Indeterminate Jacobi operators II

Abstract We consider the Jacobi operator ( T , D ( T )) associated with an indeterminate Hamburger moment problem, and present countable subsets S of the domain D ( T ) such that $${{\,\textrm{span}\,}}(S)$$ span ( S ) is dense in $$\ell ^2.$$ ℓ 2 . As an example we have $$S=\{(p_n(u))+B(u)(p_n(0)) \mid D(u)=0, u \ne 0\},$$ S = { ( p n ( u ) ) + B ( u ) ( p n ( 0 ) ) ∣ D ( u ) = 0 , u ≠ 0 } , where $$(p_n)$$ ( p n ) denotes the orthonormal polynomials of the moment problem and B , D are two of the Nevanlinna functions. It is also proved that sets like S are optimal in the sense that if one vector is removed, then the span is no longer dense.