Area Operators on Bergman Spaces

We completely characterize the boundedness of area operators from the Bergman spaces A_(α)^(p)(Bn)to the Lebesgue spaces L^(q)(S_(n))for all 0<p,q<∞.For the case n=1,some partial results were previously obtained by Wu in[Wu,Z.:Volterra operator,area integral and Carleson measures,Sci.China Math.,54,2487–2500(2011)].Especially,in the case q<p and q<s,we obtain some characterizations for the area operators to be bounded.We solve the cases left open there and extend the results to n-complex dimension.

Integration Operators on Spaces of Dirichlet Series

We first study the Volterra operator V acting on spaces of Dirichlet series.We prove that V is bounded on the Hardy space H_(0)^(p)for any 0<p≤∞,and is compact on H_(0)^(p)for 1<p≤∞.Furthermore,we show that V is cyclic but not supercyclic on H_(0)^(p)for any 0<p<∞.Corresponding results are also given for V acting on Bergman spaces H_(w,0)^(p).We then study the Volterra type integration operators T_(g).We prove that if T_(g)is bounded on the Hardy space H_(p),then it is bounded on the Bergman space H_(w)^(p).

Embedding Derivatives and Integration Operators on Hardy Type Tent Spaces

In this paper,we completely characterize the positive Borel measuresμon the unit ball B_(n)such that the differential type operator R^(m)of order m∈N is bounded from Hardy type tent space HT_(q,α)^(p)(B_(n))into L^(s)(μ)for full range of p,q,s,α.Subsequently,the corresponding compact description of differential type operator R^(m)is also characterized.As an application,we obtain the boundedness and compactness of integration operator J_(g)from HT_(q,α)^(p)(B_(n))toHT_(s,β)^(t)(B_(n)),and the methods used here are adaptable to the Hardy spaces.

Compactness of Product Operators on the Bergman Space

In this paper, we study the compactness of the product of a composition operator with another one＇s adjoint on the Bergman space. Some necessary and sufficient conditions for such operators to be compact are given.

Complex Symmetric C0-semigroups on A^2(C+)

In this paper,we study complex symmetric C0-semigroups on the Bergman space A^2(C+) of the right half-plane C+.In contrast to the classical case,we prove that the only involutive composition operator on A^2(C+) is the identity operator,and the class of J-symmetric composition operators does not coincide with the class of normal composition operators.In addition,we divide semigroups{φt}of linear fractional self-maps of C+into two classes.We show that the associated composition operator semigroup{Tt}is strongly continuous and identify its infinitesimal generator.As an application,we characterize Jσ-symmetric C0-semigroups of composition operators on A^2(C+).