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

Given scalars $a_n ( eq 0)$ 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^nBig) 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 ${|{a_n}/{a_{n+1}}|}_{ngeq 0}$ is bounded away from zero. We find that, unlike the case of weighted shifts, Shimorins 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.