Jordi Pau

On the zeros of functions in Dirichlet-type spaces

We study the sequences of zeros for functions in the Dirichlet spaces D s . Using Carleson-Newman sequences we prove that there are great similarities for this problem in the case 0 < s < 1 with that for the classical Dirichlet space.

A remark on Schatten class Toeplitz operators on Bergman spaces

Let C denote the Euclidian space of complex dimension n. For any two points z = (z1, . . . , zn) and w = (w1, . . . , wn) in C n we write 〈z, w〉 = z1w̄1 + · · · + znw̄n, and |z| = √ 〈z, z〉 = √ |z1| + · · ·+ |zn|. The set Bn = {z ∈ C : |z| < 1} is the open unit ball in C. Denote by dv the usual Lebesgue volume measure on Bn, normalized so that the volume of Bn is one. Throughout this paper we fix a real parameter α with α > −1 and write dvα(z) = cα (1 − |z|)dv(z), where cα is a positive constant chosen so that vα(Bn) = 1. The weighted Bergman space Aα(Bn) consists of those functions f holomorphic on Bn that are in the Lebesgue space L (Bn, dvα). It is a Hilbert space with inner product 〈f, g〉α = ∫

On the stable rank of Q p ∩ H ∞

Let 0 < p < 1. The stable rank of the Banach algebra Q p ∩ H ∞ is 1 if given f 1, f 2 in Q p ∩ H ∞ such that there exists g in Q p ∩ H ∞ such that f 1 + gf 2 is invertible in Q p ∩ H ∞. As a partial answer to this problem, we prove the result when f 1 is an inner function in Q p .