In the 1960s, the L-theory for the ∂-operator has become an important, indispensable part of complex analysis with influence to many other areas of mathematics. But, whereas the theory is very well developed on complex manifolds, it has been an open problem ever since to create an appropriate L-theory for the ∂-operator on singular complex spaces. In the present article, we develop such a theor...

We characterize the symbols φ for which there exists a weight w such that weighted composition operator MwCφ is compact on Bergman space Bα2. also bounded but not compact. investigate when Hilbert-Schmidt

We describe the convex set of the eigenvalues of hermitian matrices which are majorized by sum of m hermitian matrices with prescribed eigenvalues. We extend our characterization to selfadjoint nonnegative (definite) compact operators on a separable Hilbert space. We give necessary and sufficient conditions on the eigenvalue sequence of a selfadjoint nonnegative compact operator of trace class ...

In this paper we give a global characterisation of classes of ultradifferentiable functions and corresponding ultradistributions on a compact manifold X. The characterisation is given in terms of the eigenfunction expansion of an elliptic operator on X. This extends the result for analytic functions on compact manifold by Seeley [See69], and the characterisation of Gevrey functions and Gevrey u...

A function theoretic characterization is given of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation on the unit circle. When the symbol of the composition operator is univalent, compactness on BMOA is shown to be equivalent to compactness on the Bloch space, and a characterization in terms o...

We discuss compactness of the ∂-Neumann operator in the setting of weighted Lspaces on Cn. For this purpose we use a description of relatively compact subsets of Lspaces. We also point out how to use this method to show that property (P) implies compactness for the ∂-Neumann operator on a smoothly bounded pseudoconvex domain and mention an abstract functional analysis characterization of compac...

The main goal of the paper is to generalize the Duistermaat-Guillemin trace formula to the case of transversally elliptic operators on a compact foliated manifold. First, let us recall briefly the setting of the classical formula. Let P be a positive self-adjoint elliptic pseudodifferential operator of order one on a closed manifold M (for example, P = √ ∆, where ∆ is the Laplace-Beltrami opera...

On a Kähler spin manifold Kählerian twistor spinors are a natural analogue of twistor spinors on Riemannian spin manifolds. They are defined as sections in the kernel of a first order differential operator adapted to the Kähler structure, called Kählerian twistor (Penrose) operator. We study Kählerian twistor spinors and give a complete description of compact Kähler manifolds of constant scalar...

An analytic self-map φ : D → D of the open unit disk D in the complex plane induces the composition operator Cφ on H(D), the space of holomorphic functions on D, defined by Cφ(f) = f ◦φ. A basic goal in the study of composition operators is to relate function theoretic properties of φ to operator theoretic properties of Cφ. Here we review some results that characterize when φ induces a bounded ...