Reverse Cholesky factorization and tensor products of nest algebras

Paulsen, Vern I.; Woerdeman, Hugo J.

doi:10.1090/proc/13851

Cited by 3 publications

(1 citation statement)

References 12 publications

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“…The notion of tridiagonal shifts was introduced by Adams and McGuire [1]. A part of their motivation came from factorizations of positive operators on analytic Hilbert spaces [2] (also see [8]). Evidently, if b n = 0, then k is a diagonal kernel and M z is a weighted shift on H k .…”

Section: Is It Possible To Find a Quantitative Classification Of Left...mentioning

confidence: 99%

Tridiagonal shifts as compact + isometry

Das¹,

Sarkar²

2021

Preprint

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Let {a n } n≥0 and {b n } n≥0 be sequences of scalars. Suppose a n = 0 for all n ≥ 0. We consider the tridiagonal kernel (also known as band kernel with bandwidth one) aswhere D = {z ∈ C : |z| < 1}. Denote by M z the multiplication operator on the reproducing kernel Hilbert space corresponding to the kernel k. Assume that M z is left-invertible. We prove that M z = compact + isometry if and only if { bn an } n≥0 is convergent and | an an+1 | → 1.

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Section: Is It Possible To Find a Quantitative Classification Of Left...mentioning

confidence: 99%

Tridiagonal shifts as compact + isometry

Das¹,

Sarkar²

2021

Preprint

View full text Add to dashboard Cite

show abstract

Tridiagonal shifts as compact + isometry

Das

Sarkar

2022

Arch. Math.

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

Das

Sarkar

2023

Rev. Mat. Iberoam.

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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.

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Reverse Cholesky factorization and tensor products of nest algebras

Cited by 3 publications

References 12 publications

Tridiagonal shifts as compact + isometry

Tridiagonal shifts as compact + isometry

Tridiagonal shifts as compact + isometry

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

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