Abstract:
Let $X, Y$ be Banach spaces and $T : X \to Y$ be a bounded linear operator. In this paper, we initiate the study of the perturbation problems for bounded homogeneous generalized inverse $T^h$ and quasi--linear projector generalized inverse $T^H$ of $T$. Some applications to the representations and perturbations of the Moore--Penrose metric generalized inverse $T^M$ of $T$ are also given. The obtained results in this paper extend some well--known results for linear operator generalized inverses in this field.

Abstract:
Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the compactness of composition operators, we see a boundary behavior of the Bergman kernel.

Abstract:
In this paper we give an explicit description of the bounded displacement isometries of a class of spaces that includes the Riemannian nilmanifolds. The class of spaces consists of metric spaces (and thus includes Finsler manifolds) on which an exponential solvable Lie group acts transitively by isometries. The bounded isometries are proved to be of constant displacement. Their characterization gives further evidence for the author's 1962 conjecture on homogeneous Riemannian quotient manifolds. That conjecture suggests that if $\Gamma \backslash M$ is a Riemannian quotient of a connected simply connected homogeneous Riemannian manifold $M$, then $\Gamma \backslash M$ is homogeneous if and only if each isometry $\gamma \in \Gamma$ is of constant displacement. Our description of bounded isometries gives an alternative proof of an old result of J. Tits on bounded automorphisms of semisimple Lie groups. The topic of constant displacement isometries has an interesting history, starting with Clifford's use of quaternions in non--euclidean geometry, and we sketch that in a historical note.

Abstract:
Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present an explicit formula for solving the d-bar-equation $f=\dbar{g}$ on the regular part of Y, where $f$ is a d-bar-closed $(0,1)$-form with compact support. This formula will then be used to give H\"older estimates for the solution in case $Y$ is homogeneous (a cone) with an isolated singularity. Finally, a slight modification of our formula also gives an $L^2$-bounded solution operator in case Y is pure dimensional and homogeneous.

Abstract:
We establish a combinatorial formula for homogeneous moments and give some examples where it can be put to use. An application to the statistical mechanics of interacting gauged vortices is discussed.

Abstract:
The group, Drazin and Koliha-Drazin inverses are particular classes of commuting outer inverses. In this note, we use the inverse along an element to study some spectral conditions related to these inverses in the case of bounded linear operators on a Banach space.

Abstract:
Let $D$ be a bounded homogeneous domain in $\mbb{C}^n$ and let $\Gamma$ be a cyclic discrete subgroup of the automorphism group of $D$. It is shown that the complex space $D/\Gamma$ is Stein.

Abstract:
In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in C^n. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.

Abstract:
Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property.