Estimates of Approximation Numbers, Nuclearity of the Resolvent of a Third‐Order Singular Differential Operator, and the Completeness of Its Root Vectors

ABSTRACT In this paper, we examine a third‐order singular differential operator in the space , with a coefficient which is a continuous function on and exhibits potential growth at infinity. We prove the existence of the operator's resolvent and establish the maximal regularity of the solutions to the equation associated with the operator. We also prove the compactness property of the resolvent and obtain two‐sided bounds for the singular numbers ( s ‐numbers). These bounds provide the rate at which the resolvent of the operator under consideration can be approximated by finite‐dimensional linear operators. To the best of our knowledge, we establish for the first time the nuclearity of the resolvent for a third‐order differential operator and demonstrate the completeness of the operator's root vectors in the case of an unbounded domain with a rapidly growing coefficient at infinity.