Tridiagonal shifts as compact + isometry

TL;DR

This paper characterizes when the multiplication operator on a specific tridiagonal kernel Hilbert space decomposes into a compact operator plus an isometry, based on the asymptotic behavior of the kernel coefficients.

Contribution

It provides a precise characterization of the structure of the multiplication operator as compact plus isometry in terms of the kernel's coefficient sequences.

Findings

The operator is compact plus isometry if and only if the coefficient ratios satisfy certain asymptotic conditions.

The conditions involve the limits of ratios of the sequences $b_n/a_n$ and $a_n/a_{n+1}$.

The result links the spectral properties of the operator to the asymptotic behavior of the kernel coefficients.

Abstract

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 $|\frac{a_n}{a_{n+1}}| \rightarrow 1$.