Closable Operators

Definition

Closable operators are densely defined linear operators on a Banach space that can be extended to a closed operator. This means that if the operator's graph is not closed, it can still be made closed by appropriately defining its extension, which often involves finding a limit for sequences in the graph. Understanding closable operators is essential in functional analysis as they help in the study of operator theory, particularly in relation to closed and densely defined operators.

5 Must Know Facts For Your Next Test

1. A closable operator can be associated with a unique closed operator, which serves as its closure.
2. If an operator is closable, then it has a closure that can be studied using standard techniques from functional analysis.
3. The concept of closable operators plays a critical role in spectral theory, particularly when examining self-adjoint operators.
4. In many cases, identifying whether an operator is closable involves looking at the behavior of sequences within its graph and checking for convergence.
5. The closure of a closable operator may yield important properties like continuity and boundedness that are crucial for applications in differential equations.

Review Questions

• How does the concept of closable operators relate to the notion of a closed operator in functional analysis?
  • Closable operators are inherently connected to closed operators because every closable operator has an associated closed operator known as its closure. The main difference lies in that while closed operators have their graphs closed from the beginning, closable operators may need adjustment through their closure process to ensure that their graphs become closed. This relationship is significant since studying closable operators allows us to apply techniques applicable to closed operators.
• Discuss the implications of an operator being densely defined when considering its closability.
  • An operator being densely defined indicates that its domain is dense in the underlying Banach space, meaning that any point in that space can be approximated by points from the domain. This property is crucial for closability because it ensures that sequences converging within this domain can lead to limits that also belong to the space, which helps in determining if the operator's closure will yield a well-defined limit point. Thus, knowing an operator is densely defined gives insights into its potential to be closable.
• Evaluate how understanding closable operators enhances our ability to work with spectral properties of linear operators.
  • Understanding closable operators significantly enhances our ability to study spectral properties because they allow us to extend linear operators while maintaining desirable characteristics like boundedness and continuity. In spectral theory, many results depend on operators being closed or having a clear spectral decomposition. By recognizing whether an operator is closable, we can identify if it possesses spectral properties similar to those of self-adjoint or normal operators, enabling more profound insights into their eigenvalues and corresponding eigenvectors.

Related terms

Closed Operator: An operator is called closed if its graph is a closed subset of the product space formed by the domain and the codomain.

Densely Defined Operator: An operator is densely defined if its domain is dense in the space, meaning that every element in the space can be approximated by elements from the domain.

Graph of an Operator: The graph of an operator is the set of all pairs (x, Tx) where x is in the domain of the operator and Tx is its corresponding output.