The Fredholm properties of Toeplitz operators on the Bergman space A have been well-known for continuous symbols since the 1970s. We investigate the case p = 1 with continuous symbols under a mild additional condition, namely that of the logarithmic vanishing mean oscillation in the Bergman metric. Most differences are related to boundedness properties of Toeplitz operators acting on A that ari...

Let T be the Toeplitz algebra on the Bergman space La(B, dv) of the unit ball in C. We show that the image of T in the Calkin algebra satisfies the double commutant relation: π(T ) = {π(T )}′′. This is a surprising result, for it is the opposite of what happens in the Hardy-space case [16,17].

In his 1993 paper [10], Kristian Seip characterizes sampling and interpolation sequences for the space A−n and gives an outline of the proof of the corresponding theorem for the Bergman space A2. Using his techniques, we provide a complete proof of the result concerning sampling sequences not only for A2, but also for Ap, 1 ≤ p < ∞.

We present an overview of uniformly discrete sequences and their relationship to other important types of sets, such as uniformly separated sequences, sets of interpolation and sampling, and zero-sets for Bergman spaces. Introduction A sequence Γ = {zk} in the unit disk D is said to be uniformly discrete if ρ(zj , zk) = ∣∣∣∣ zj − zk 1− zjzk ∣∣∣∣ ≥ δ for some fixed δ > 0 and all j 6= k. The supr...

In this paper, we study general extremal problems for non-vanishing functions in Bergman spaces. We show the existence and uniqueness of solutions to a wide class of such problems. In addition, we prove certain regularity results: the extremal functions in the problems considered must be in a Hardy space, and in fact must be bounded. We conjecture what the exact form of the extremal function is...