🎭Operator Theory Unit 2 – Spectrum of Linear Operators

Key Concepts and Definitions

• Linear operators map elements from one vector space to another while preserving linear combinations
• The spectrum of a linear operator $T$ is the set of all complex numbers $\lambda$ for which the operator $T - \lambda I$ is not invertible
• Resolvent set consists of all complex numbers $\lambda$ for which $T - \lambda I$ is invertible
  • The resolvent operator $R(\lambda, T) = (T - \lambda I)^{-1}$ exists for all $\lambda$ in the resolvent set
• Spectral radius $r(T)$ is the supremum of the absolute values of the elements in the spectrum of $T$
• Compact operators have the property that the image of any bounded set is relatively compact
• Self-adjoint operators satisfy $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y$ in the Hilbert space
• Normal operators commute with their adjoint, i.e., $TT^* = T^*T$

Types of Linear Operators

• Bounded operators have a finite operator norm, i.e., $\|T\| = \sup\{\|Tx\| : \|x\| = 1\} < \infty$
• Unbounded operators have an infinite operator norm and are defined on a dense subspace of the Hilbert space
• Closed operators have the property that if a sequence $\{x_n\}$ converges to $x$ and $\{Tx_n\}$ converges to $y$, then $Tx = y$
• Symmetric operators satisfy $\langle Tx, y \rangle = \langle x, Ty \rangle$ for all $x, y$ in the domain of $T$
  • Self-adjoint operators are symmetric operators that are defined on a dense domain and have no proper symmetric extensions
• Positive operators satisfy $\langle Tx, x \rangle \geq 0$ for all $x$ in the Hilbert space
• Unitary operators preserve inner products, i.e., $\langle Ux, Uy \rangle = \langle x, y \rangle$ for all $x, y$ in the Hilbert space
• Projection operators satisfy $P^2 = P$ and are self-adjoint

Spectral Theory Basics

• The spectrum of a linear operator can be divided into three parts: point spectrum, continuous spectrum, and residual spectrum
• Point spectrum (discrete spectrum) consists of eigenvalues, which are complex numbers $\lambda$ for which there exists a non-zero vector $x$ such that $Tx = \lambda x$
• Continuous spectrum consists of complex numbers $\lambda$ for which $T - \lambda I$ is not invertible, but the range of $T - \lambda I$ is dense in the Hilbert space
• Residual spectrum consists of complex numbers $\lambda$ for which $T - \lambda I$ is not invertible, and the range of $T - \lambda I$ is not dense in the Hilbert space
• The spectral theorem states that every normal operator on a Hilbert space has a unique spectral decomposition
• Spectral mapping theorem relates the spectrum of a function of an operator to the function applied to the spectrum of the operator
• Functional calculus allows the definition of functions of operators using the spectral decomposition

Continuous and Discrete Spectra

• Continuous spectrum arises when the operator has no eigenvalues, but the resolvent operator is not continuous
  • Examples include the position operator and the momentum operator in quantum mechanics
• Discrete spectrum consists of isolated eigenvalues with finite multiplicities
  • Compact self-adjoint operators on a Hilbert space have a purely discrete spectrum
• Absolutely continuous spectrum is a subset of the continuous spectrum and is related to the Lebesgue decomposition of measures
• Singular continuous spectrum is a subset of the continuous spectrum that is not absolutely continuous
• Mixed spectrum occurs when an operator has both continuous and discrete parts in its spectrum
• The spectral measure associated with a self-adjoint operator can be decomposed into discrete, absolutely continuous, and singular continuous parts

Eigenvalues and Eigenvectors

• Eigenvalues are complex numbers $\lambda$ for which there exists a non-zero vector $x$ (eigenvector) such that $Tx = \lambda x$
• Eigenvectors corresponding to distinct eigenvalues are linearly independent
• The set of all eigenvectors corresponding to an eigenvalue $\lambda$, along with the zero vector, forms the eigenspace of $\lambda$
  • The dimension of the eigenspace is called the geometric multiplicity of the eigenvalue
• The algebraic multiplicity of an eigenvalue is the order of the root $\lambda$ in the characteristic polynomial of the operator
• For self-adjoint operators, the geometric and algebraic multiplicities of each eigenvalue are equal
• Eigenvectors of a self-adjoint operator corresponding to different eigenvalues are orthogonal
• Spectral radius formula: $r(T) = \lim_{n \to \infty} \|T^n\|^{1/n}$, where $r(T)$ is the spectral radius of $T$

Spectral Decomposition

• The spectral theorem states that every normal operator on a Hilbert space has a unique spectral decomposition
  • For a self-adjoint operator $T$, there exists a unique resolution of the identity $\{E(\lambda)\}$ such that $T = \int_{\mathbb{R}} \lambda dE(\lambda)$
• The spectral decomposition of a compact self-adjoint operator $T$ is given by $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$, where $\{\lambda_n\}$ are the eigenvalues and $\{e_n\}$ are the corresponding orthonormal eigenvectors
• Functional calculus allows the definition of functions of operators using the spectral decomposition
  • For a self-adjoint operator $T$ and a measurable function $f$, $f(T) = \int_{\mathbb{R}} f(\lambda) dE(\lambda)$
• Spectral decomposition can be used to solve operator equations and to study the behavior of operators under perturbations
• The spectral decomposition of unitary operators involves a resolution of the identity on the unit circle
• Spectral decomposition is a powerful tool for diagonalizing matrices and operators, which simplifies many calculations

Applications in Quantum Mechanics

• In quantum mechanics, observables are represented by self-adjoint operators on a Hilbert space
• The spectrum of an observable corresponds to the possible outcomes of a measurement of that observable
  • Eigenvalues represent the discrete outcomes, while the continuous spectrum represents a range of possible outcomes
• The spectral decomposition of an observable is closely related to the measurement process and the collapse of the wave function
• The time evolution of a quantum system is described by a unitary operator, which can be expressed using the spectral decomposition of the Hamiltonian
• The spectral theorem is crucial for understanding the properties of quantum systems and the interpretation of quantum mechanics
• Quantum mechanics relies heavily on the theory of linear operators and their spectra, making it an essential application of spectral theory

Advanced Topics and Open Problems

• Spectral theory can be extended to unbounded operators, which arise naturally in quantum mechanics (e.g., position and momentum operators)
• The study of pseudospectra provides insight into the behavior of operators under perturbations and the stability of numerical algorithms
• Spectral theory for non-self-adjoint operators is an active area of research, with applications in non-Hermitian quantum mechanics and PT-symmetric systems
• The spectral theory of random matrices has found applications in various fields, including quantum chaos and number theory
• Spectral theory on Banach spaces and more general topological vector spaces is an ongoing area of research
• The relationship between the spectrum of an operator and its numerical range is an important topic, with the famous Toeplitz-Hausdorff theorem as a key result
• The study of spectral theory for operators on manifolds and in non-commutative geometry is an active area of research, with connections to physics and geometry
• Open problems in spectral theory include the characterization of spectra for specific classes of operators, the study of spectral properties under various perturbations, and the development of efficient numerical methods for computing spectra