Boundedness and Compactness of Hankel Operators on Large Fock Space

We introduce the BMO spaces and use them to characterize complex-valued functions f such that the big Hankel operators H f and H f ¯ are both bounded or compact from a weighted large Fock space F p ϕ into a weighted Lebesgue space L p ϕ when 1 ≤ p < ∞ .

Bergman spaces with exponential type weights

For $1\le p<\infty $ 1 ≤ p < ∞ , let $A^{p}_{\omega }$ A ω p be the weighted Bergman space associated with an exponential type weight ω satisfying $$ \int _{{\mathbb{D}}} \bigl\vert K_{z}(\xi ) \bigr\vert \omega (\xi )^{1/2} \,dA(\xi ) \le C \omega (z)^{-1/2}, \quad z\in {\mathbb{D}}, $$ ∫ D | K z ( ξ ) | ω ( ξ ) 1 / 2 d A ( ξ ) ≤ C ω ( z ) − 1 / 2 , z ∈ D , where $K_{z}$ K z is the reproducing kernel of $A^{2}_{\omega }$ A ω 2 . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight $\omega _{*}$ ω ∗ . As an application, we prove the boundedness of the Bergman projection on $L^{p}_{\omega }$ L ω p , identify the dual space of $A^{p}_{\omega }$ A ω p , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from $A^{p}_{\omega }$ A ω p into $A^{q}_{\omega }$ A ω q , $1\le p,q<\infty $ 1 ≤ p , q < ∞ , such as Toeplitz and (big) Hankel operators.

OPERATORS ON BERGMAN SPACES

Bergman space theory has been at the core of complex analysis research for many years. Indeed, Hardy spaces are related to Bergman spaces. The popularity of Bergman spaces increased when functional analysis emerged. Although many researchers investigated the Bergman space theory by mimicking the Hardy space theory, it appeared that, unlike their cousins, Bergman spaces were more complex in different aspects. The issue of invariant subspace constitutes one common problem in mathematics that is yet to be resolved. For Hardy spaces, each invariant subspace for shift operators features an elegant description. However, the method for formulating particular structures for the large invariant subspace of shift operators upon Bergman spaces is still unknown. This paper aims to characterize bounded Hankel operators involving a vector-valued Bergman space compared to other different vector value Bergman spaces.

Compact Hankel Operators Between Distinct Hardy Spaces and Commutators

The paper is devoted to the study of compactness of Hankel operators acting between distinct Hardy spaces generated by Banach function lattices. We prove an analogue of Hartman’s theorem characterizing compact Hankel operators in terms of properties of their symbols. As a byproduct we give an estimation of the essential norm of such operators. Furthermore, compactness of commutators and semicommutators of Toeplitz operators for unbounded symbols is discussed.

Asymmetric Truncated Hankel Operators: Rank One, Matrix Representation

Asymmetric truncated Hankel operators are the natural generalization of truncated Hankel operators. In this paper, we determine all rank one operators of this class. We explore these operators on finite-dimensional model spaces, in particular, their matrix representation. We also give their matrix representation and the one for asymmetric truncated Toeplitz operators in the case of model spaces associated to interpolating Blaschke products.