6.4 Symmetric and self-adjoint unbounded operators
3 min read•august 16, 2024
Symmetric and self- unbounded operators are crucial in operator theory. They're defined by specific properties involving inner products and domains. While all self-adjoint operators are symmetric, the reverse isn't always true.
These operators have real-valued spectra and are essential in quantum mechanics and PDEs. Self-adjoint operators, in particular, possess spectral resolutions, making them invaluable for modeling physical systems and solving complex mathematical problems.
Symmetric vs Self-Adjoint Operators
Definitions and Properties
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- Symmetric unbounded operator T on Hilbert space H satisfies ⟨Tx,y⟩ = ⟨x,Ty⟩ for all x, y in D(T)
- Adjoint T* of unbounded operator T defined on D(T*) = {y ∈ H : x ↦ ⟨Tx,y⟩ bounded on D(T)}
- Self-adjoint unbounded operator T satisfies D(T) = D(T*) and Tx = T*x for all x in D(T)
- D(T) of symmetric or self-adjoint unbounded operator forms dense linear subspace of H
- Symmetric and self-adjoint operators classified as closed operators
- Graphs represent closed subsets of H × H
- of self-adjoint unbounded operator exists as subset of real line
- Not necessarily true for symmetric operators
Key Distinctions
- Self-adjoint operators always symmetric, but symmetric operators not always self-adjoint
- Domain distinction differentiates self-adjoint and symmetric operators
- T: D(T) = D(T*)
- : D(T) ⊆ D(T*) with possible strict inclusion
- always exists for self-adjoint operators
- May not exist for symmetric operators
- (n+, n-) of symmetric operator T defined as dimensions of ker(T* ± iI)
- T self-adjoint if and only if both deficiency indices equal zero
- Symmetric operator T possesses self-adjoint extensions if deficiency indices equal
- Indices (0,0) indicate T already self-adjoint
- establishes bijective correspondence between self-adjoint extensions of symmetric operator and unitary operators between deficiency subspaces
Spectral Properties of Self-Adjoint Operators
Spectrum Characteristics
- Spectrum of self-adjoint unbounded operator T forms closed subset of real line
- Spectral theorem for self-adjoint unbounded operators
- T represented as integral T = ∫ λ dE(λ) with respect to spectral measure E
- Spectral measure E of self-adjoint operator T decomposes Hilbert space H
- Orthogonal subspaces correspond to different spectrum parts
- Spectrum components of self-adjoint operator T
- (eigenvalues)
- (non-eigenvalue points where (T - λI)^(-1) exists as unbounded operator)
- (empty for self-adjoint operators)
Essential Spectrum and Weyl's Criterion
- of T includes non-isolated eigenvalues of finite multiplicity
- characterizes essential spectrum using singular sequences
- λ in essential spectrum if sequence {xn} in D(T) exists with:
- ||xn|| = 1
- xn weakly converges to 0
- ||(T-λI)xn|| → 0
- λ in essential spectrum if sequence {xn} in D(T) exists with:
Applications of Self-Adjoint Operators
Quantum Mechanics
- Observables in quantum mechanics represented by self-adjoint operators on Hilbert space of wave functions
- Hamiltonian operator H (total energy of quantum system) exemplifies unbounded self-adjoint operator
- Time-independent Schrödinger equation Hψ = Eψ presents eigenvalue problem for self-adjoint Hamiltonian operator
- Spectral theorem for self-adjoint operators provides mathematical foundation for measurement postulate in quantum mechanics
- establishes one-to-one correspondence between self-adjoint operators and strongly continuous one-parameter unitary groups
- Crucial for describing time evolution in quantum systems
Partial Differential Equations and Sturm-Liouville Theory
- Elliptic operators (Laplacian) often realized as self-adjoint unbounded operators on appropriate function spaces
- Sturm-Liouville theory heavily relies on spectral properties of self-adjoint operators
- Deals with certain second-order differential operators
- Applications in various fields (vibrating strings, heat conduction, quantum mechanics)
Key Terms to Review (21)
Adjoint: The adjoint of an operator is a concept that captures how the operator interacts with the inner product structure of a space. Essentially, for a given linear operator, its adjoint is defined such that it satisfies a specific relationship involving inner products, reflecting a kind of symmetry in the action of the operator and its adjoint. This concept plays a crucial role when discussing symmetric and self-adjoint unbounded operators, providing insights into their properties and applications.
Cayley Transform: The Cayley Transform is a mathematical concept that relates to the transformation of operators, particularly in the context of unbounded operators. It provides a way to map a densely defined symmetric operator into a bounded operator on a Hilbert space, which is crucial for analyzing self-adjoint extensions and spectral properties of these operators.
Closed Operator: A closed operator is a linear operator defined on a subset of a Hilbert space that has the property that if a sequence of points converges in the space and the corresponding images under the operator converge, then the limit point is also in the operator's range. This concept is essential for understanding how operators behave in various contexts, including their domains and relationships with unbounded linear operators.
Continuous Spectrum: The continuous spectrum refers to the set of values (often real numbers) that an operator can take on, without any gaps, particularly in relation to its eigenvalues. This concept is crucial in distinguishing between different types of spectra, such as point and residual spectra, and plays a key role in understanding various properties of operators.
David Hilbert: David Hilbert was a prominent German mathematician who made significant contributions to various fields of mathematics, particularly in the areas of functional analysis and operator theory. His work laid the foundational principles for understanding infinite-dimensional spaces and self-adjoint operators, which are crucial in modern mathematical physics and analysis.
Deficiency indices: Deficiency indices are a pair of non-negative integers that characterize the extension properties of a symmetric operator. Specifically, they provide information on the dimensions of the deficiency spaces associated with an operator, which helps determine whether the operator can be extended to a self-adjoint operator or if it has self-adjoint extensions.
Denseness condition: The denseness condition refers to the requirement that the domain of a symmetric or self-adjoint operator must be dense in the Hilbert space it acts upon. This means that every element in the Hilbert space can be approximated arbitrarily closely by elements from the domain of the operator, ensuring that the operator has well-defined action throughout the space. This property is crucial for ensuring the validity of various results in functional analysis and operator theory.
Domain: In the context of operator theory, the domain of an operator refers to the set of elements for which the operator is defined and can be applied. Understanding the domain is crucial because it determines where the operator behaves in a well-defined manner, especially when dealing with unbounded linear operators, as they can have more complex and nuanced behaviors compared to bounded operators.
Essential spectrum: The essential spectrum of an operator refers to the set of complex numbers that can be viewed as 'limiting' points of the spectrum of the operator, representing the 'bulk' of the spectrum that is not influenced by compact perturbations. It captures the behavior of the operator at infinity and is crucial in distinguishing between discrete eigenvalues and continuous spectrum.
John von Neumann: John von Neumann was a Hungarian-American mathematician, physicist, and computer scientist who made significant contributions to various fields, including operator theory, quantum mechanics, and game theory. His work laid the foundation for much of modern mathematics and theoretical physics, particularly in the context of functional analysis and the mathematical formulation of quantum mechanics.
Operator Norms: Operator norms are a way to measure the 'size' or 'magnitude' of a bounded linear operator between normed spaces. They help quantify how much an operator can stretch or compress vectors from one space to another. Understanding operator norms is crucial when working with adjoint operators and symmetric or self-adjoint unbounded operators, as they provide insight into the behavior of these operators in terms of stability and convergence.
Point Spectrum: The point spectrum of an operator consists of the set of eigenvalues for that operator, specifically those values for which the operator does not have a bounded inverse. These eigenvalues are significant as they correspond to vectors in the Hilbert space that are annihilated by the operator minus the eigenvalue times the identity operator.
Range: The range of a linear operator is the set of all possible outputs that can be produced by applying the operator to all inputs in its domain. This concept is crucial for understanding how operators behave and interact with functions, as it reflects the extent to which an operator can transform elements from one space to another. The range provides insights into properties such as injectivity and surjectivity, which are important in defining bounded operators, analyzing polar decompositions, and understanding symmetric and self-adjoint unbounded operators.
Residual Spectrum: The residual spectrum of a bounded linear operator consists of those points in the spectrum that are not eigenvalues and do not belong to the point spectrum. It represents the part of the spectrum where the operator fails to be invertible but has a non-empty resolvent set. Understanding this spectrum is essential when analyzing operators, especially in distinguishing between different types of spectra and their implications.
Self-adjoint operator: A self-adjoint operator is a linear operator on a Hilbert space that is equal to its own adjoint. This property ensures that the operator has real eigenvalues and allows for various important results in functional analysis and quantum mechanics. Self-adjoint operators have deep connections with spectral theory, stability, and physical observables.
Self-adjointness condition: The self-adjointness condition refers to a property of certain linear operators where the operator is equal to its adjoint, meaning that for a given operator \( A \), it holds that \( A = A^* \). This condition is essential in understanding the behavior of unbounded operators in Hilbert spaces, as it guarantees real eigenvalues and a complete set of eigenfunctions, which are crucial for applications in quantum mechanics and differential equations.
Spectral resolution: Spectral resolution refers to the process of decomposing a self-adjoint operator into a family of projections associated with its eigenvalues and eigenvectors. This concept is crucial for understanding how operators can be analyzed in terms of their spectra, allowing one to study the properties and behavior of symmetric and self-adjoint unbounded operators more effectively. It forms a fundamental link between the abstract theory of operators and their applications in functional analysis.
Spectrum: In operator theory, the spectrum of an operator refers to the set of values (complex numbers) for which the operator does not have a bounded inverse. It provides important insights into the behavior of the operator, revealing characteristics such as eigenvalues, stability, and compactness. Understanding the spectrum helps connect various concepts in functional analysis, particularly in relation to eigenvalues and the behavior of compact and self-adjoint operators.
Stone's Theorem: Stone's Theorem is a fundamental result in functional analysis that provides a framework for understanding the spectral properties of self-adjoint operators through functional calculus. It essentially states that any bounded self-adjoint operator can be represented via continuous functions on its spectrum, allowing us to extend the notion of functions acting on operators. This theorem is crucial for dealing with both bounded and unbounded self-adjoint operators, especially when considering their spectral characteristics.
Symmetric operator: A symmetric operator is a type of linear operator defined on a domain in a Hilbert space that satisfies the condition \( \langle Ax, y \rangle = \langle x, Ay \rangle \) for all elements \( x \) and \( y \) in its domain. This property ensures that the operator is self-adjoint when it is equal to its adjoint and plays a critical role in the study of unbounded linear operators and their spectral properties.
Weyl's Criterion: Weyl's Criterion is a fundamental result in operator theory that provides a characterization of the spectrum of self-adjoint operators. Specifically, it states that a bounded linear operator on a Hilbert space is self-adjoint if and only if its spectral measure is concentrated on the real line. This criterion plays a critical role in understanding the properties of symmetric and self-adjoint unbounded operators, as it relates to the nature of their spectrum and the existence of eigenvalues.