Spectral aspects of the Berezin transform

Ioos, Louis; Kaminker, Victoria; Polterovich, Leonid; Shmoish, Dor

doi:10.5802/ahl.63

Cited by 10 publications

(32 citation statements)

References 38 publications

(34 reference statements)

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“…where λ E 1 > 0 is the first positive eigenvalue of the Laplacian (1.14) associated with the unique metric h E ∈ Met(E) satisfying the weak Hermite-Einstein equation (1.24). This shows in particular the exponential convergence of the iterations of the dynamical system T Ep , extending the results of [21,Th. 4.4,Rem.…”

Section: Introductionsupporting

confidence: 83%

“…In Section 3, we extend to the setting of quantization in stages our study in [21] of Berezin-Toeplitz quantization as a theory of quantum measurements. In particular, we introduce and study in Section 3.1 a natural notion of a Berezin transform B Ep associated with the Berezin-Toeplitz quantization of a holomorphic Hermitian vector bundle (E, h E ) over (X, p ω).…”

Section: Introductionmentioning

confidence: 99%

“…with lowest eigenvalue λ 0 = 0 associated with the constant function. The main result of [21,Th. 3.1] then shows that for every integer k ∈ N, the following asymptotic estimate as p → +∞ holds,…”

Section: Introductionmentioning

confidence: 99%

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Quantization of symplectic fibrations and canonical metrics

Ioos¹,

Polterovich²

2021

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We relate Berezin-Toeplitz quantization of higher rank vector bundles to quantum-classical hybrid systems and quantization in stages of symplectic fibrations. We apply this picture to the analysis and geometry of vector bundles, including the spectral gap of the Berezin transform and the convergence rate of Donaldson's iterations towards balanced metrics on stable vector bundles. We also establish refined estimates in the scalar case to compute the rate of Donaldson's iterations towards balanced metrics on Kähler manifolds with constant scalar curvature.

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Section: Introductionsupporting

confidence: 83%

Section: Introductionmentioning

confidence: 99%

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Quantization of symplectic fibrations and canonical metrics

Ioos¹,

Polterovich²

2021

Preprint

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“…In the present paper, we study the spectral properties of the Berezin transform of a certain class of so-called orbit POVMs, whose construction was briefly described in the Preliminaries and Remark 6.7 of [8]. It is a representation-theoretic construction following ideas first introduced by Perelomov in [9, p. 223].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The Spectrum of the Berezin transform for Gelfand pairs

Shmoish¹

2021

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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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“…It has since found applications in many areas of mathematics and mathematical physics. See [6,38,44,46,47,53,65,71,77,84,85] for some expository work about the Berezin transform in mathematical analysis and [39,40,48,50,57,68,70,85] for some applications of the Berezin transform in quantization and mathematical physics. The purpose of this article is to give a brief survey of the mathematical theory of the Berezin transform, especially regarding its applications to operator theory.…”

Section: Introductionmentioning

confidence: 99%