Indeterminate Jacobi operators
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866911193263767552 |
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| author | Berg, Christian Szwarc, Ryszard |
| author_facet | Berg, Christian Szwarc, Ryszard |
| contents | 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 $\ell^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_00586 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Indeterminate Jacobi operators Berg, Christian Szwarc, Ryszard Functional Analysis Complex Variables 47B25, 47B36, 44A60 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 $\ell^2$. |
| title | Indeterminate Jacobi operators |
| topic | Functional Analysis Complex Variables 47B25, 47B36, 44A60 |
| url | https://arxiv.org/abs/2301.00586 |