AI 中文总结
本文构造反例证明加权Bergman空间中Beurling型定理成立的临界值为$\alpha=1$，并推广预定曲率构造，将常数改进为$1$。
AI 中文摘要
对于每个$\alpha>1$，我们构造一个有限集$A\subset\mathbb{D}$，使得基于零点的不变子空间$I_A$不满足游荡子空间性质。因此，Beurling型定理在加权Bergman空间$A^2_\alpha$上成立当且仅当$-1<\alpha\le1$，证实了Shimorin的一个猜想。作为进一步应用，我们将Hedenmalm和Perdomo的预定曲率构造推广到所有$\alpha>1$，从而将其结果中的常数$\alpha_0\approx1.04$替换为$1$。
英文摘要
For every $α>1$ we construct a finite set $A\subset\mathbb{D}$ such that the zero-based invariant subspace $I_A$ fails the wandering subspace property. Consequently, the Beurling-type theorem holds on the weighted Bergman space $A^2_α$ if and only if $-1<α\le1$, confirming a conjecture of Shimorin. As a further application, we extend the prescribed-curvature construction of Hedenmalm and Perdomo to all $α>1$, thereby replacing the constant $α_0\approx1.04$ in their result by $1$.
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