Differentiation, implicit functions, and applications to generalized well-posedness

Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces

Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.

On the domain of implicit functions in a projective limit setting without additional norm estimates

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain $D$ which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse functions theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

Hypercyclic differentiation operators

A classical theorem due to G.D. Birkhoff states that there exists an entire function whose translates approximate any given entire function, as accurately as desired, over any ball of the complex plane. We show this result may be generalized to the space of entire functions of compact bounded type defined on a Banach space with separable dual.

广义复对称复合算子及其应用

Generalized complex symmetric composition operators with applications

本文刻画了多圆盘和单位圆盘上加权复合-微分算子的复对称性与自伴性条件，研究了广义加权复合微分算子的复对称结构，并探讨了复合算子Berezin值域的凸性及几何解释。

Hardy空间$H^2(\mathbb{D}^n)$上的广义加权复合算子

Generalized weighted composition operators on Hardy space $H^2(\mathbb{D}^n)$

本文研究多圆盘Hardy空间上广义加权复合算子的复对称性，给出Hermitian性与正规性的显式条件，并确定算子及其伴随的核。

加权Bergman空间上的广义加权复合微分算子

Generalized weighted composition-differentiation operators on weighted Bergman spaces

本研究在加权Bergman空间$\mathcal{A}^2_α(\mathbb{D})$中探究广义加权复合微分算子$L_{n, ψ, ϕ}$的复对称性，推导其$C_{μ,η}$-对称的充要条件，得到算子为埃尔米特和正规的显式条件，并分析该对称下算子的谱性质与伴随算子的核。