The Berezin transform and invariant differential operators

Unterberger, André; Upmeier, Harald

doi:10.1007/bf02101491

Cited by 99 publications

(93 citation statements)

References 22 publications

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“…For the particular case of A = T , the Toeplitz calculus, the link transform T * T is known as the Berezin transform B R,ν , and the function T * T =: b ν R has been computed in [24]:…”

Section: Bounded Symmetric Domains Of Complex and Real Typementioning

confidence: 99%

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Engliš

Upmeier

2010

Math. Z.

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Abstract. An analogue of the star product, familiar from deformation quantization, is studied in the setting of real bounded symmetric domains. The analogue turns out to be a certain invariant operator, which one might call star restriction, from functions on the complexification of the domain into functions on the domain itself. In particular, we establish the usual (i.e. semiclassical) asymptotic expansion of this star restriction, and further obtain a real-variable analogue of a theorem of Arazy and Ørsted concerning the analogous expansion for the Berezin transform. Covariant Calculi on Complex and Real Symmetric DomainsLet D = G/K be an irreducible bounded symmetric domain in C d in the HarishChandra realization, with G the identity connected component of the group of all biholomorphic self-maps of D and K the stabilizer of the origin; K can also be realized as the automorphism group Aut(Z) of the Hermitian Jordan triple Z ≈ C where dz stands for the Lebesgue measure, K(z, w) is the ordinary (unweighted) Bergman kernel of D, and c ν is a normalizing constant to make dµ ν a probability measure. The space, where J g denotes the complex Jacobian of the mapping g. (In general, if ν/p is not an integer, then U (ν) is only a projective representation due to the ambiguity in the choice of the power J g −1 (z) ν/p .) This situation will henceforth be called the complex bounded case.In addition to bounded symmetric domains, we will also consider the complex flat case of a Hermitian vector space In most cases, such calculi can be built by the recipewhere dµ 0 is a G-invariant measure on D, and A ζ is a family of operators in(One calls such a family a covariant operator field on D. One also usually normalizes the measure dµ 0 so that A 1 is the identity operator.) Note that in view of the transitivity of the action of G on D, any covariant operator field is uniquely determined by its value A 0 at the origin ζ = 0. The best known examples of such calculi are the Toeplitz calculus T and the Weyl calculus W, corresponding to T 0 = ·, 1 1 (the projection onto the constants) and W 0 f (z) = f (−z) (the reflection operator), respectively [2]. For the complex flat case, W is just the well-known Weyl calculus from the theory of pseudodifferential operators, see e.g. [12].Given a covariant operator calculus A, the associated star product * on functions on D is defined byWhile f * g is a well-defined object for some calculi (e.g. for A = W, at least on C d and rank one symmetric domains, see [4]), in most cases (e.g. for A = T , the Toeplitz calculus), it makes sense only for very special functions f, g and (7) is then usually understood as an equality of asymptotic expansions as the Wallach parameter ν tends to infinity. For instance, for A = T , it was shown in [6] that for any f, g ∈ C ∞ (D) with compact support,as ν → ∞, for some bilinear differential operators C j (not depending on f, g and ν).(The assumption of compact support can be relaxed, cf.[9].) We can thus define f * g as the formal power seriesInterpreting ν as th...

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“…For the particular case of A = T , the Toeplitz calculus, the link transform T * T is known as the Berezin transform B R,ν , and the function T * T =: b ν R has been computed in [24]:…”

Section: Bounded Symmetric Domains Of Complex and Real Typementioning

confidence: 99%

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Engliš

Upmeier

2010

Math. Z.

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“…An application of the stationary phase again implies that the Berezin transform has an asymptotic expansion I + α −1 Δ + · · · as h 0, where Δ is the invariant Laplacian on Ω; the higher-order coefficients have been computed explicitly by Unterberger and Upmeier [30]. For the Berezin quantization, the same total set (43) can be used as in the preceding example (with the same proof).…”

Section: Bounded Symmetric Domainsmentioning

confidence: 99%

Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

Engliš¹

2006

Rev. Mat. Complut.

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The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L 2 -spaces of holomorphic functions (weighted Bergman spaces). In both cases, the construction basically uses only the fact that these spaces have a reproducing kernel. We explore the possibilities of using other function spaces with reproducing kernels instead, such as L 2 -spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic functions, and so on. Both positive and negative results are obtained.

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“…This formula reduces proving (5.4) to proving the equality 5) and the latter is just a straightforward computation.…”

Section: Asymptotic Expansion Of the Q-berezin Transformmentioning

confidence: 99%

“…Fortunately, even in the classical setting there is an alternative way to define covariant symbols. Namely, the map operator on a weighted Bergman space → covariant symbol, a function on the ball is, roughly speaking, the adjoint of the map function on the ball → Toeplitz operator on a weighted Bergman space with respect to certain SU(n, 1)-invariant inner products in the spaces of functions and operators (see [5]). The significance of the Toeplitz and covariant calculi is of course well known and has been intensively studied.…”

Section: Introductionmentioning

confidence: 99%

Berezin transform on the quantum unit ball

Shklyarov

Zhang

2003

Journal of Mathematical Physics

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We introduce and study, in the framework of a theory of quantum Cartan domains, a q-analogue of the Berezin transform on the unit ball. We construct q-analogues of weighted Bergman spaces, Toeplitz operators and covariant symbol calculus. In studying the analytical properties of the Berezin transform we introduce also the q-analogue of the SU (n, 1)-invariant Laplace operator (the Laplace-Beltrami operator) and present related results on harmonic analysis on the quantum ball.. These are applied to obtain an analogue of one result by A. Unterberger and H. Upmeier. An explicit asymptotic formula expressing the q-Berezin transform via the q-Laplace-Beltrami operator is also derived. At the end of the paper, we give an application of our results to basic hypergeometric q-orthogonal polynomials.

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The Berezin transform and invariant differential operators

Cited by 99 publications

References 22 publications

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Toeplitz quantization and asymptotic expansions for real bounded symmetric domains

Berezin and Berezin-Toeplitz Quantizations for General Function Spaces

Berezin transform on the quantum unit ball

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