In this paper, we prove embedding theorems for the Möbius invariant space \mathcal {Q}_p on the open unit ball of \mathbb {C}^n into logarithmic tent spaces in the Bergman metric. [ABSTRACT FROM AUTHOR]
In this paper, we study the dynamics of an operator \mathcal T naturally associated to the so-called Collatz map , which maps an integer n \geq 0 to n / 2 if n is even and 3n + 1 if n is odd. This operator \mathcal T is defined on certain weighted Bergman spaces \mathcal B ^2 _\omega of analytic functions on the unit disk. Building on previous work of Neklyudov, we show that \mathcal T is hypercyclic on \mathcal B ^2 _\omega, independently of whether the Collatz Conjecture holds true or not. Under some assumptions on the weight \omega, we show that \mathcal T is actually ergodic with respect to a Gaussian measure with full support, and thus frequently hypercyclic and chaotic. [ABSTRACT FROM AUTHOR]
It is well-known that the research of linear combination of composition operators has become a topic of increasing interest. Recently, Choe, Koo and Wang proved that the compactness of combinations composition operators induced by the symbols satisfying the condition (CNC) implies that each difference is compact on the weighted Bergman space. Motivated by that work, in this paper, we discuss which difference is compact on the weighted Bergman space when the coefficients do not satisfy the condition (CNC). [ABSTRACT FROM AUTHOR]