Closable Operator

Definition

A closable operator is a linear operator between two Hilbert spaces that can be extended to a closed operator, meaning that its graph can be closed in the product space. This concept is crucial in understanding the relationship between closability and the existence of adjoint operators, as it helps to establish whether an operator can be associated with a well-defined adjoint under certain conditions.

5 Must Know Facts For Your Next Test

1. An operator is closable if there exists a closed extension of that operator, meaning you can define a closure that includes all limits of sequences from the operator's original graph.
2. The closure of a closable operator plays an essential role in ensuring that the adjoint exists and behaves well under certain mathematical operations.
3. If an operator is densely defined, it significantly influences its closability because it implies that the operator's domain is large enough to allow for closure.
4. A closable operator may have non-closed graphs, yet still be extensible to closed operators, showcasing the importance of understanding the properties of their graphs.
5. Understanding whether an operator is closable has implications in functional analysis, particularly when dealing with unbounded operators and spectral theory.

Review Questions

• How does the concept of a closable operator relate to the idea of a closed operator?
  • A closable operator relates to a closed operator in that it can be extended to become closed. Specifically, if an operator is closable, this means there exists a closed operator whose graph includes all limits of sequences from the original operator's graph. This connection highlights why understanding which operators are closable is fundamental when considering the closure properties and behavior of operators in functional analysis.
• Discuss why the density of the domain of an operator affects its closability.
  • The density of the domain plays a crucial role in determining whether an operator is closable because it ensures that every point in the Hilbert space can be approximated by points from within the domain. If an operator has a dense domain, it allows for limits of sequences derived from this domain to potentially converge within the space, thereby facilitating closure. Without this density, certain sequences may not converge to elements within the operator's range, making it impossible to extend it to a closed form.
• Evaluate how understanding closable operators impacts the existence and properties of adjoint operators in spectral theory.
  • Understanding closable operators is vital for determining the existence and properties of adjoint operators since only closable operators can reliably yield well-defined adjoints. When an operator is closable, its closure leads to an associated adjoint that preserves important characteristics such as continuity and boundedness. This connection directly influences spectral theory because many results depend on how operators relate through their adjoints; thus, recognizing whether an operator is closable lays the groundwork for deeper exploration into its spectral properties and behavior.

Related terms

Closed Operator: An operator is closed if its graph is closed in the product space of the Hilbert space with itself, indicating that the operator maps convergent sequences to convergent sequences.

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 image under the operator, providing a visual representation of the operator's action.

Adjoint Operator: An adjoint operator is an operator that represents the 'dual' action of a given linear operator, defined such that it preserves the inner product structure between Hilbert spaces.