不定雅可比算子
Indeterminate Jacobi operators
AI总结：
本文研究不定Hamburger矩问题对应的亏指数(1,1)对称雅可比算子，刻画其定义域及自伴扩张定义域中正交多项式与第二类多项式序列线性组合的归属，并证明算子定义域是一族显式有界算子的公共值域。
摘要：
我们研究与不定Hamburger矩问题相关的雅可比算子$(T,D(T))$，即$\u2113^2$中由作用在仅有限项非零的复序列子空间上的雅可比矩阵取闭包所定义的算子。众所周知，该算子是亏指数为$(1,1)$的对称算子。对复数$z$，设$\u200b\bmath{p}_z, \bmath{q}_z$分别表示对应于正交多项式$p_n$和第二类多项式$q_n$的平方可和序列$(p_n(z))$与$(q_n(z))$。我们确定了复数$u,v$对应的$\bmath{p}_u,\bmath{p}_v,\bmath{q}_u,\bmath{q}_v$的线性组合是否属于$D(T)$，或属于$T$在$\u2113^2$中自伴扩张的定义域。结果依赖于与该矩问题相关的四个二元Nevanlinna函数。我们还证明了$D(T)$是一族显式构造的$\u2113^2$上有界算子的公共值域。
英文摘要：
We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmetric with deficiency indices (1,1). For a complex number z let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences (p_n(z)) and (q_n(z)) corresponding to the orthonormal polynomials p_n and polynomials q_n of the second kind. We determine whether linear combinations of $\mathfrak{p}_u,\mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for complex u,v belong to D(T) or to the domain of the self-adjoint extensions of T in $\ell^2$. The results depend on the four Nevanlinna functions of two variables associated with the moment problem. We also show that D(T) is the common range of an explicitly constructed family of bounded operators on $\ell^2$.
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