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An example for the holomorphic sectional curvature of the Bergman metric

Żywomir Dinew (2010)

Annales Polonici Mathematici

We study the behaviour of the holomorphic sectional curvature (or Gaussian curvature) of the Bergman metric of planar annuli. The results are then utilized to construct a domain for which the curvature is divergent at one of its boundary points and moreover the upper limit of the curvature at that point is maximal possible, equal to 2, whereas the lower limit is -∞.

Asymptotic behavior of the sectional curvature of the Bergman metric for annuli

Włodzimierz Zwonek (2010)

Annales Polonici Mathematici

We extend and simplify results of [Din 2010] where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly to [Din 2010] the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two, whereas its infimum is equal to -∞ .

Berezin and Berezin-Toeplitz quantizations for general function spaces.

Miroslav Englis (2006)

Revista Matemática Complutense

The standard Berezin and Berezin-Toeplitz quantizations on a Kähler manifold are based on operator symbols and on Toeplitz operators, respectively, on weighted L2-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 L2-spaces of harmonic functions, Sobolev spaces, Sobolev spaces of holomorphic...

Bloch-to-Hardy composition operators

Evgueni Doubtsov, Andrei Petrov (2013)

Open Mathematics

Let φ be a holomorphic mapping between complex unit balls. We characterize those regular φ for which the composition operators C φ: f ↦ f ○ φ map the Bloch space into the Hardy space.

Bounded Toeplitz and Hankel products on weighted Bergman spaces of the unit ball

Małgorzata Michalska, Maria Nowak, Paweł Sobolewski (2010)

Annales Polonici Mathematici

We prove a sufficient condition for products of Toeplitz operators T f T , where f,g are square integrable holomorphic functions in the unit ball in ℂⁿ, to be bounded on the weighted Bergman space. This condition slightly improves the result obtained by K. Stroethoff and D. Zheng. The analogous condition for boundedness of products of Hankel operators H f H * g is also given.

Complex symmetry of Toeplitz operators on the weighted Bergman spaces

Xiao-He Hu (2022)

Czechoslovak Mathematical Journal

We give a concrete description of complex symmetric monomial Toeplitz operators T z p z ¯ q on the weighted Bergman space A 2 ( Ω ) , where Ω denotes the unit ball or the unit polydisk. We provide a necessary condition for T z p z ¯ q to be complex symmetric. When p , q 2 , we prove that T z p z ¯ q is complex symmetric on A 2 ( Ω ) if and only if p 1 = q 2 and p 2 = q 1 . Moreover, we completely characterize when monomial Toeplitz operators T z p z ¯ q on A 2 ( 𝔻 n ) are J U -symmetric with the n × n symmetric unitary matrix U .

Effective local finite generation of multiplier ideal sheaves

Dan Popovici (2010)

Annales de l’institut Fourier

Let ϕ be a psh function on a bounded pseudoconvex open set Ω n , and let ( m ϕ ) be the associated multiplier ideal sheaves, m . Motivated by global geometric issues, we establish an effective version of the coherence property of ( m ϕ ) as m + . Namely, given any B Ω , we estimate the asymptotic growth rate in m of the number of generators of ( m ϕ ) | B over 𝒪 Ω , as well as the growth of the coefficients of sections in Γ ( B , ( m ϕ ) ) with respect to finitely many generators globally defined on Ω . Our approach relies on proving asymptotic integral...

Espace de Dixmier des opérateurs de Hankel sur les espaces de Bergman à poids

Romaric Tytgat (2015)

Czechoslovak Mathematical Journal

Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes f ¯ tels que l’opérateur de Hankel H f ¯ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten 𝒮 p quand p tend vers 1 et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs. Abstract....

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