Agler-Commutant Lifting on an Annulus

McCullough, Scott; Sultanic, Saida

doi:10.1007/s00020-012-1948-x

Cited by 3 publications

(1 citation statement)

References 24 publications

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“…When r = q, this Hilbert function space is known as the Hardy space on A q denoted by H 2 (A q ) and the reproducing kernel S Aq (•, •) := S Aq (•, •; q) is called the Szegő kernel of A q [56,64]. The kernel (1.4) with a parameter r > 0 is considered as a weighted Szegő kernel of A q [57] and H 2 r (A q ) is the reproducing kernel Hilbert space (RKHS) [3] with respect to S Aq (•, •; r) [52,53]. We call r the weight parameter in this paper.…”

Section: Weighted Szegő Kernel and Gaf On An Annulusmentioning

confidence: 99%

Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Katori¹,

Shirai²

2020

Preprint

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On an annulus A q := {z ∈ C : q < |z| < 1} with a fixed q ∈ (0, 1), we study a Gaussian analytic function (GAF) and its zero set which defines a point process on A q called the zero-point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by r > 0 is given by the weighted Szegő kernel of A q with the weight parameter r studied by Mccullough and Shen. The GAF and the zero-point process have symmetry associated with the q-inversion of coordinate z ↔ q/z and the parameter change r ↔ q 2 /r, and when r = q they are invariant under conformal transformations which preserve A q . Conditioning the GAF by giving zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of Mccullough and Shen but the weight parameter r is changed depending on the given zeros. Then we prove that the zero-point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit q → 0, a simpler but still non-trivial PDPP is obtained on a punctured unit disk D × := D \ {0}. In the further limit r → 0 the present PDPP is reduced to the determinantal point process on D studied by Peres and Virág.

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Section: Weighted Szegő Kernel and Gaf On An Annulusmentioning

confidence: 99%

Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Katori¹,

Shirai²

2020

Preprint

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Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes

Katori

Shirai

2022

Commun. Math. Phys.

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On an annulus $${{\mathbb {A}}}_q :=\{z \in {{\mathbb {C}}}: q< |z| < 1\}$$ A q : = { z ∈ C : q < | z | < 1 } with a fixed $$q \in (0, 1)$$ q ∈ ( 0 , 1 ) , we study a Gaussian analytic function (GAF) and its zero set which defines a point process on $${{\mathbb {A}}}_q$$ A q called the zero point process of the GAF. The GAF is defined by the i.i.d. Gaussian Laurent series such that the covariance kernel parameterized by $$r >0$$ r > 0 is identified with the weighted Szegő kernel of $${{\mathbb {A}}}_q$$ A q with the weight parameter r studied by McCullough and Shen. The GAF and the zero point process are rotationally invariant and have a symmetry associated with the q-inversion of coordinate $$z \leftrightarrow q/z$$ z ↔ q / z and the parameter change $$r \leftrightarrow q^2/r$$ r ↔ q 2 / r . When $$r=q$$ r = q they are invariant under conformal transformations which preserve $${{\mathbb {A}}}_q$$ A q . Conditioning the GAF by adding zeros, new GAFs are induced such that the covariance kernels are also given by the weighted Szegő kernel of McCullough and Shen but the weight parameter r is changed depending on the added zeros. We also prove that the zero point process of the GAF provides a permanental-determinantal point process (PDPP) in which each correlation function is expressed by a permanent multiplied by a determinant. Dependence on r of the unfolded 2-correlation function of the PDPP is studied. If we take the limit $$q \rightarrow 0$$ q → 0 , a simpler but still non-trivial PDPP is obtained on the unit disk $${\mathbb {D}}$$ D . We observe that the limit PDPP indexed by $$r \in (0, \infty )$$ r ∈ ( 0 , ∞ ) can be regarded as an interpolation between the determinantal point process (DPP) on $${{\mathbb {D}}}$$ D studied by Peres and Virág ($$r \rightarrow 0$$ r → 0 ) and that DPP of Peres and Virág with a deterministic zero added at the origin ($$r \rightarrow \infty $$ r → ∞ ).

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Herglotz's representation and Carathéodory's approximation

Bhattacharyya,

Bhowmik,

Kumar

2024

Bulletin of London Math Soc

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Herglotz's representation of holomorphic functions with positive real part and Carathéodory's theorem on approximation by inner functions are two well‐known classical results in the theory of holomorphic functions on the unit disc. We show that they are equivalent. On a multi‐connected domain , a version of Heglotz's representation is known. Carathéodory's approximation was not known. We formulate and prove it and then show that it is equivalent to the known form of Herglotz's representation. Additionally, it also enables us to prove a new Heglotz's representation in the style of Korányi and Pukánszky. Of particular interest is the fact that the scaling technique of the disc is replaced by Carathéodory's approximation theorem while proving this new form of Herglotz's representation. Carathéodory's approximation theorem is also proved for operator‐valued functions on a multi‐connected domain.

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Agler-Commutant Lifting on an Annulus

Cited by 3 publications

References 24 publications

Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Zeros of the i.i.d. Gaussian Laurent series on an annulus: weighted Szegő kernels and permanental-determinantal point processes

Zeros of the i.i.d. Gaussian Laurent Series on an Annulus: Weighted Szegő Kernels and Permanental-Determinantal Point Processes

Herglotz's representation and Carathéodory's approximation

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