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 $|\frac{ a_n}{ a_{ n+1}}| \rightarrow 1$.
Categories: math.FA math.CV math.OA
