A Compactification of the Space of Expanding Maps on the Circle

We show the space of expanding Blaschke products on S¹ is compactified by a sphere of invariant measures, reminiscent of the sphere of geodesic currents for a hyperbolic surface. More generally, we develop a dynamical compactification for the Teichmüller space of all measure preserving topological covering maps of S¹. Research supported in part by the NSF.     

加权Bergman空间上解析Toeplitz算子的换位及相似

本文在加权Bergman 空间A_α²(D)(α>-1) 上证明了由n-Blaschke 因子诱导的解析Toeplitz算子相似于n个Bergman 移位的直接和, 刻画了一类解析Toeplitz 算子的换位并研究了两个解析Toeplitz 算子的相似性.     

Expanding a Finite Blaschke Product

It is shown that a finite Blaschke product with finite poles, has a nonzero residue. The proofs for the two types of Blaschke products are essentially different.     

Reducing Subspaces of Toeplitz Operators on $N_ϕ$-Type Quotient Modules on the Torus

In this paper, we prove that the Toeplitz operator with finite Blaschke product symbol $S_{ψ(z)}$ on $N_ϕ$ has at least $m$ non-trivial minimal reducing subspaces, where $m$ is the dimension of $H^2(Γ_ω) ⊖ ϕ(ω)H^2 (Γ_ω)$. Moreover, the restriction of $S_{ψ(z)}$ on any of these minimal reducing subspaces is unitary equivalent to the Bergman shift $M_z$.     

An example of a maximal unimodular Dirichlet algebra

We construct an example of a maximal unimodular Dirichlet algebra whose one-point Gleason parts are dense in the maximal ideal space. The main statements were announced in [1] and now we provide their complete proofs.     

A note on uniform approximation by the indestructibles

We modify an idea in an earlier paper to enlarge the collection of inner functions that are known to be in the uniform closure of the indestructible Blaschke products.     

Surfaces with isotropic Blaschke tensor in S 3

Let M2 be an umbilic-free surface in the unit sphere S3. Four basic invariants of M2 under the Moebius transformation group of S3 are Moebius metric g, Blaschke tensor A, Moebius second fundamental form B and Moebius form Φ. We call the Blaschke tensor is isotropic if there exists a smooth function λ such that A = λg. In this paper, We classify all surfaces with isotropic Blaschke tensor in S3.     

On the Blaschke isoparametric hypersurfaces in the unit sphere

Given an immersed submanifold x ： M^M → S^n in the unit sphere S^n without umbilics, a Blaschke eigenvalue of x is by definition an eigenvalue of the Blaschke tensor of x. x is called Blaschke isoparametric if its Mobius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the MSbius geometry of submanifolds. When n = 4, the corresponding classification theorem was given by the authors. In this paper, we are able to complete the corresponding classification for n = 5. In particular, we shall prove that all the Blaschke isoparametric hypersurfaces in S^5 with more than two distinct Blaschke eigenvalues are necessarily Mobius isoparametric.     

On -norms of meromorphic functions with fixed poles

We study boundedness of the differentiation and embedding operators in the shift-coinvariant subspaces generated by Blaschke products with sparse zeros, that is, in the spaces of meromorphic functions with fixed poles in the lower half-plane endowed with -norm. We answer negatively the question of K.M. Dyakonov about the necessity of the condition for the boundedness of the differentiation on . Our main tool is a construction of an unconditional basis of rational fractions in .

     

Multiplication and division by inner functions in the space of Bloch functions

A subspace of the Hardy space is said to have the -property if whenever and is an inner function with . We let denote the space of Bloch functions and the little Bloch space. Anderson proved in 1979 that the space does not have the -property. However, the question of whether or not ( ) has the -property was open. We prove that for every the space does not have the -property.

We also prove that if is any infinite Blaschke product with positive zeros and is a Bloch function with , as , then the product is not a Bloch function.