Let $varphi(z)=z^m, z in mathbb{U}$, for some positive integer $m$, and $C_varphi$ be the composition operator on the Bergman space $mathcal{A}^2$ induced by $varphi$. In this article, we completely determine the point spectrum, spectrum, essential spectrum, and essential norm of the operators $C^*_varphi C_varphi, C_varphi C^*_varphi$ as well as self-commutator and anti-self-commutators of $C_...

The importance of recognizing repeating structures in web applications has generated a large body of work on algorithms for mining the HTML Document Object Model (DOM). A restricted tree edit distance metric, called the Restricted Top Down Metric (RTDM), is most suitable for DOM mining as well as less computationally expensive than the general tree edit distance. Given two trees with input size...

We characterise the boundedness of a Toeplitz operator on the Bergman space with a L symbol. We also prove that the compactness of a Toeplitz operator on the Bergman space with a L symbol is completely determined by the boundary behaviour of its Berezin transform. This result extends known results in the cases when the symbol is either a positive L function, an L∞ function or a general BMO func...

In these lectures we plan to present a survey of certain aspects of harmonic analysis on a Heisenberg nilmanifold Γ\Hn. Using Weil-Brezin-Zak transform we obtain an explicit decomposition of L(Γ\H) into irreducible subspaces invariant under the right regular representation of the Heisenberg group. We then study the Segal-Bargmann transform associated to the Laplacian on a nilmanifold and charac...

Let M1 and M2 be two Kähler manifolds. We call M1 and M2 relatives if they share a non-trivial Kähler submanifold S, namely, if there exist two holomorphic and isometric immersions (Kähler immersions) h1 : S → M1 and h2 : S → M2. Moreover, two Kähler manifolds M1 and M2 are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds S1 and S2 ...

A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Θ s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian...

We prove that the Bergman kernel function associated to a finitely connected domain Ω in the plane is given as a rational combination of only three basic functions of one complex variable: an Alhfors map, its derivative, and one other function whose existence is deduced by means of the field of meromorphic functions on the double of Ω. Because many other functions of conformal mapping and poten...

In this paper we consider a class of weighted integral operators on L2(0,∞) and show that they are unitarily equivalent to little Hankel operators between weighted Bergman spaces of the right half plane. We use two parameters α, β ∈ (−1,∞) and involve two weights to define Bergman spaces of the domain and range of the little Hankel operators. We obtained conditions for the Hankel integral opera...

Tropical varieties play an important role in algebraic geometry. The Bergman complex B(M) and the positive Bergman complex B+(M) of an oriented matroid M generalize to matroids the notions of the tropical variety and positive tropical variety associated to a linear ideal. Our main result is that if A is a Coxeter arrangement of type Φ with corresponding oriented matroid MΦ, then B (MΦ) is dual ...