Large Bergman spaces: invertibility, cyclicity, and subspaces of arbitrary index

Inner functions, invariant subspaces and cyclicity in $\mathcal{P}^t(μ)$-spaces

We study the invariant subspaces generated by inner functions for a class of $\mathcal{P}^t(μ)$-spaces which can be identified as spaces of analytic functions in the unit disk $\mathbb{D}$, where $μ$ is a measure supported in the closed unit disk and $\mathcal{P}^t(μ)$ is the span of analytic polynomials in the usual Lebesgue space $L^t(μ)$. Our measures define a range of spaces somewhere in between the Hardy and the Bergman spaces, and our results are thus a mixture of results from these two theories. For a large class of measures $μ$ we characterize the cyclic inner functions, and exhibit so…

Bergman 空间与 Fock 空间上的 Douglas 问题

The Douglas question on the Bergman and Fock spaces

本文研究 Bergman 空间与 Fock 空间上由正测度诱导的 Toeplitz 算子的可逆性，利用 Berezin 变换和反向 Carleson 条件给出若干充要条件，并证明两类空间上的 Douglas 问题在一般情形下答案均为否定。

解析函数空间中的移位循环性

Shift-cyclicity in analytic function spaces

本综述探讨解析函数空间中移位循环函数的判定问题，涵盖Hardy、Dirichlet型、完全Pick及Bergman等空间，总结其共性、差异及开放问题，并关联至扩张完备性与黎曼猜想。

Bloch空间中的平移不变子空间

Shift invariant subspaces in the Bloch space

本文研究Bloch空间中平移算子的弱星闭不变子空间，证明有界解析函数可分解为循环因子与满足持久性性质的真不变子空间生成因子，奇异内函数起关键作用，并指出与Bergman空间描述的显著差异，同时给出非循环的可逆函数，回答了九十年代初的开放问题。

径向加权Besov空间中通过弱*序列循环性实现的循环性

Cyclicity via weak$^\ast$ sequential cyclicity in Radially weighted Besov spaces

该研究针对径向加权Besov空间中循环性与弱*序列循环性的差异，给出无零点有界全纯函数循环性的对数条件，提出弱*序列循环乘子的比较原理，为非完全Pick空间下的循环性现象提供新认知。

加权 Bergman 空间上复合算子的复对称性

Complex symmetry of composition operators on weighted Bergman spaces

本文研究了加权 Bergman 空间上复合算子的复对称性，证明了复对称算子的不动点存在性，并揭示了复对称性与循环性的紧密联系，给出了非旋转椭圆自同构下算子非复对称的阶的条件。

准共形符号与投影复合算子

Quasiconformal symbols and projected composition operators

本文研究了在加权伯格曼空间上具有准共形符号的投影复合算子的可逆性，探讨了符号在不同区域内的共形性质及加权类别的影响。

A transformation rule for the index of commuting operators

In the setting of several commuting operators on a Hilbert space one defines the notions of invertibility and Fredholmness in terms of the associated Koszul complex. The index problem then consists of computing the Euler characteristic of such a special type of Fredholm complex. In this paper we investigate transformation rules for the index under the holomorphic functional calculus. We distinguish between two different types of index results: 1) A global index theorem which expresses the index in terms of the degree function of the "symbol" and the locally constant index function of the "coor…