Fine spectra of upper triangular triple-band matrices over the sequence space $\ell_{p}$, $(0<p<\infty)$

The operator $A(r,s,t)$ on sequence space on $\ell_p$ is defined $A(r,s,t)x=(rx_{k-1}+sx_{k}+tx_{k+1})^{\infty}{k=0}$ where $x=(x_k)\in \ell{p}$, with $(0<p<1)$. The main purpose of this paper is to determine the fine spectrum with respect to the Goldberg's classification of the operator $A(r,s,t)$ defined by a triple sequential band matrix over the sequence space $\ell_p$. Additionally, we give the approximate point spectrum, defect spectrum and compression spectrum of the matrix operator $A(r,s,t)$ over the space $\ell_{p}$.