$\Lambda ^2$-Fibonacci sequence spaces

Abstract

We introduce a new matrix $\hat{{\mathfrak {F}}}_{\Lambda ^2}$ obtained by the product of the double band Fibonacci matrix $\hat{{\mathfrak {F}}}$ and the $\Lambda ^2$ matrix. We construct new Banach sequence spaces $\ell _p^{\lambda ^2}(\hat{{\mathfrak {F}}})$ and $\ell _\infty ^{\lambda ^2}(\hat{{\mathfrak {F}}})$ using this new matrix, and study their topological and inclusion properties. We further obtain the basis of $\ell _p^{\lambda ^2}(\hat{{\mathfrak {F}}})$ and compute the $\alpha$-, $\beta$- and $\gamma$-duals of $\ell _p^{\lambda ^2}(\hat{{\mathfrak {F}}})$ and $\ell _\infty ^{\lambda ^2}(\hat{{\mathfrak {F}}}).$ Some characterization results related to the matrix classes $(\ell _p^{\lambda ^2}(\hat{{\mathfrak {F}}}),{\mathfrak {Y}})$ are stated and proved, where ${\mathfrak {Y}}$ is any of the space $\ell _{\infty },$ c,  $c_0$ or $\ell _1.$ The criteria for compactness of certain matrix operators from the space $\ell _p^{\lambda ^2}(\hat{{\mathfrak {F}}})$ to the space ${\mathfrak {Y}}=\{\ell _{\infty },c,c_0,\ell _1\}$ are obtained.