标题： 算子的收敛性及其亏指数$(k,k)$的自伴扩张 显示英文标题

标题： Convergence of operators with deficiency indices $(k,k)$ and of their self-adjoint extensions

摘要： 我们考虑一个可分希尔伯特空间 $\mathcal{H}$上的闭对称算子序列 $\{A_n\}_{n=1}^\infty$。 假设所有 $A_n$都有相等的亏指数 $(k,k)$，因此自伴扩张 $\{B_n\}_{n=1}^\infty$存在，并根据冯·诺依曼扩张理论由 $\mathcal{H}$上的部分等距算子 $\{U_n\}_{n=1}^\infty$参数化。 在对$A_n$的两种不同收敛性假设下，我们给出了$B_n$的强预解收敛与$U_n$的强收敛之间的精确联系。 显示英文摘要

摘要： We consider an abstract sequence $\{A_n\}_{n=1}^\infty$ of closed symmetric operators on a separable Hilbert space $\mathcal{H}$. It is assumed that all $A_n$'s have equal deficiency indices $(k,k)$ and thus self adjoint extensions $\{B_n\}_{n=1}^\infty$ exist and are parametrized by partial isometries $\{U_n\}_{n=1}^\infty$ on $\mathcal{H}$ according to von Neumann's extension theory. Under two different convergence assumptions on the $A_n$'s we give the precise connection between strong resolvent convergence of the $B_n$'s and strong convergence of the $U_n$'s.