AI 中文总结
该研究在希尔伯特空间上构造反例，否定了逆生成元问题，证明了克兰克-尼科尔森格式在算子范数下的不稳定性，反例源于有限维构造及绍德尔乘子方法。
AI 中文摘要
我们在希尔伯特空间上给出了逆生成元问题的否定解。更确切地说，我们在希尔伯特空间$H$上构造了一个具有稠密值域的有界算子$A$，它生成一个有界、强稳定的$C_0$-半群，而$A^{-1}$不生成$C_0$-半群。我们还构造了一个指数稳定生成元$A$，满足$0\in\rho(A)$，使得其逆半群无界且至少按双对数增长。对于该生成元，每个凯莱变换都满足普通克雷斯（Kreiss）预解条件，但既不是强克雷斯有界的，也不是幂有界的；其幂满足双对数下界。因此，对于任意固定步长的长时间计算，以及在任意固定最终时刻的网格细化下，克兰克-尼科尔森（Crank--Nicolson）格式在算子范数下均不稳定。我们的反例源于一个共同的有限维构造：对于$\alpha\in(0,1)$，我们利用$\mathbb{C}^{2n}$的显式基，其部分和投影是一致有界的，且其无条件常数与$n^\alpha$相当；随后，我们利用模按双指数衰减的特征值序列，将反例对应的矩阵构造为关于这些基的绍德尔（Schauder）乘子。
英文摘要
We give a negative solution to the inverse generator problem on Hilbert spaces. More precisely, we construct a bounded operator $A$ with dense range on a Hilbert space $H$ that generates a bounded, strongly stable $C_0$-semigroup, while $A^{-1}$ does not generate a $C_0$-semigroup. We also construct an exponentially stable generator $A$ with $0 \in ρ(A)$ such that the inverse semigroup is unbounded and grows at least double logarithmically. For the latter generator, every Cayley transform satisfies the ordinary Kreiss resolvent condition but is neither strongly Kreiss bounded nor power bounded. Moreover, its powers satisfy a doubly logarithmic lower bound. Therefore, the Crank--Nicolson scheme is unstable in operator norm both for every fixed step size over long times and under mesh refinement at any fixed final time. Our counterexamples are deduced from a common finite-dimensional construction. For $α\in(0,1)$, we use explicit bases of $\mathbb C^{2n}$ whose partial-sum projections are uniformly bounded and whose unconditionality constants are comparable to $n^α$. The matrices underlying the counterexamples are then obtained as Schauder multipliers with respect to these bases, using a sequence of eigenvalues whose moduli decay doubly exponentially.
Comments25 pages, added Theorem 1.6 and a Lean 4 certificate
