The space of spectral measures is a homogeneous reductive space

We prove that the space of spectral measures on a W^*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.     

Stability of integral operators on a space of homogeneous type

In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels have certain Hölder regularity and mild singularity near the diagonal. We show that given any , the ‐stability of the operator is equivalent for different , where I stands for the identity operator.     

Decay of matrix coefficients on reductive homogeneous spaces of spherical type

Let \(Z\) be a homogeneous space \(Z=G/H\) of a real reductive Lie group \(G\) with a reductive subgroup \(H\) . The investigation concerns the quantitative decay of matrix coefficients on \(Z\) under the assumption that \(Z\) is of spherical type, that is, minimal parabolic subgroups have open orbits on \(Z\) .     

Naturally reductive metrics on homogeneous systems

We show that naturally reductive (indefinite) metrics on homogeneous systems are determined by nondegenerate invariant forms of their tangent Lie triple algebras. By using this we obtain the decomposition theorem of homogeneous systems. Furthermore, we show that a naturally reductive Riemannian homogeneous space is irreducible if and only if its tangent Lie triple algebra is simple.     

Naturally reductive Riemannian homogeneous structures on some classes of generic submanifolds in complex space forms

We prove that the universal covering spaces of the generic submanifolds of C P _n and of C H _n are naturally reductive homogeneous spaces by determining explicitly tensor fields defining naturally reductive homogeneous structures on them.     

Naturally reductive homogeneous Finsler spaces

In this paper, we study naturally reductive Finsler metrics. We first give a sufficient and necessary condition for a Finsler metric to be naturally reductive with respect to certain transitive group of isometries. Then we study in detail the left invariant naturally reductive metrics on compact Lie groups and give a method to construct the non-Riemannian ones. Further, we give a classification of left invariant naturally reductive metrics on nilpotent Lie groups. Finally, we give a classification of all the naturally reductive Finsler spaces of dimension less or qual to 4. As applications, we obtain some rigidity theorems about naturally reductive Finsler metrics. Namely, any left invariant non-symmetric naturally reductive Finsler metric on a compact simple Lie group or an indecomposable nilpotent Lie group must be Riemannian. On the other hand, we provide a very convenient method to construct non-symmetric Berwald spaces which are neither Riemannian nor locally Minkowskian, a kind of spaces which are sought after in the book by Bao et al. (An introduction to Riemann–Finsler geometry, GTM 200, 2000).     

Deformation of proper actions on reductive homogeneous spaces

Let G be a real reductive Lie group and H a closed reductive subgroup of G. We investigate the deformation of standard compact quotients of G/H, that is, of quotients of G/H by discrete groups Γ that are uniform lattices in some closed reductive subgroup L of G acting properly and cocompactly on G/H. For L of real rank 1, we prove that after a small deformation in G, such a group Γ keeps acting properly discontinuously and cocompactly on G/H. More generally, we prove that the properness of the action of any convex cocompact subgroup of L on G/H is preserved under small deformations, and we extend this result to reductive homogeneous spaces G/H over any local field. As an application, we obtain compact quotients of SO(2n, 2)/U(n, 1) by Zariski-dense discrete subgroups of SO(2n, 2) acting properly discontinuously.     

Average sampling and reconstruction in a reproducing kernel subspace of homogeneous type space

In this paper, we first introduce a reproducing kernel subspace of , where is a homogeneous type space. Then we consider average sampling and reconstruction of signals in the reproducing kernel subspace of . We show that signals in the reproducing kernel subspace of could be stably reconstructed from its average samples taken on a relatively‐separated set with small gap. Exponential convergence is established for the iterative approximation‐projection reconstruction algorithm.     

The boundedness of sublinear operators on Morrey-Herz spaces over the homogeneous type space

In this article, the Morrey-Herz spaces over the homogeneous type basic space are introduced; moreover, several theorems about the boundedness of some sublinear operators on these Morrey-Herz spaces are established, which extend the related known results.