On some study of the Fine Spectra of $n$-th band triangular matrices

It has been observed that for the 2nd and 3rd band lower triangular matrices $B(r,s)$ and $B(r,s,t)$, only the boundary of the spectrum gives the continuous spectrum while the rest of the entire interior region gives the residual spectrum over the sequence spaces $c_0$, $l_p$ and $bv_p$. The main focus of our present study is to investigate the possibilities of the occurrence of the similar kinds of behavior for the cases of $n (\ge4)$ band lower triangular matrices over the sequence spaces $c_0$, $l_p$ and $bv_p$. The outcomes depicts that not only the boundary part but a finite set from the interior region of the spectrum is included in the continuous spectrum while the same set is excluded from the residual spectrum. In this context, we have proved an interesting result regarding the image of the closed unit disk $|z|\le 1$ under a polynomial of degree $n\ge 1$ which plays the key role in our study. Similar studies has also been done for the sequence spaces $c$, $l_1$, $bv$ and $l_\infty$. Upper triangular matrices has also been investigated for some sequence spaces.