  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.
