Tridiagonal kernels and left-invertible operators with applications to Aluthge transforms
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Susmita Das [1] ; Jaydeb Sarkar [1]
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[1] Indian Statistical Institute
Indian Statistical Institute
India
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 39, Nº 2, 2023, págs. 397-437
- Idioma: inglés
- DOI: 10.4171/RMI/1403
- Enlaces
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Resumen
Given scalars (≠0)an(=0) and bn, ≥0n≥0, the tridiagonal kernel or band kernel with bandwidth 11 is the positive definite kernel k on the open unit disc D defined by k(z,w)=n=0∑∞((an+bnz)zn)((aˉn+bˉnwˉ)wˉn)(z,w∈D).
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 0=0b0=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.
