Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms

TL;DR

This paper studies tridiagonal kernels on the unit disk, characterizes when shift operators are left-invertible, explores their models, and develops methods for Aluthge transforms, revealing structural preservation conditions.

Contribution

It introduces a new class of tridiagonal kernels, characterizes left-invertibility of associated shift operators, and analyzes their models and Aluthge transforms, highlighting structural invariance conditions.

Findings

Shift operator $M_z$ is left-invertible iff $ig|a_n/a_{n+1}ig|$ is bounded away from zero.

Shimorin's models preserve tridiagonality only under specific conditions ($b_0=0$ or $M_z$ is a weighted shift).

Develops computational methods for Aluthge transforms, noting limitations of Shimorin models in certain cases.

Abstract

Given scalars $a_n (\neq 0)$ and $b_n$, $n \geq 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $\mathbb{D}$ defined by \[ k(z, w) = \sum_{n=0}^\infty \Big((a_n + b_n z)z^n\Big) \Big((\bar{a}_n + \bar{b}_n \bar{w}) \bar{w}^n \Big) \qquad (z, w \in \mathbb{D}). \] This defines a reproducing kernel Hilbert space $\mathcal{H}_k$ (known as tridiagonal space) of analytic functions on $\mathbb{D}$ with $\{(a_n + b_nz) z^n\}_{n=0}^\infty$ as an orthonormal basis. We consider shift operators $M_z$ on $\mathcal{H}_k$ and prove that $M_z$ is left-invertible if and only if $\{|{a_n}/{a_{n+1}}|\}_{n\geq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin's models for left-invertible operators fail to bring to the foreground the tridiagonal structure of shifts. In fact, the tridiagonal structure of a kernel $k$, as above, is preserved under Shimorin model if and only if $b_0=0$ or that $M_z$ is a weighted shift. We prove concrete classification results concerning invariance of tridiagonality of kernels, Shimorin models, and positive operators. We also develop a computational approach to Aluthge transforms of shifts. Curiously, in contrast to direct kernel space techniques, often Shimorin models fails to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.