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Indeterminate Jacobi operators | | Christian Berg
; Ryszard Szwarc
; | | Date: |
2 Jan 2023 | | Abstract: | 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 $ell^2$. | | Source: | arXiv, 2301.00586 | | Services: | Forum | Review | PDF | Favorites | |
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