6.2 Closed operators and closable operators
4 min readâĒaugust 16, 2024
Unbounded operators can be tricky, but closed and closable operators help us make sense of them. These special types let us extend familiar ideas about bounded operators to the wild world of unbounded ones.
Closed operators have graphs that are closed sets, while closable ones can be extended to closed operators. This distinction is key for working with unbounded operators in quantum mechanics and other areas of math and physics.
Closed Operators in Hilbert Spaces
Definition and Properties
Top images from around the web for Definition and Properties
On the Spectrum of Hilbert Matrix Operator | Integral Equations and Operator Theory View original
Is this image relevant?
Adjoint Representation [The Physics Travel Guide] View original
Is this image relevant?
Hilbert space - Wikipedia View original
Is this image relevant?
On the Spectrum of Hilbert Matrix Operator | Integral Equations and Operator Theory View original
Is this image relevant?
Adjoint Representation [The Physics Travel Guide] View original
Is this image relevant?
1 of 3
Top images from around the web for Definition and Properties
On the Spectrum of Hilbert Matrix Operator | Integral Equations and Operator Theory View original
Is this image relevant?
Adjoint Representation [The Physics Travel Guide] View original
Is this image relevant?
Hilbert space - Wikipedia View original
Is this image relevant?
On the Spectrum of Hilbert Matrix Operator | Integral Equations and Operator Theory View original
Is this image relevant?
Adjoint Representation [The Physics Travel Guide] View original
Is this image relevant?
1 of 3
- Closed operator T on Hilbert space H constitutes a with a closed graph in the product space H Ã H
- Graph of operator T encompasses the set {(x, Tx) : x â dom(T)}, where dom(T) represents T's domain
- Closed operators extend the concept of continuous linear operators to unbounded operators
- Bounded closed operators occur if and only if their domain spans the entire Hilbert space
- Identity operator and zero operator serve as examples of closed operators on any Hilbert space (identity maps each element to itself, zero maps all elements to zero)
- Closed operators exhibit stability properties (sum of two closed operators with identical domains remains closed)
- Adjoint of a densely defined closed operator maintains closure
Importance and Applications
- Closed operators play a crucial role in functional analysis and quantum mechanics
- Allow for rigorous treatment of unbounded operators in physical theories
- Provide a framework for studying spectral properties of operators
- Enable the formulation of theorems on operator extensions and restrictions
- Facilitate the analysis of differential operators in partial differential equations
- Contribute to the development of perturbation theory for linear operators
- Support the study of semigroups of operators in evolution equations
Closable Operators vs Closed Operators
Defining Closable Operators
- possesses a closed extension, allowing its graph to extend to a closed operator's graph
- Operator T qualifies as closable if and only if its graph's closure represents the graph of some operator, termed T's closure
- Every closed operator maintains closability, but not all closable operators achieve closure
- Domain of a closable operator consistently forms a proper subset of its closure's domain
- Closable operators characterized by the property: if a sequence in the domain converges to zero and its image converges, the image's limit must equal zero
- Sum and composition of closable operators may not preserve closability, highlighting the property's delicate nature
- Understanding closable operators proves crucial for extending an operator's domain while preserving essential properties
Comparing Closed and Closable Operators
- Closed operators represent a subset of closable operators
- Closable operators offer more flexibility in domain extension compared to closed operators
- Closed operators maintain stability under certain operations (addition, composition) that closable operators may not
- Closable operators can be "completed" to form closed operators through the closure process
- Closed operators directly satisfy certain continuity properties, while closable operators may require extension
- Both types of operators play important roles in functional analysis and operator theory
- Understanding the distinction aids in classifying and analyzing unbounded operators in various mathematical contexts
Identifying Closed or Closable Operators
Verification Techniques
- Determine closed nature of unbounded operator T by verifying: for any sequence {xn} in dom(T) with xn â x and Txn â y, x â dom(T) and Tx = y hold true
- Check closability condition: xn â 0 and Txn â y implies y = 0 for any sequence {xn} in dom(T)
- Examine operator's domain and range to identify potential closedness or closability issues
- Utilize operator-adjoint relationship: T proves closable if T* maintains dense definition
- Consider operator's specific properties (symmetry, self-adjointness) for insights into closed or closable nature
- Investigate operator behavior on domain boundary to detect potential closedness issues
- Employ counterexamples to disprove closedness or closability when applicable
Examples and Applications
- Differential operators on suitable function spaces often exhibit closedness or closability
- Multiplication operators in L2 spaces can be analyzed for closedness based on the multiplier function's properties
- Integral operators may be closable but not closed, depending on the kernel function
- Symmetric operators in quantum mechanics are often closable but may not be self-adjoint
- Laplacian operator on smooth functions with compact support proves closable but not closed in L2 spaces
- Momentum operator in quantum mechanics exemplifies a closed unbounded operator
- Closure analysis of the time derivative operator in evolution equations reveals important physical properties
Closure of Closable Operators
Properties and Construction
- Closure of closable operator T, denoted TĖ, represents the smallest closed extension of T
- TĖ's domain encompasses all x â H for which a sequence {xn} in dom(T) exists where xn â x and {Txn} converges in H
- Closure of a closable operator preserves important properties (linearity, boundedness on domain) of the original operator
- Closure study can reveal information about the operator's spectrum and resolvent
- Closure of a symmetric operator always maintains symmetry but may not achieve self-adjointness
- Understanding operator closure proves crucial in studying unbounded operators in quantum mechanics and mathematical physics
- Closure process can sometimes uncover hidden symmetries or conservation laws in physical systems
Applications and Implications
- Closure analysis aids in extending partially defined operators to maximal domains
- Studying closure helps in understanding the behavior of unbounded operators in limit processes
- Closure properties impact the spectral theory of unbounded operators
- In quantum mechanics, closure analysis relates to the completeness of observables
- Closure examination proves useful in determining essential self-adjointness of symmetric operators
- Understanding closure aids in the formulation of domain problems for differential operators
- Closure analysis contributes to the study of operator semigroups and their generators
Key Terms to Review (16)
Application in differential equations: Application in differential equations refers to the use of differential equations to model real-world phenomena and solve problems related to change, motion, and dynamic systems. This concept is crucial as it connects theoretical mathematics with practical scenarios in fields such as physics, engineering, biology, and economics, enabling the analysis of systems governed by rates of change.
Application in Quantum Mechanics: Application in quantum mechanics refers to the use of mathematical frameworks, specifically operator theory, to model and analyze quantum systems. This concept is critical for understanding how operators, such as observables and states, interact in a quantum environment. The properties of these operators, including their spectrum and closure, directly influence the behavior and predictions of quantum mechanical systems.
Banach space: A Banach space is a complete normed vector space, meaning that it is a vector space equipped with a norm that allows for the measurement of vector lengths and distances, and every Cauchy sequence in the space converges to a limit within that space. This concept is fundamental in functional analysis as it provides a framework for studying various operators and their properties in a structured way.
Bounded closed operator: A bounded closed operator is a type of linear operator defined between two Banach spaces that is both bounded and closed. This means that it satisfies the condition of being continuous (bounded) and has the property that its graph is closed in the product space of the two Banach spaces. Understanding this term is crucial as it links the concepts of operator continuity and the behavior of sequences within functional analysis.
Characterization of closability: Characterization of closability refers to the criteria used to determine whether a densely defined operator can be closed, meaning its graph is a closed subset of the product space. This concept is vital in understanding the behavior of operators, as closed operators have properties that allow for better analysis and solution methods in functional analysis.
Closable Operator: A closable operator is a type of linear operator that can be extended to a closed operator on a Hilbert space or a Banach space. This means that there exists a closed extension of the operator, which allows for the preservation of certain properties such as continuity and boundedness. Understanding closable operators is important in operator theory as they relate to the broader concepts of closed operators, densely defined operators, and the extensions of linear maps.
Closed Graph Theorem: The Closed Graph Theorem states that if a linear operator between two Banach spaces has a closed graph, then the operator is bounded. This theorem is essential because it provides a connection between the topological property of the graph of an operator and the algebraic property of boundedness. Understanding this relationship helps in analyzing linear operators and determining whether they are continuous or not, especially when dealing with closed and closable operators.
Closed range: The closed range of an operator is the set of all output values that can be achieved by applying the operator to its domain, and this set is closed in the target space. This property is essential in understanding how operators behave, especially when considering their continuity and boundedness. Closed range provides insights into the structure of operators, particularly in relation to their adjoints and the conditions under which they can be classified as Fredholm operators.
Closure of an operator: The closure of an operator is the smallest closed extension of that operator, meaning it includes all the limit points of sequences in the operator's domain. This concept is essential in understanding the behavior of operators in Hilbert and Banach spaces, particularly when dealing with unbounded operators. The closure can provide insight into whether an operator can be extended to a larger domain while still preserving closedness.
Continuity of closed operators: Continuity of closed operators refers to the property that ensures the continuity of the operator when it is defined on a closed subset of a Banach space. This concept is essential for understanding how operators behave with respect to convergence, particularly in functional analysis, where closed operators play a crucial role in the study of linear mappings between spaces.
Difference between closed and closable operators: Closed operators are linear operators defined on a dense domain in a Hilbert space, where the graph of the operator is closed in the product space of the Hilbert space. In contrast, closable operators are those that may not be closed but possess a closure that is also a linear operator, allowing them to be extended to closed operators. Understanding these distinctions is essential as they relate to the behavior and properties of operators in functional analysis.
Graph of a closed operator: The graph of a closed operator is the set of all pairs $(x, Tx)$ where $x$ is in the domain of the operator $T$, and $Tx$ is the corresponding output in the codomain. This concept connects directly to closed operators as it encapsulates both the input and output in a structured manner, revealing properties like continuity and limiting behavior. Understanding the graph helps in recognizing whether an operator is closed or not, which is essential for further analysis in functional analysis.
Hahn-Banach Theorem: The Hahn-Banach Theorem is a fundamental result in functional analysis that allows the extension of bounded linear functionals defined on a subspace of a normed vector space to the entire space, without increasing their norm. This theorem plays a crucial role in connecting linear operators with bounded linear operators and serves as a foundation for understanding the behavior of unbounded linear operators and their domains, as well as closed and closable operators.
Linear operator: A linear operator is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that if you take any two vectors and add them, then apply the operator, it's the same as applying the operator to each vector individually and then adding the results. Understanding linear operators is crucial because they form the backbone of many concepts in functional analysis, especially in relation to closed and closable operators, as well as their applications in differential equations.
Necessity of continuity for closed operators: The necessity of continuity for closed operators refers to the requirement that a closed operator must be continuous when defined on a Banach space. This property is crucial because it ensures that the operator preserves limits of sequences, which is essential for various applications in functional analysis. Without this continuity, the operator might not behave predictably, undermining its effectiveness in solving equations and representing physical systems.
Unbounded Closed Operator: An unbounded closed operator is a linear operator that is defined on a dense domain in a Hilbert or Banach space and is closed in the sense that its graph is a closed subset of the product space. This means that if a sequence of points in the domain converges to a point, and the images under the operator converge to some limit, then this limit must also lie in the operator's range. Understanding these operators is crucial when discussing the extension of operators and their spectral properties.