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

Abstract

Given scalars $a_n(\ne 0)$ and $b_n$, $n\ge 0$, the tridiagonal kernel or band kernel with bandwidth $1$ is the positive definite kernel $k$ on the open unit disc $\mathbb{D}$ defined by

This defines a reproducing kernel Hilbert space ${\mathcal{H}}_k$ (known as tridiagonal space) of analytic functions on $\mathbb{D}$ with $\{(a_n+b_{n}z)z^n{\}}_{n=0}^{\infty }$ 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 $\{|a_n/a_{n+1}|{\}}_{n\ge 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorin 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 models 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 fail to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.