5.1 Closed operators
7 min read•august 21, 2024
Closed operators extend the concept of continuity for linear operators, playing a crucial role in spectral theory. They form the foundation for understanding unbounded operators, essential in functional analysis and quantum mechanics.
These operators are defined on subsets of normed vector spaces, with their graphs closed in the product space. Closed operators' properties, like domain closure and graph structure, provide powerful tools for analyzing operator behavior and spectral characteristics.
Definition of closed operators
- Closed operators form a crucial concept in spectral theory, extending the notion of continuity for linear operators
- These operators play a significant role in understanding unbounded operators and their properties in functional analysis
Domain and range
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- Closed operators defined on a subset of a normed vector space
- Domain closure property requires the graph of the operator to be closed in the product space
- Range of a not necessarily closed, unlike bounded operators
- Importance of domain in determining operator properties (continuity, )
Closure of operators
- Closure of an operator extends its domain to include limit points
- Process involves taking the closure of the operator's graph in the product space
- Closed operators remain unchanged under the closure operation
- Closure operation preserves linearity but may alter other properties
Graph of closed operators
- Graphs of closed operators provide a geometric interpretation of their properties
- Understanding the graph structure aids in analyzing operator behavior and spectral characteristics
Graphical representation
- Graph of an operator T defined as the set of points (x, Tx) in the product space X × Y
- Closed operators have closed graphs in the product topology
- Vertical line test used to determine if a graph represents a function
- Importance of graph in spectral theory and functional analysis
Properties of closed graphs
- Closed graphs imply continuity for everywhere defined operators ()
- Graph closedness preserved under algebraic operations (addition, scalar multiplication)
- Relationship between graph closedness and operator boundedness
- Implications of closed graphs for spectral properties and operator domains
Closable operators
- Closable operators represent a broader class of operators that can be extended to closed operators
- Understanding closability helps in dealing with unbounded operators in functional analysis
Conditions for closability
- Densely defined operator T is closable if its adjoint T* is densely defined
- Closability equivalent to the existence of a closed extension
- Necessary and sufficient condition involves the closure of the graph
- Importance of closability in defining self-adjoint extensions
Minimal closed extension
- defined as the smallest closed operator extending the original operator
- Construction involves taking the closure of the operator's graph
- Uniqueness of the minimal closed extension for closable operators
- Applications in defining self-adjoint extensions of symmetric operators
Adjoint of closed operators
- Adjoint operators play a crucial role in spectral theory and quantum mechanics
- Understanding adjoints of closed operators extends the concept from bounded to unbounded operators
Existence of adjoint
- Adjoint of a densely defined closed operator always exists
- Construction of adjoint domain using the graph of the original operator
- Relationship between closedness of an operator and its adjoint
- Importance of in ensuring the existence of adjoint
Properties of adjoint
- Adjoint of a closed operator is closed
- Relationship between the spectrum of an operator and its adjoint
- Adjoint of sum and product of operators (under certain conditions)
- Role of adjoints in defining self-adjoint and normal operators
Closed unbounded operators
- extend the theory of bounded operators to handle important physical phenomena
- These operators are essential in quantum mechanics and partial differential equations
Examples in functional analysis
- Differential operators (d/dx) on suitable function spaces
- Multiplication operators on L^2 spaces
- in various function spaces
- Momentum and position operators in quantum mechanics
Spectral properties
- may include point, continuous, and residual parts
- and for closed unbounded operators
- Spectral theorem for self-adjoint unbounded operators
- Relationship between spectral properties and operator resolvent
Closed operators vs bounded operators
- Comparison between closed and bounded operators highlights the broader applicability of closed operators
- Understanding these differences is crucial for applying operator theory in various fields
Similarities and differences
- Bounded operators always closed, but closed operators not necessarily bounded
- Closed operators share some properties with bounded operators (graph closedness)
- Spectral theory more complex for closed unbounded operators
- Importance of domain considerations in closed operator theory
Domain considerations
- Bounded operators defined on entire space, closed operators may have proper subspace domains
- Dense domain crucial for many properties of closed operators
- Domain of closed operator may not be easily characterized
- Relationship between domain and range for closed operators
Closed operator theorems
- Fundamental theorems involving closed operators form the backbone of functional analysis
- These theorems provide powerful tools for analyzing operator properties and solving equations
Closed graph theorem
- States that a closed everywhere defined linear operator between Banach spaces is bounded
- Implications for proving boundedness of operators
- Relationship between closedness and continuity for linear operators
- Applications in proving well-posedness of differential equations
Open mapping theorem
- Surjective continuous linear operator between Banach spaces has open mapping property
- Relationship to closed graph theorem and Banach-Steinhaus theorem
- Applications in proving existence of solutions for operator equations
- Importance in functional analysis and operator theory
Applications of closed operators
- Closed operators find extensive applications in various branches of mathematics and physics
- Understanding these applications helps in appreciating the importance of closed operator theory
Differential operators
- as closed operators on suitable function spaces
- Laplacian and its powers as closed unbounded operators
- Applications in partial differential equations and spectral theory
- Relationship between differential operators and boundary value problems
Integral operators
- as closed operators under certain conditions
- and their closedness properties
- Applications in integral equations and inverse problems
- Relationship between integral and differential operators
Closed operators in Hilbert spaces
- setting provides additional structure for studying closed operators
- Many important operators in quantum mechanics are closed operators in Hilbert spaces
Self-adjoint closed operators
- Self-adjoint operators as closed operators equal to their own adjoints
- Spectral theorem for self-adjoint operators in Hilbert spaces
- Relationship between self-adjointness and real spectrum
- Importance in quantum mechanics for representing observables
Normal closed operators
- Normal operators commute with their adjoints
- Spectral theorem for normal operators in Hilbert spaces
- Relationship between normality and spectral properties
- Applications in functional calculus and operator algebras
Perturbation theory for closed operators
- Perturbation theory studies how small changes in operators affect their properties
- Understanding perturbations is crucial for analyzing stability of physical systems
Relatively bounded perturbations
- Definition of for closed operators
- for self-adjoint operators under relatively bounded perturbations
- Stability of essential spectrum under relatively compact perturbations
- Applications in quantum mechanics and scattering theory
Stability of closedness
- Conditions under which closedness is preserved under perturbations
- Small bounded perturbations preserve closedness
- Relatively bounded perturbations with small relative bound preserve closedness
- Importance in studying robustness of physical models
Numerical range of closed operators
- Numerical range provides important information about operator properties
- Extends the concept from bounded to closed unbounded operators
Definition and properties
- Numerical range defined as the set of complex numbers (Tx, x) for unit vectors x
- Convexity of numerical range (Toeplitz-Hausdorff theorem)
- Relationship between numerical range and spectrum for normal operators
- Applications in studying operator norms and spectral properties
Relation to spectrum
- Spectrum of closed operator contained in the closure of its numerical range
- Numerical range contains all eigenvalues of the operator
- Relationship between numerical range and resolvent set
- Applications in estimating spectral bounds and operator norms
Resolvent of closed operators
- Resolvent plays a crucial role in spectral theory and operator equations
- Understanding resolvent properties is essential for analyzing closed operators
Definition and properties
- Resolvent defined as (λI - T)^(-1) for λ in the resolvent set
- Analytic properties of the resolvent function
- First and second resolvent identities
- Applications in spectral theory and operator equations
Spectral mapping theorem
- Relationship between spectrum of T and spectrum of functions of T
- Statement of for closed operators
- Applications in functional calculus for closed operators
- Importance in quantum mechanics and operator algebras
Closed operators in quantum mechanics
- Quantum mechanics relies heavily on the theory of closed operators in Hilbert spaces
- Understanding these operators is crucial for formulating and solving quantum mechanical problems
Observables as closed operators
- Physical observables represented by
- Position and momentum operators as unbounded closed operators
- Hamiltonian operator as a self-adjoint closed operator
- Importance of domain considerations in defining quantum observables
Self-adjointness in physics
- Self-adjointness ensures real eigenvalues for physical observables
- Stone's theorem relating self-adjoint operators to one-parameter unitary groups
- Spectral theorem for self-adjoint operators and its physical interpretation
- Applications in quantum measurement theory and uncertainty principles
Key Terms to Review (36)
Adjoint of Closed Operators: The adjoint of a closed operator is a fundamental concept in functional analysis, referring to a specific operator that captures the behavior of the original closed operator in a dual space. The adjoint operator plays a crucial role in extending the properties of closed operators and is essential for establishing relationships between different spaces in spectral theory. Understanding the adjoint helps in analyzing the spectrum of operators and their associated eigenvalues and eigenvectors.
Adjoint operator: An adjoint operator is a linear operator that corresponds to another operator in a specific way, defined through the inner product in a Hilbert space. The adjoint of an operator captures important properties like symmetry and self-adjointness, making it essential for understanding the structure and behavior of linear operators. The concept of adjoint operators is central to various properties and classifications of operators, influencing their relationships with closed, bounded, and continuous linear operators.
Banach space: A Banach space is a complete normed vector space, meaning it is a vector space equipped with a norm that allows for the measurement of vector lengths, and every Cauchy sequence in this space converges to a limit within the space. This concept is fundamental in functional analysis as it provides a structured setting for various mathematical problems, linking closely with operators, spectra, and continuity.
Boundedness: Boundedness refers to a property of operators or functions that limits their output values within a specified range, ensuring that there exists a constant such that the operator or function does not grow indefinitely. This concept is crucial in various contexts, as it implies stability and predictability, particularly when analyzing operators in Hilbert spaces, closed operators, and symmetric operators. Understanding boundedness is key to exploring the resolvent set and determining the continuity and behavior of linear operators.
C^*: In the context of closed operators, c^* refers to the adjoint operator of a closed operator c. The adjoint operator captures important properties of the original operator and provides insights into its behavior, particularly in Hilbert spaces. Understanding c^* is crucial as it relates to the closure of operators and their spectral properties, impacting how they can be used in functional analysis.
Closable Operator: 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.
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 continuous. This theorem connects the properties of closed operators and continuous linear operators, providing a crucial link for understanding how boundedness can be inferred from the closedness of the graph in functional analysis.
Closed operator: A closed operator is a linear operator defined on a dense subset of a Hilbert space, which has the property that if a sequence of points converges in the Hilbert space and the image of that sequence under the operator also converges, then the limit is in the range of the operator. This concept is critical when discussing properties of unbounded self-adjoint operators and their adjoints, as it ensures that certain limits and continuity conditions are satisfied in functional analysis.
Closed Unbounded Operators: Closed unbounded operators are a type of linear operator defined on a dense domain within a Hilbert space that are closed in the sense that their graph is a closed set in the product space of the Hilbert space and its dual. These operators can be quite important because they often arise in quantum mechanics and partial differential equations, where they serve as generalizations of bounded operators.
Closedness: Closedness refers to a property of certain operators in functional analysis, where a given operator is considered closed if its graph is a closed subset of the product space of the domain and codomain. This concept is crucial because it relates to the well-behaved nature of operators and plays a significant role in the study of convergence and continuity in functional spaces.
Compact Operator: A compact operator is a linear operator between Banach spaces that maps bounded sets to relatively compact sets. This means that when you apply a compact operator to a bounded set, the image will not just be bounded, but its closure will also be compact, making it a powerful tool in spectral theory and functional analysis.
D(a): The notation d(a) represents the domain of a closed operator 'a' in the context of functional analysis. This concept is crucial because it helps determine the set of elements for which the operator is defined, influencing properties such as boundedness and continuity. Understanding d(a) not only aids in analyzing closed operators but also connects to the broader framework of operator theory, particularly in the study of self-adjoint operators and spectral properties.
Dense domain: A dense domain is a subset of a Hilbert space such that its closure is the entire space, meaning that every element in the space can be approximated arbitrarily closely by elements from the dense domain. This concept is crucial when dealing with unbounded self-adjoint operators and closed operators, as it ensures that these operators can act on a rich enough set of functions to produce meaningful spectral results and analytical properties.
Differential Operator: A differential operator is a mathematical operator that involves the differentiation of a function. In spectral theory, differential operators are crucial as they appear in the formulation of differential equations that describe various physical and mathematical phenomena. Understanding their properties, such as self-adjointness and closedness, helps in analyzing the spectral characteristics of linear operators, particularly in relation to boundary value problems and their solutions.
Discrete Spectrum: A discrete spectrum refers to a set of isolated eigenvalues of an operator, often associated with bounded self-adjoint operators in Hilbert spaces. This concept highlights the specific points in the spectrum where the operator has eigenvalues and relates to physical systems where these isolated points represent quantized energy levels, particularly in quantum mechanics.
Essential Spectrum: The essential spectrum of an operator is the set of points in the spectrum that cannot be isolated eigenvalues of finite multiplicity. This means it captures the 'bulk' behavior of the operator, especially in infinite-dimensional spaces, and reflects how the operator behaves under perturbations. Understanding the essential spectrum is crucial for analyzing stability and the spectral properties of various operators, especially in contexts like unbounded self-adjoint operators and perturbation theory.
First Closed Operator Theorem: The First Closed Operator Theorem is a fundamental result in functional analysis that establishes conditions under which a densely defined linear operator is closed if it is closed in the graph topology. This theorem emphasizes the importance of closed operators in the study of unbounded operators, providing essential criteria for determining their properties and behavior.
Fredholm Integral Operators: Fredholm integral operators are a special class of linear operators defined on function spaces, characterized by their representation as integral equations involving a kernel function. They play a crucial role in the study of compact operators and are closely connected to closed operators due to their boundedness and specific properties related to invertibility and the index.
Graph of a closed operator: The graph of a closed operator is the set of all pairs $(x, Tx)$ where $x$ belongs to the domain of the operator $T$, and $Tx$ is its corresponding image in the codomain. This concept helps in understanding the behavior of closed operators, particularly in relation to their continuity and the limits of sequences. The graph can be visualized as a subset in the product space formed by the domain and codomain, providing insights into the properties of the operator and its impact on function spaces.
Hilbert space: A Hilbert space is a complete inner product space that provides the framework for many areas in mathematics and physics, particularly in quantum mechanics and functional analysis. It allows for the generalization of concepts such as angles, lengths, and orthogonality to infinite-dimensional spaces, making it essential for understanding various types of operators and their spectral properties.
Kato-Rellich Theorem: The Kato-Rellich Theorem is a result in spectral theory that provides conditions under which the essential spectrum of a self-adjoint operator remains unchanged under certain perturbations. This theorem is significant in understanding how small changes in operators can affect their eigenvalues and spectra, particularly in the context of unbounded self-adjoint operators and their resolvents.
Laplacian operator: The Laplacian operator is a second-order differential operator that plays a crucial role in mathematical physics and spectral theory, defined as the divergence of the gradient of a function. It measures how much a function deviates from being constant and is widely used in problems involving heat conduction, wave propagation, and potential theory. Understanding the Laplacian operator is essential when dealing with closed operators and inequalities related to the geometry of spaces.
Minimal closed extension: A minimal closed extension of a closed operator is the smallest closed extension that contains the original operator and has the same domain. It is essential in understanding how closed operators behave and in determining their closure properties within a larger space. This concept ensures that we can analyze operators in a more extensive context without losing essential characteristics of their original domains.
Multiplication Operator: The multiplication operator is a linear operator that takes a function and multiplies it by a fixed scalar or another function, effectively transforming the input while preserving linearity. This operator plays a crucial role in the context of closed operators, as it can often generate new closed operators when applied to functions in a Hilbert space. Understanding how this operator interacts with other operators and functions is essential for grasping deeper concepts in spectral theory.
Normal Closed Operators: Normal closed operators are bounded linear operators on a Hilbert space that commute with their adjoint and have a closed graph. They combine properties of normality and closure, meaning they maintain certain mathematical structures that are critical in spectral theory. Their significance lies in their well-defined spectral properties and the ease with which they can be analyzed through functional calculus.
Numerical range of closed operators: The numerical range of a closed operator is the set of all complex numbers that can be expressed as the inner product of the operator acting on a vector with that vector itself, specifically for unit vectors. It provides insight into the behavior and properties of the operator, linking to essential aspects such as spectrum and stability. Understanding the numerical range helps in analyzing operator norms and their spectral characteristics, making it a crucial concept in spectral theory.
Operator Norm: The operator norm is a way to measure the size or 'magnitude' of a bounded linear operator on a normed space. It essentially quantifies how much the operator can stretch or shrink vectors, providing a consistent means to compare different operators. This concept connects to various important areas, including how operators behave on closed spaces, the significance of trace class operators, and the overall structure of bounded linear operators on Hilbert spaces.
Range Theorem: The Range Theorem states that for a closed operator defined on a Banach space, the range of the operator is closed if its adjoint is densely defined. This concept connects to the behavior of linear operators and their ranges, playing a critical role in understanding the properties and structure of closed operators. It provides insights into the conditions under which the image of an operator remains well-behaved, linking the operator's properties to functional analysis.
Relatively bounded perturbations: Relatively bounded perturbations refer to modifications made to an operator that do not significantly alter its essential properties, particularly regarding its behavior at infinity. This concept is crucial when analyzing the stability and spectral characteristics of operators, as it allows for the examination of how small changes influence their closedness and spectrum.
Resolvent of Closed Operators: The resolvent of a closed operator is a crucial mathematical tool that helps in analyzing the properties and behaviors of operators in functional analysis. It is defined as the operator-valued function $(A -
ho I)^{-1}$ for complex numbers $
ho$ that are not in the spectrum of the operator $A$, where $A$ is a closed operator and $I$ is the identity operator. The resolvent provides insights into the spectral properties of the operator and helps to determine eigenvalues and eigenvectors.
Self-adjoint closed operators: Self-adjoint closed operators are a special type of linear operator defined on a Hilbert space that satisfy certain properties, ensuring they are both closed and equal to their adjoint. These operators play a crucial role in spectral theory, particularly because they guarantee real eigenvalues and have well-defined spectral decompositions. Their closed nature ensures that limits of convergent sequences of their domains also lie within the domain, maintaining the integrity of their operations.
Self-adjoint operator: A self-adjoint operator is a linear operator defined on a Hilbert space that is equal to its own adjoint, meaning that it satisfies the condition $$A = A^*$$. This property ensures that the operator has real eigenvalues and a complete set of eigenfunctions, making it crucial for understanding various spectral properties and the behavior of physical systems in quantum mechanics.
Spectral Mapping Theorem: The spectral mapping theorem is a fundamental result in spectral theory that relates the spectra of a bounded linear operator and a function of that operator. Specifically, if a function is applied to an operator, the spectrum of the resulting operator can be determined from the original spectrum, highlighting how the properties of the operator transform under functional calculus. This theorem connects various concepts, including closed operators, resolvent sets, and the behavior of resolvents under perturbation.
Spectrum of closed unbounded operators: The spectrum of closed unbounded operators refers to the set of complex numbers that characterize the behavior of such operators in a Hilbert space. It provides crucial insights into the properties and structure of these operators, particularly regarding their resolvent and the associated eigenvalues. Understanding the spectrum helps in determining whether an operator is invertible and in analyzing its spectral properties, which play a vital role in various applications within functional analysis.
Sturm-Liouville Operators: Sturm-Liouville operators are a class of differential operators that arise in the study of boundary value problems, typically represented in the form $$L[y] = -(p(x)y')' + q(x)y$$. These operators play a vital role in spectral theory, particularly in understanding the properties of self-adjoint operators and their eigenvalue problems, which are directly connected to deficiency indices and the characterization of closed operators.
Volterra Integral Operators: Volterra integral operators are linear integral operators defined by an integral of the form $$(Tf)(x) = \int_{a}^{x} K(x,t) f(t) \, dt$$, where $K(x,t)$ is the kernel of the operator, $f$ is a function in a suitable function space, and $x$ varies over the interval of integration. These operators play a significant role in functional analysis and spectral theory, particularly concerning closed operators and their properties, such as compactness and continuity.