Antiholomorphic Symbols Research Articles

English

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Essential Norm Estimates for Little Hankel Operators with Anti Holomorphic Symbols on Weighted Bergman Spaces of the Unit Ball

We give estimates for the essential norm of a little Hankel operator with anti holomorphic symbol on weighted Bergman spaces of the unit ball in terms of the Bloch semi-norm of its symbol function.

Essential Norm Estimates for Little Hankel Operators with Anti Holomorphic Symbols on the weighted Bergman Space of the Unit Disk

We give estimates for the essential norm of little Hankel operators with anti holomorphic symbols on weighted Bergman spaces of the unit disk in terms of the Bloch semi norm of its symbol function.

English

On complete pseudoconvex Reinhardt domains in ℂ2, we show that there is no nonzero Hankel operator with anti-holomorphic symbol that is Hilbert-Schmidt. In the proof, we explicitly use the pseudoconvexity property of the domain. We also present two examples of unbounded non-pseudoconvex domains in ℂ2 that admit nonzero Hilbert-Schmidt Hankel operators with anti-holomorphic symbols. In the first example the Bergman space is finite dimensional. However, in the second example the Bergman space is infinite dimensional and the Hankel operator $${H_{{{\bar z}_1}{{\bar z}_2}}}$$ is Hilbert-Schmidt.

Hilbert–Schmidt Hankel Operators with Anti-holomorphic Symbols on Complex Ellipsoids

On a complex ellipsoid in \({\mathbb{C}^n}\) , we show that there is no nonzero Hankel operator with an anti-holomorphic symbol that is Hilbert–Schmidt.

Hankel Operators on Fock Spaces and Related Bergman Kernel Estimates

Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on ℂ n . The weights defining these Hilbert spaces are radial and subject to a mild smoothness condition. In addition, it is assumed that the weights decay at least as fast as the classical Gaussian weight. The main result of the paper says that a Hankel operator on such a Fock space is bounded if and only if the symbol belongs to a certain BMOA space, defined via the Berezin transform. The latter space coincides with a corresponding Bloch space which is defined by means of the Bergman metric. This characterization of boundedness relies on certain precise estimates for the Bergman kernel and the Bergman metric. Characterizations of compact Hankel operators and Schatten class Hankel operators are also given. In the latter case, results on Carleson measures and Toeplitz operators along with Hormander’s L 2 estimates for the $\bar{\partial}$ operator are key ingredients in the proof.

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures

We compute the norm of pointwise multiplication operators, Toeplitz and Big Hankel operators with antiholomorphic symbols, defined on Besov spaces. These norms will be given in terms of Carleson measures for Besov spaces related to the symbol.

Hankel operators and the Stieltjes moment problem

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μ n the image measure in C n of μ ⊗ σ n under the map ( t , ξ ) ↦ t ξ , where σ n is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L 2 ( μ n ) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n = 1 , we prove that there are nontrivial Hilbert–Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.

A note on Schatten‐class membership of Hankel operators with antiholomorphic symbols on generalized Fock‐spaces

AbstractIn this paper we investigate Hankel operators with anti‐holomorphic L2‐symbols on generalized Fock spaces Am2 in one complex dimension. The investigation of the mentioned operators was started in [4] and [3]. Here, we show that a Hankel operator with anti‐holomorphic L2‐symbol is in the Schatten‐class Sp if and only if the symbol is a polynomial with degree N satisfying 2N < m and p > . The result has been proved independently before in the recent work [2], which also considers the case of several complex variables. However, in addition to providing a different proof for the result the present work shows that the methodology developed in [4] and [3] can be adopted in order to work to characterize Schatten‐class membership. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Continuity and Schatten–von Neumann p-class membership of Hankel operators with anti-holomorphic symbols on (generalized) Fock spaces

In this paper we investigate Hankel operators H f ¯ : A m 2 → A m 2 ⊥ with anti-holomorphic symbols f ¯ = ∑ k = 0 ∞ b k z ¯ k ∈ L 2 ( C , | z | m ) , where A m 2 are general Fock spaces. We will show that H f ¯ is not continuous if the corresponding symbol is not a polynomial f ¯ = ∑ k = 0 N b k z ¯ k . For polynomial symbols we will give necessary and sufficient conditions for continuity and compactness in terms of N and m. For monomials z ¯ k we will give a complete characterization of the Schatten–von Neumann p-class membership for p > 0 . Namely in case 2 k < m the Hankel operators H z ¯ k are in the Schatten–von Neumann p-class iff p > 2 m / ( m − 2 k ) ; and in case 2 k ⩾ m they are not in the Schatten–von Neumann p-class.

Hankel operators with antiholomorphic symbols on the Fock space

We consider Hankel operators of the formHz¯k:Fm:={f:f is entire and∫Cn|f(z)|2e−|z|m&gt;∞}→L2(e−|z|m)H_{\overline {z}^k}: \mathcal {F}^m:=\{f : f \mbox { is entire and} \int _{\mathbb {C}^n}|f(z)|^2e^{-|z|^m}&gt;\infty \}\rightarrow L^2(e^{-|z|^m}). Herek,m,n∈Nk,m,n \in \mathbb {N}. We show that in the case of one complex dimension the Hankel operators are compact but not Hilbert-Schmidt ifm&gt;2km&gt;2k.

Images of Hankel operators on the polydisk

We study images of Hankel operators on the Bergman or Hardy space of the poly-disk. We give characterisations of pairs of antiholomorphic symbols inducing Hankel operators whose images are mutually orthogonal.

SOME REMARKS ON CCR AND WEYL OPERATORS

Using the symbol calculus developed by Berezin, Krée and Rączka, it is shown that the algebra generated by the canonical commutation relation (CCR) is splitted into creation part and annhilation part with holomorphic and anti-holomorphic symbols respectively. Further, in the Weyl representation of CCR, three Weyl operators will constitute an irreducible set of the full CCR-algebra if some number theoretic conditions are satisfied.

Möbius invariant Qpspaces associated with the Green’s function on the unit ball of Cn

In this paper, function spaces Q(p)(B) and Q(p,o)(B), associated with the Green's function, are defined and studied for the unit ball B of C-n. We prove that Q(p)(B) and Q(p,o)(B) are Mobius invariant Banach spaces and that Q(p)(B) = Bloch(B), Q(p,o)(B) = B-o(B) (the little Bloch space) when 1 < p < n/(n - 1),Q(1) = BMOA(partial derivative B) and Q(1,0)(B) = VMOA(partial derivative B). This fact makes it possible for us to deal with BMOA and Bloch space in the same way. And we give necessary and sufficient conditions on boundedness (and compactness) of the Hankel operator with antiholomorphic symbols relative to Q(p)(B) (and Q(p,o)(B)). Moreover, other properties about the above spaces and \phi(z)(omega)\,phi(z)(omega) is an element of Aut(B), are obtained.

Möbius invariant besov p-spaces and hankel operators in the bergman space on the ball in C n

In this paper, we extend the ddinition of Besov p-spaces to higher dimensions. We characterize these spaces and solve a conjecture given by Zhu. We also prove that for n ≥ 2, the Schatten ideal S p of the Bergman space on the unit ball in C n contains nonzero Hankel operators with antiholomorphic symbols if and only if p>2n.

Hilbert-Schmidt Hankel Operators on the Bergman Space

We show that there are no nonzero Hilbert-Schmidt Hankel operators on the Bergman space of the open unit ball in ${{\mathbf {C}}^n}$ with antiholomorphic symbols when $n \geq 2$.

Hilbert-Schmidt Hankel operators on the Bergman space

We show that there are no nonzero Hilbert-Schmidt Hankel operators on the Bergman space of the open unit ball in C n {{\mathbf {C}}^n} with antiholomorphic symbols when n ≥ 2 n \geq 2 .