AI 中文总结
本文研究 Bergman 空间上 Toeplitz 算子的超循环性，通过直接求解预解方程和绕数论证，在三对角情形下给出精确充要条件，消除了先前限制。
AI 中文摘要
我们研究了 Bergman 空间 $L_{A}^{2}(\mathbb{D})$ 上具有形式 $\Psi(z) = \gamma\bar{z}+\psi(z)$ 的符号的 Toeplitz 算子的超循环性，其中 $\gamma \in \mathbb{C} \setminus \{0\}$ 且 $\psi$ 在闭单位圆盘 $\overline{\mathbb{D}}$ 的开邻域上解析。我们的方法绕过了经典的 Hardy 空间技巧（再生核线性组合、Nevanlinna 分解），通过直接求解由 Bergman 投影产生的积分-微分预解方程。一个关键的绕数论证表明，对于反方向符号（$|\gamma| > \sup_{z \in \overline{\mathbb{D}}} |\psi'(z)|$），符号的像 $\Psi(\mathbb{D})$ 包含在 $T_\Psi$ 的点谱中。在三对角情形 $\Psi(z) = a\bar{z}+b+cz$ 中，我们通过将递推系数与符号椭圆长轴上的旋转 Favard 谱测度联系起来，完全解决了长期存在的特征向量完备性问题。结合自交换子正性，这建立了超循环性的无条件、精确的充要刻画：$T_\Psi$ 是超循环的当且仅当 $|a| > |c|$ 且 $\Psi(\mathbb{D})$ 与单位圆盘及其外部均相交，完全消除了先前文献中的 $(3+\sqrt{2})$ 限制。
英文摘要
We investigate the hypercyclicity of Toeplitz operators on the Bergman space $L_{A}^{2}(\mathbb{D})$ with symbols of the form $Ψ(z) = γ\bar{z}+ψ(z)$, where $γ\in \mathbb{C} \setminus \{0\}$ and $ψ$ is analytic on an open neighborhood of the closed unit disc $\overline{\mathbb{D}}$. Our approach bypasses the classical Hardy space techniques (reproducing kernel linear combinations, Nevanlinna factorization) by directly solving the integro-differential resolvent equation arising from the Bergman projection. A key winding number argument shows that for sense-reversing symbols ($|γ| > \sup_{z \in \overline{\mathbb{D}}} |ψ'(z)|$), the symbol's image $Ψ(\mathbb{D})$ is contained in the point spectrum of $T_Ψ$. In the tridiagonal case $Ψ(z) = a\bar{z}+b+cz$, we fully resolve the longstanding eigenvector completeness problem by linking the recurrence coefficients to a rotated Favard spectral measure on the major axis of the symbol's ellipse. Combined with self-commutator positivity, this establishes an unconditional, exact necessary and sufficient characterization of hypercyclicity: $T_Ψ$ is hypercyclic if and only if $|a| > |c|$ and $Ψ(\mathbb{D})$ intersects both the unit disc and its exterior, completely eliminating the $(3+\sqrt{2})$ restriction of previous literature.
