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

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.