Closure of an operator

Definition

The closure of an operator is the smallest closed extension of that operator, encompassing all limit points of sequences generated by the operator's action on a dense subset. This concept connects deeply with closed and closable operators, as well as the behavior of unbounded operators and their adjoints. Understanding the closure helps in analyzing the properties and domains of these operators, shedding light on their continuity and boundedness within a functional analysis framework.

5 Must Know Facts For Your Next Test

1. The closure of an operator is formed by considering all limit points of sequences obtained from applying the operator to elements within its domain.
2. If an operator is closed, it is equal to its closure, meaning that it does not need further extensions to capture its limits.
3. A densely defined operator can be closed or closable, which impacts how we work with unbounded operators.
4. The closure plays a crucial role when dealing with adjoints, particularly because the adjoint of a closed operator is also closed.
5. When dealing with unbounded operators, understanding their closures helps in establishing conditions under which they are densely defined and their domains are well-defined.

Review Questions

• How does the concept of closure relate to closed and closable operators?
  • The concept of closure directly relates to whether an operator is closed or closable. If an operator is closed, then it equals its closure, meaning no additional points are needed to capture its action fully. Conversely, if an operator is closable, it has a closure that provides a closed extension of the original operator. This relationship highlights how we analyze operators in functional analysis regarding their behavior and properties in relation to limit points.
• In what ways does the closure of an unbounded operator influence its adjoint?
  • The closure of an unbounded operator significantly influences its adjoint by ensuring that if the original operator is closed, then its adjoint will also be a closed operator. When we take the closure of an unbounded operator, we can analyze its domain and establish conditions under which it maintains boundedness. This interplay between closures and adjoints becomes crucial when working with unbounded operators, especially in Hilbert spaces where inner product structures are involved.
• Evaluate how understanding the closure of an operator aids in determining the properties of densely defined operators.
  • Understanding the closure of an operator is essential for evaluating properties such as boundedness and continuity in densely defined operators. It allows us to identify whether an operator can be extended to a closed one or if it can be approximated by other well-defined operators. This knowledge aids in constructing solutions to differential equations and analyzing spectral properties within functional analysis, making it foundational for tackling more complex problems related to unbounded operators.

Related terms

Closed operator: An operator is called closed if its graph is a closed set in the product space of its domain and range.

Closability: An operator is closable if it has a closed extension, which means its closure exists and can be defined.

Adjoint operator: The adjoint of an operator is defined in relation to its domain and range, capturing how the operator interacts with inner products in Hilbert spaces.