An extension of an operator is a new operator that acts on a larger domain while agreeing with the original operator on its initial domain. This concept is crucial because it allows us to expand the reach of operators, making them applicable in broader contexts, which is especially important when dealing with closed and closable operators. Understanding extensions helps in analyzing the continuity and boundedness properties of operators within functional analysis.
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An extension of an operator may be unique if certain conditions are met, particularly when dealing with closed operators.
The existence of extensions often hinges on properties like density and continuity in the context of Hilbert or Banach spaces.
An operator can have multiple extensions; however, not all extensions retain the same properties as the original operator.
In many cases, an extension must still respect the original operator's boundedness or continuity to maintain functional relevance.
Understanding extensions is key in applications such as differential operators, where extending their domains can lead to well-posed problems.
Review Questions
How does the concept of extending an operator enhance our understanding of closed and closable operators?
Extending an operator helps illustrate how we can work with closed and closable operators in a more flexible manner. Closed operators maintain their properties under extension, allowing for new domains where they can still function correctly. For closable operators, finding extensions often leads to defining them as closed, which simplifies analysis and provides deeper insights into their behavior across different settings.
Discuss the implications of having multiple extensions for a single operator in relation to its boundedness and continuity.
When an operator has multiple extensions, it's essential to consider how these extensions might differ in terms of boundedness and continuity. Not every extension will preserve these properties; therefore, identifying which extensions maintain boundedness becomes crucial for ensuring that the operator remains useful in applied contexts. The ability to classify these extensions influences how we apply operators in various mathematical problems.
Evaluate the role of density conditions in determining whether an extension of an operator can be closed or not.
Density conditions are pivotal in establishing whether an extension of an operator can be closed because they relate to the behavior of sequences within the domain. If the original domain is dense in a larger space, it provides a pathway for sequences converging under the original operator to reflect closure in its extension. This relationship helps determine if an extended operator will continue to function effectively without losing the original's essential characteristics, thus influencing its practical application in functional analysis.
An operator whose graph is a closed set in the product space of the domain and codomain, ensuring the limit of converging sequences remains within the operator's mapping.
Closability: The property of an operator that allows it to be extended to a closed operator, meaning it can be redefined over a larger domain without losing its essential characteristics.