2023/01/02 by Christian Berg, Berg, Christian, Ryszard Szwarc +1 · 1 citation
Mathematics · Physics and Astronomy · #44A60 #47B25 #47B36 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical functions and polynomials #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2301.00586
openalex publication_date 2023/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in ℓ2 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 \mathfrakpz, \mathfrakqz denote the square summable sequences (pn(z)) and (qn(z)) corresponding to the orthonormal polynomials pn and polynomials qn of the second kind. We determine whether linear combinations of \mathfrakpu,\mathfrakpv,\mathfrakqu,\mathfrakqv for complex u,v belong to D(T) or to the domain of the self-adjoint extensions of T in ℓ2. 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 ℓ2.