Abstract
Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.
Highlights
In operator theory, the study of commuting operators remains an important topic [2]
Let A and B be symmetric operators in a Hilbert space H and let D be a dense linear manifold in H such that D is contained in the domain of A, B, AB and BA and such that ABx = BAx for all x ∈ D
We mainly study the commutativity of Jacobi operators, i. e. the operators generated by in nite Jacobi matrices
Summary
The study of commuting operators remains an important topic [2]. We recall that two self-adjoint (possibly unbounded) operators (or, more generally, two normal operators) are said to commute if their spectral projections commute. Some necessary and su cient conditions for self-adjointness of Jacobi operators are studied. He proved the following su cient conditions for the commutativity of two operators: Theorem, Nelson .
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