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

Given scalars a_n(≠ 0) and b_n, n≤0, the tridiagonal kernel or band kernel with bandwidth 1 is the positive definite kernel k on the open unit disc D defined by ... This defines a reproducing kernel Hilbert space H_k (known as tridiagonal space) of analytic functions on D with ... as an orthonormal basis. We consider shift operators M_z on H_k and prove that M_z is left-invertible if and only if {/a_n/a_n+1/}_n≤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 b₀ = 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 tridi- agonal Aluthge transforms of shifts defined on tridiagonal spaces. [ABSTRACT FROM AUTHOR]