Isometries of the product of composition operators on weighted Bergman space

In this paper, the necessary and sufficient conditions for the product of composition operators to be isometry are obtained on weighted Bergman space. With the help of a counter example we also proved that unlike on [Formula: see text] and [Formula: see text] the composition operator on [Formula: see text] induced by an analytic self-map on [Formula: see text] with fixed origin need not be of norm one. We have generalized the Schwartz’s [Composition operators on [Formula: see text], thesis, University of Toledo (1969)] well-known result on [Formula: see text] which characterizes the almost multiplicative operator on [Formula: see text]

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.

Products of Toeplitz Operators on the 2-Analytic Bergman Space

Let f and g be bounded functions, and let T f and T g be Toeplitz operators on A 2 2 D . We show that if the product T f T g equals zero and one of f and g is a radial function satisfying a Mellin transform condition, then the other function must be zero.

Compact Differences of Composition Operators on Bergman Spaces Induced by Doubling Weights

Bounded and compact differences of two composition operators acting from the weighted Bergman space $$A^p_\omega $$ A ω p to the Lebesgue space $$L^q_\nu $$ L ν q , where $$0<q<p<\infty $$ 0 < q < p < ∞ and $$\omega $$ ω belongs to the class "Equation missing" of radial weights satisfying two-sided doubling conditions, are characterized. On the way to the proofs a new description of q-Carleson measures for $$A^p_\omega $$ A ω p , with $$p>q$$ p > q and "Equation missing", involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of q-Carleson measures for the classical weighted Bergman space $$A^p_\alpha $$ A α p with $$-1<\alpha <\infty $$ - 1 < α < ∞ to the setting of doubling weights. The case "Equation missing" is also briefly discussed and an open problem concerning this case is posed.