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

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 $$ k(z, w) = \sum\_{n=0}^\infty \big((a\_n + b\_n z) z^n\big) \big((\bar{a}\_n + \bar{b}\_n \bar{w}) \bar{w}^n \big) \quad (z, w \in \mathbb{D}). $$ 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 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.