Closability of Closed Compact Linear Operators

This paper investigates the closability of closed compact linear operators on Hilbert spaces. It establishes that under certain conditions, every closed compact linear operator is closable. Additionally, it investigates the properties of closable operators and highlights their stability under limits and algebraic operations. We establish a non-closability criterion based on sequences in the domain and demonstrate that the limit of bounded compact operators need not be closable. Moreover, we examine the behavior of closability when operators are added, restricted, or composed with isometries. These results prove that closability is not always preserved under such constructions. Thus, it provides a framework for understanding its role in the analysis of closed compact operators. The analysis relies on spectral theory, the closed graph theorem, Fredholm theory, and von Neumann’s theorem. Consequently, the findings extend known properties of bounded compact operators and provide a clearer understanding of closability and the spectral behavior of closed compact linear operators.