}}$ and $\ell _{\infty}(\mathcal {R})=(\ell _{\infty})_{\mathcal {R}}$ have been defined in (Braha and Mansour in J. Math. Anal. Appl. 543:128902, 2025) by employing the matrix $\mathcal {R}$, where $\mathcal {R}=(R_{jk})$ is a triangle defined by $$\begin{aligned} R_{jk}=\left \{ \textstyle\begin{array}{c@{\quad}c@{\quad}l} \dfrac{(R_{k}+R_{k+1})(R_{j-k}+R_{j-k+1})}{R_{j+3}-R_{j+1}}& , & (0 \leq k\leq j), \ 0& , &(j< k), \end{array}\displaystyle \right . \end{aligned}$$ for all $j,k\in \mathbb{N}_{0}$. Here, $R_{k}\in \mathbb{R}=\text{the set of real numbers}$, denotes the $k^{\textrm{th}}$ Riordan number $(k\in \mathbb{N}_{0}=\{0,1,2,\ldots \})$ and in this paper, we wish to characterize certain matrix classes $(\mathfrak {U}(\mathcal {R}), \mathfrak {V})$, where $\mathfrak {U}=\{\ell _{p},\ell _{\infty}\}$ and $\mathfrak {V}\in \{\ell _{\infty},c,c_{0},\ell _{1}\}$. Moreover, certain conditions for compactness of matrix operators on the spaces $\ell _{p}(\mathcal {R})$ and $\ell _{\infty}(\mathcal {R})$ are determined.