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

Article Type

Research Article

Publication Title

Revista Matematica Iberoamericana

Abstract

Given scalars an(≠0) and bn, 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 {equation presented} This defines a reproducing kernel Hilbert space Hk (known as tridiagonal space) of analytic functions on D with {(an + bnz)zn}∞n=0 as an orthonormal basis. We consider shift operators Mz on Hk and prove that Mz is left-invertible if and only if {|an/an+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 Mz 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.

First Page

397

Last Page

437

DOI

https://10.4171/RMI/1403

Publication Date

1-1-2023

Comments

Open Access, Gold, Green

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