  Let $\{a_n\}_{n\geq 0}$ and $\{b_n\}_{n\geq 0}$ be sequences of scalars.
Suppose $a_n \neq 0$ for all $n \geq 0$. We consider the tridiagonal kernel
(also known as band kernel with bandwidth one) as \[ k(z, w) =
\sum_{n=0}^\infty ((a_n + b_n z)z^n) \overline{(({a}_n + {b}_n {w}) {w}^n)}
\qquad (z, w \in \mathbb{D}), \] where $\mathbb{D} = \{z \in \mathbb{C}: |z| <
1\}$. Denote by $M_z$ the multiplication operator on the reproducing kernel
Hilbert space corresponding to the kernel $k$. Assume that $M_z$ is
left-invertible. We prove that $M_z =$ compact $+$ isometry if and only if
$|\frac{b_n}{a_n}-\frac{b_{n+1}}{a_{n+1}}|\rightarrow 0$ and
$|\frac{a_n}{a_{n+1}}| \rightarrow 1$.
