This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symmetric domain. These results generalize a recent result on the Hilbert unit ball due to the author and his coauthor but also cover all irreducible bounded symmetric domains. Moreover, we obtain two trace formulae and a new integral estimate related to the Forelli-Rudin estimate. The key ingredient of the proofs involves the function theory on the bounded symmetric domain and the spectrum estimate of Bergman-type and and Szegö-type operators.
We characterize the membership of Hankel operators with general symbols in the Schatten Classes $S^p,\, p\in(0,1),$ of the large Bergman spaces $A^2_ω$. The case $p\geq 1$ was proved by Lin and Rochberg.
A full description of the membership in the Schatten ideal $S_ p(A^2_ω)$ for $0<p<\infty$ of the Toeplitz operator acting on large weighted Bergman spaces is obtained.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<\infty$ of the Hankel operators $H_ f$ and $H_{\bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.
We consider the problem of $L^1$ (un)boundedness for a wide class of orthogonal projections, including Bergman projections on domains in complex manifolds and Szegö projections on abstract CR manifolds.
Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half-space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form ${\cal{S}} := \{z \in \mathbb{O} \mid 0 < \Re(z) < d\}$ from which we derive by a limit argument considering $d \to \infty$ the Szegö…
Let $P_γ$ be the orthogonal projection from the space $L ^2 (\mathbb{B}_n, dv_γ)$ to the standard weighted Bergman space $L_a ^2 (\mathbb{B}_n, dv_γ)$. In this paper, we characterize the Schatten $p$ class membership of the commutator $[M_f, P_γ]$ when $\frac{2n}{n + 1 + γ} < p < \infty$. In particular, if $\frac{2n}{n + 1 + γ} < p < \infty$, then we show that $[M_f, P_γ]$ is in the Schatten $p$ class if and only if the mean oscillation MO${}_γ(f)$ is in $ L^p(\mathbb{B}_n, dζ)$ where $dζ$ is the Möbius invariant measure on $\mathbb{B}_n.$ This answers a question recently raised by K. Zhu.
We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy spac…
We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space $A^2_ω$, where $ω$ is a radial weight satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from $A^p_ω$ to $A^q_v$.
We characterize operator-theoretic properties (boundedness, compactness, and Schatten class membership) of Toeplitz operators with positive measure symbols on Bergman spaces of holomorphic hermitian line bundles over Kähler Cartan-Hadamard manifolds in terms of geometric or operator-theoretic properties of measures.
Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class …
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_φ$ on Bergman type spaces $A_ω^p $ with double weight $ω$. Let $X=\{u\in H(D): uC_φ:A_ω^p\to A_ω^p \mbox{ is bounded}\}.$ For some regular weights $ω$, we obtain that $X=H^\infty$ if and only if $φ$ is a finite Blaschke product.
A natural class of weighted Bergman spaces on the symmetrized polydisc is isometrically embedded as a subspace in the corresponding weighted Bergman space on the polydisc. We find an orthonormal basis for this subspace. It enables us to compute the kernel function for the weighted Bergman spaces on the symmetrized polydisc using the explicit nature of our embedding. This family of kernel functions include the Szegö and the Bergman kernel on the symmetrized polydisc.
We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(ω)$ belong to the Schatten class $S_p$, $0<p<\infty$.
The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 α(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this si…
The exact range of the positive real parameter $p$ is determined so that the multiplications by polynomial functions acting on Bergman spaces over complex ellipsoids are Schatten $p$-summable.
In this paper, we characterize bounded, compact or Schatten class weighted composition operators acting on Bergman spaces with the exponential type weights. Moreover, we give the proof of the necessary part for the boundedness of composition operators on large weighted Bergman spaces given by Kriete and Maccluer in 1992.
In this paper, we study Toeplitz operators on the weighted harmonic Bergman spaces with nonnegative symbols, the weights we choose here are Muckenhoupt A_2 weights. Results obtained include characterizations of bounded Toeplitz operators, compact Toeplitz operators, invertible Toeplitz operators and Toeplitz operators in the Schatten classes.
In this paper we study the Bergman-Toeplitz operator $T_ψ$ induced by $ψ(w) = K_Ω^{-α}(w,w)d_Ω^β(w)$ with $α, β\geq 0$ acting from a weighted $L^p$-space $L_a^p(Ω)$ to another one $L_a^q(Ω)$ on a large class of pseudoconvex domains of finite type. In the case $1 < p \leq q < \infty$, the following results are established: \\ - Necessary and sufficient conditions for boundedness, which generalize the recent results obtained by Khanh, Liu and Thuc.\\ - Upper and lower estimates for essential norm, in particular, a criterion for compactness.\\ - A characterization of Schatten class membership of …
Bergman-type integral operators are classical operators in complex analysis and operator theory. Recently, the first author and his collaborator \cite{DiW} completely characterized the $L^p$-$L^q$ boundedness of Bergman-type integral operators $K_α,K_α^+$ and the $L^p$-$L^q$ compactness of $K_α$ on the unit ball. In this paper, we will use a substantially new method to completely characterize the $L^p$-$L^q$ compactness of $K_α^+,$ but also prove that the $L^p$-$L^q$ compactness of operators $K_α,K_α^+$ is in fact equivalent. Moreover, we completely characterize Schatten class and Macaev class…
In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{ϕ,ψ}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.
We study the Carleson measures and the Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip's results from the unit disc of $\mathbb C$ to the unit ball of $\mathbb C^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.
For a Dirichlet series g, we study the Volterra operator Tg of symbol g, acting on a class of weighted Hilbert spaces of Dirichlet series. We obtain sufficient / necessary conditions for Tg to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of Tg in Schatten classes. We also relate the boundedness of Tg to the boundedness of a multiplicative Hankel form of symbol g.
In this paper we introduce the Schatten class of operators and the Berezin transform of operators in the quaternionic setting. The first topic is of great importance in operator theory but it is also necessary to study the second one because we need the notion of trace class operators, which is a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S-spectrum for quaternionic operators, which is the not…
In this paper, we focus on the weighted Bergman spaces $A_φ^{p}$ in $\mathbb{D}$ with $φ\in\mathcal{W}_{0}$. We first give characterizations of those finite positive Borel measures $μ$ in $\mathbb{D}$ such that the embedding $A_φ^{p}\subset L_μ^{q}$ is bounded or compact for $0<p,q<\infty$. Then we describe bounded or compact Toeplitz operators $T_μ$ from one Bergman space $A_φ^{p}$ to another $A_φ^{q}$ for all possible $0<p,q<\infty$. Finally, we characterize Schatten class Toeplitz operators on $A_φ^{2}$.
In this note we prove that the ball is a maximiser of some Schatten $p$-norms of the Riesz potential operators among all domains of a given measure in $\mathbb R^{d}$. In particular, the result is valid for the polyharmonic Newton potential operator, which is related to a nonlocal boundary value problem for the poly-Laplacian extending the one considered by M. Kac in the case of the Laplacian, so we obtain and isoperimetric inequalities for its eigenvalues as well, namely, analogues of Rayleigh-Faber-Krahn and Hong-Krahn-Szegö inequalities.
Hankel operators with anti-holomorphic symbols are studied for a large class of weighted Fock spaces on $\cn$. 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.…
