A Brief Introduction to Berezin–Toeplitz Operators on Compact Kähler Manifolds

Floch, Yohann Le

doi:10.1007/978-3-319-94682-5

Cited by 14 publications

(13 citation statements)

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“…Let us briefly recall the construction of this quantization (see [BMS94,LF18,Sch10] for preliminaries). Let X be a quantizable closed Kähler manifold with dim C X = d, and let (L, h) be a holomorphic Hermitian line bundle as above.…”

Section: Berezin Transform Vs Laplace-beltrami Operatormentioning

confidence: 99%

Spectral aspects of the Berezin transform

Ioos¹,

Kaminker²,

Polterovich³

et al. 2020

Annales Henri Lebesgue

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We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian inner products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kähler manifolds with constant scalar curvature, and yield exponential convergence of Donaldson's iterations to the fixed point. Furthermore, viewing POVMs as data clouds, we study their spectral features via geometry of measure metric spaces and the diffusion distance.

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Section: Berezin Transform Vs Laplace-beltrami Operatormentioning

confidence: 99%

Spectral aspects of the Berezin transform

Ioos¹,

Kaminker²,

Polterovich³

et al. 2020

Annales Henri Lebesgue

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“…eq. ( 43) in [9] and Definition 5.1.1, Example 7.1.8 and Theorem 7.2.1 in [4]) In this case, Theorem 2 tells us that…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…Here we focus on the Berezin-Toeplitz quantization procedure, introduced for the first time by Berezin in [1]. In fact, we restrict our attention to the Berezin-Toeplitz quantization of closed Kähler manifolds [1,2,3,4]. Such a quantization is defined by a sequence of positive surjective linear maps T : C ∞ (M ) → L (H ) with T (1) = Id.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Spectrum of the Berezin transform for Gelfand pairs

Shmoish¹

2021

Preprint

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We discuss the Berezin transform, a Markov operator associated to positive-operator valued measures (POVMs). We consider the class of so-called orbit POVMs, constructed on the quotient space Ω = G/K of a compact group G by its subgroup K. We restrict attention to the case where (G, K) is a Gelfand pair and derive an explicit formula for the spectrum of the Berezin transform in terms of the characters of the irreducible unitary representations of G. We then specialize our results to the case study G = SU(2) and K S 1 , and find the spectra of orbit POVMs on S 2 . We confirm previous calculations by Zhang and Donaldson of the spectrum of the standard quantization of S 2 coming from Kähler geometry. Then, we make a couple of conjectures about the oscillations in the sequence of eigenvalues, and prove them in the simplest case of second-highest weight vector. Finally, for low weights, we prove that the corresponding orbit POVMs on S 2 violate the axioms of a Berezin-Toeplitz quantization. Contents 1 Introduction and Main Results 2 Spectrum of the Berezin Transform of an Orbit POVM 2.1 The Berezin Transform as a Convolution Operator . . . . . . .

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“…More importantly, we make explicit the dependence in f and g of the remainders in the sense of (P1)-(P3). We refer the reader to lecture notes [LF16] by Y. Le Floch for a skillfully written exposition of our Theorem 1.1 in the Kähler case and useful preliminaries.…”

Section: | −mentioning

confidence: 99%