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

Abstract

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 k(z, w) = Σ_n=0^∞ ((a_n + b_nz) zⁿ) ((ā_n + b̄_nw) wⁿ), (z, w ∈ D). This defines a reproducing kernel Hilbert space H_k (known as tridiagonal space) of analytic functions on D with { (a_n + b_nz) zⁿ }_n=0^∞ 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 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, Shimorin models often fail to yield tridiagonal Aluthge transforms of shifts defined on tridiagonal spaces.