For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$, we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $, respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$, written as $H(\varphi )$ and $H(\overline \varphi )$ and equipped with appropriate inner products, are called sub-Bergman spaces.

We prove the following three results in the paper: for $-1<\alpha \le 0$, the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$, we have $H(\varphi )=H(\overline \varphi )=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$, we have $D^2_\varphi (A^2_\alpha )= D^2_{\overline \varphi }(A^2_\alpha )=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

We prove that for $0<p<+\infty $ and $-1<\alpha <+\infty ,$ a conformal map defined on the unit disk belongs to the weighted Bergman space $A_{\alpha }^p$ if and only if a certain integral involving the hyperbolic distance converges.

The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_{\unicode[STIX]{x1D6FC}}^{p}$ of the derivative of $B$ or the derivative of functions in its model space $H^{2}\ominus \mathit{BH}^{2}$ is investigated.

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

For an analytic function $f$ on the unit disk $\mathbb{D}$, we show that the ${{L}^{2}}$ integral mean of $f$ on $\text{c}\,\text{}\,\text{ }\!\!|\!\!\text{ z }\!\!|\!\!\text{ }\,\text{}\,\text{r}$ with respect to the weighted area measure ${{\left( 1\,-\,|z{{|}^{2}} \right)}^{\alpha }}dA\left( z \right)$ is a logarithmically convex function of $r$ on $\left( c,\,1 \right)$, where $-3\,\le \,\alpha \,\le \,0\,\text{and}\,\text{c}\,\in \,[\,0,\,1)$. Moreover, the range $[-3,\,0]$ for $\alpha $ is best possible. When $c\,=\,0$, our arguments here also simplify the proof for several results we obtained in earlier papers.

If $B$ is the Blachke product with zeros $\{{{z}_{n}}\},\,\text{then}\,\left| {B}'\left( z \right) \right|\,\le \,{{\Psi }_{B}}\left( z \right)$, where

$${{\Psi }_{B}}\,=\,\sum\limits_{n}{\frac{1-{{\left| {{z}_{n}} \right|}^{2}}}{{{\left| 1-{{\overline{z}}_{n}}z \right|}^{2}}}.}$$

Moreover, it is a well-known fact that, for $0\,<\,p\,<\,\infty $,

$${{M}_{p\left( r,{B}' \right)}}\,=\,{{\left( \frac{1}{2\pi }\,\int_{0}^{2\pi }{{{\left| {B}'\left( \text{r}{{\text{e}}^{i\theta }} \right) \right|}^{p}}d\theta } \right)}^{{1}/{p}\;}},\,0\,\le \,r\,<\,1,$$

is bounded if and only if ${{M}_{p}}\left( r,\,{{\Psi }_{B}} \right)$ is bounded. We find a Blaschke product ${{B}_{0}}$ such that ${{M}_{p}}\left( r,\,{{{{B}'}}_{0}} \right)$ and ${{M}_{p}}\left( r,{{\Psi }_{{{B}_{0}}}} \right)$ are not comparable for any $\frac{1}{2}\,<\,p\,<\,\infty $. In addition, it is shown that, if $0\,<\,p\,<\,\infty$, $B$ is a Carleson-Newman Blaschke product and a weight $\omega $ satisfies a certain regularity condition, then

$${{\int }_{\mathbb{D}}}{{\left| {B}'\left( z \right) \right|}^{p}}\omega \left( z \right)dA\left( z \right)~\asymp {{\int }_{\mathbb{D}}}{{\Psi }_{B}}{{\left( z \right)}^{p}}\omega \left( z \right)dA\left( z \right),$$

where $d\,A\left( z \right)$ is the Lebesgue area measure on the unit disc.

We use a generalised Nevanlinna counting function to compute the Hilbert–Schmidt norm of a composition operator on the Bergman space $L_{a}^{2}(\mathbb{D})$ and weighted Bergman spaces $L_{a}^{1}(\text{d}A_{\unicode[STIX]{x1D6FC}})$ when $\unicode[STIX]{x1D6FC}$ is a nonnegative integer.

For $0<p<\infty$ and $-2\leq {\it\alpha}\leq 0$ we show that the $L^{p}$ integral mean on $r\mathbb{D}$ of an analytic function in the unit disk $\mathbb{D}$ with respect to the weighted area measure $(1-|z|^{2})^{{\it\alpha}}\,dA(z)$ is a logarithmically convex function of $r$ on $(0,1)$.

We show that the modulus of the Bergman kernel $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}B(z, \zeta )$ of a general homogeneous Siegel domain of type II is ‘almost constant’ uniformly with respect to $z$ when $\zeta $ varies inside a Bergman ball. The control is expressed in terms of the Bergman distance. This result was proved by A. Korányi for symmetric Siegel domains of type II. Subsequently, R. R. Coifman and R. Rochberg used it to establish an atomic decomposition theorem and an interpolation theorem by functions in Bergman spaces $A^p$ on these domains. The atomic decomposition theorem and the interpolation theorem are extended here to the general homogeneous case using the same tools. We further extend the range of exponents $p$ via functional analysis using recent estimates.

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

We obtain necessary and sufficient conditions for the compactness of differences of composition operators acting on the weighted Bergman spaces in the unit ball. A representation of a composition operator as a finite sum of composition operators modulo compact operators is also studied.

An analytic function f in the unit disc belongs to F(p,q,s), if

is uniformly bounded for all a ∈ . Here is the Green function of , and φa(z)=(a−z)/(1−āz). It is shown that for 0 < γ < ∞ and |w|=1 the singular inner function exp(γ(z+w)/(z−w)) belongs to F(p,q,s), 0<s≤1, if and only if . Moreover, it is proved that, if 0<s<1, then an inner function belongs to the Möbius invariant Besov-type space for some (equivalently for all) p > max{s,1−s} if and only if it is a Blaschke product whose zero sequence {zn} satisfies .

We study the membership of Blaschke products in Lipschitz spaces, especially for interpolating Blaschke products and for those whose zeros lie in a Stolz angle. We prove several theorems that complement or extend the earlier works of Ahern and the author.

We study the boundedness properties of Rudin–Forelli-type operators associated to tubular domains over symmetric cones. As an application, we give a characterization of the topological dual space of the weighted Bergman space .

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

Let $G$ be the abelian Lie group $\mathbb{R}^n\times\mathbb{R}^k/\mathbb{Z}^k$, acting on the complex space $X=\mathbb{R}^{n+k}\times\ri G$. Let $F$ be a strictly convex function on $\mathbb{R}^{n+k}$. Let $H$ be the Bergman space of holomorphic functions on $X$ which are square-integrable with respect to the weight $e^{-F}$. The $G$-action on $X$ leads to a unitary $G$-representation on the Hilbert space $H$. We study the irreducible representations which occur in $H$ by means of their direct integral. This problem is motivated by geometric quantization, which associates unitary representations with invariant Kähler forms. As an application, we construct a model in the sense that every irreducible $G$-representation occurs exactly once in $H$.