functional analysis unit 8 study guides

Key Concepts and Definitions

• Spectrum of a bounded linear operator $T$ on a Banach space $X$ denoted as $\sigma(T)$ consists of all $\lambda \in \mathbb{C}$ such that $T - \lambda I$ is not invertible
• Point spectrum $\sigma_p(T)$ includes all eigenvalues of $T$ where $Tx = \lambda x$ for some nonzero $x \in X$
• Continuous spectrum $\sigma_c(T)$ contains all $\lambda \in \sigma(T)$ such that $T - \lambda I$ is injective and has dense range, but is not surjective
• Residual spectrum $\sigma_r(T)$ consists of all $\lambda \in \sigma(T)$ such that $T - \lambda I$ is injective but its range is not dense in $X$
  • For self-adjoint operators on a Hilbert space, the residual spectrum is always empty
• Resolvent set $\rho(T)$ is the complement of the spectrum in $\mathbb{C}$ where $(T - \lambda I)^{-1}$ exists and is bounded
• Spectral radius $r(T)$ is defined as $r(T) = \sup{|\lambda| : \lambda \in \sigma(T)}$ and satisfies $r(T) \leq |T|$

Spectral Theory Fundamentals

• Spectral theory studies the properties and decompositions of linear operators based on their spectra
• For a bounded self-adjoint operator $T$ on a Hilbert space $H$, the spectrum $\sigma(T)$ is a nonempty compact subset of $\mathbb{R}$
  • The point spectrum $\sigma_p(T)$ consists of eigenvalues, and the corresponding eigenvectors form an orthonormal basis for $H$
• Spectral theorem for compact self-adjoint operators states that $T$ has a countable set of eigenvalues ${\lambda_n}$ converging to 0, and $H$ has an orthonormal basis of eigenvectors ${e_n}$
  • $T$ can be represented as $Tx = \sum_{n=1}^{\infty} \lambda_n \langle x, e_n \rangle e_n$ for all $x \in H$
• Functional calculus allows extending continuous functions $f$ on $\sigma(T)$ to bounded operators $f(T)$ on $H$
  • For a self-adjoint operator $T$, $f(T)$ is defined as $f(T)x = \sum_{n=1}^{\infty} f(\lambda_n) \langle x, e_n \rangle e_n$ for all $x \in H$
• Spectral measures and projections provide a more general framework for representing operators using integrals with respect to a measure (spectral integral)

Types of Spectra

• Approximate point spectrum $\sigma_{ap}(T)$ includes all $\lambda \in \mathbb{C}$ for which there exists a sequence ${x_n}$ in $X$ with $|x_n| = 1$ and $|(T - \lambda I)x_n| \to 0$
• Compression spectrum $\sigma_{com}(T)$ consists of all $\lambda \in \mathbb{C}$ such that the range of $T - \lambda I$ is not dense in $X$
• Essential spectrum $\sigma_{ess}(T)$ is the set of all $\lambda \in \mathbb{C}$ for which $T - \lambda I$ is not Fredholm (index not defined)
  • For self-adjoint operators, $\sigma_{ess}(T) = \sigma(T) \setminus \sigma_d(T)$, where $\sigma_d(T)$ is the set of isolated eigenvalues with finite multiplicity
• Discrete spectrum $\sigma_d(T)$ includes all isolated eigenvalues with finite algebraic multiplicity
• Weyl spectrum $\sigma_w(T)$ is the complement of all $\lambda \in \mathbb{C}$ for which $T - \lambda I$ is Fredholm with index 0
  • For self-adjoint operators, $\sigma_w(T) = \sigma_{ess}(T)$

Spectral Mapping Theorem

• Spectral mapping theorem relates the spectrum of $f(T)$ to the spectrum of $T$ for a continuous function $f$ on $\sigma(T)$
  • $\sigma(f(T)) = f(\sigma(T)) = {f(\lambda) : \lambda \in \sigma(T)}$
• For polynomials $p$, the spectral mapping theorem holds without any restrictions: $\sigma(p(T)) = p(\sigma(T))$
• For analytic functions $f$ on an open set containing $\sigma(T)$, the spectral mapping theorem also holds: $\sigma(f(T)) = f(\sigma(T))$
  • This includes functions like $e^z$, $\sin(z)$, and $\cos(z)$
• For general continuous functions $f$, the spectral mapping theorem may not hold for the full spectrum, but it does hold for the approximate point spectrum and surjectivity spectrum

Resolvent and Spectral Projections

• Resolvent operator $R(\lambda, T) = (T - \lambda I)^{-1}$ is a bounded operator defined for all $\lambda$ in the resolvent set $\rho(T)$
  • $R(\lambda, T)$ is an analytic function of $\lambda$ on $\rho(T)$ and satisfies the resolvent identity: $R(\lambda, T) - R(\mu, T) = (\mu - \lambda)R(\lambda, T)R(\mu, T)$
• Spectral projections $P_{\lambda}$ associated with an isolated eigenvalue $\lambda$ of finite multiplicity are defined using the Cauchy integral formula: $P_{\lambda} = \frac{1}{2\pi i} \oint_{\Gamma} R(z, T) dz$
  • $\Gamma$ is a simple closed curve enclosing $\lambda$ and no other points of $\sigma(T)$
  • $P_{\lambda}$ is a projection onto the eigenspace corresponding to $\lambda$ and commutes with $T$
• Riesz projections generalize spectral projections to isolated subsets of the spectrum with finite algebraic multiplicity
• Spectral projections and Riesz projections allow decomposing the space $X$ into invariant subspaces corresponding to different parts of the spectrum

Compact Operators and Their Spectra

• Compact operators are a class of bounded linear operators that map bounded sets to relatively compact sets
  • For a compact operator $T$ on an infinite-dimensional Banach space $X$, $0 \in \sigma(T)$
• Spectrum of a compact operator consists of a countable set of eigenvalues with 0 as the only possible accumulation point
  • Each nonzero eigenvalue has finite algebraic multiplicity
• Fredholm alternative theorem characterizes the solvability of the equation $(I - T)x = y$ for a compact operator $T$
  • Either $(I - T)$ is invertible, or the equation has a non-trivial solution in the null space of $(I - T)$
• Spectral theory for compact operators is particularly well-developed, with the spectrum consisting only of the point spectrum and {0}
  • Compact self-adjoint operators on a Hilbert space have an orthonormal basis of eigenvectors (Hilbert-Schmidt theorem)
• Singular value decomposition (SVD) for compact operators generalizes the eigenvalue decomposition to non-self-adjoint operators
  • SVD provides a canonical form for compact operators using singular values and singular vectors

Applications in Functional Analysis

• Spectral theory is used to study differential operators, such as the Laplace operator or Schrödinger operator in quantum mechanics
  • Eigenvalues and eigenfunctions of these operators provide information about the behavior of solutions to PDEs and the energy levels of quantum systems
• Spectral theory is essential for understanding the behavior of linear dynamical systems, such as the heat equation or wave equation
  • The spectrum of the associated operator determines the stability and long-term behavior of the system
• Spectral methods are used for numerical approximation of PDEs by expanding the solution in terms of eigenfunctions of a suitable operator
  • This leads to efficient and accurate numerical schemes for solving various problems in physics and engineering
• Spectral theory is applied in operator algebras, such as C*-algebras and von Neumann algebras, to study their structure and representations
  • The spectrum of an element in a C*-algebra provides information about its properties and the structure of the algebra
• Spectral theory is used in the study of Banach algebras, particularly in understanding the relationship between the spectrum and the algebraic properties of elements
  • The spectral radius formula connects the spectral radius of an element to its algebraic properties

Problem-Solving Techniques

• To find the spectrum of a bounded operator, first determine the point spectrum by solving the eigenvalue equation $Tx = \lambda x$
  • Then, investigate the properties of $T - \lambda I$ for $\lambda$ not in the point spectrum to classify the remaining spectral types
• For self-adjoint operators on a Hilbert space, use the spectral theorem to decompose the operator and the space using eigenvalues and eigenvectors
  • This simplifies the analysis of the operator and allows for the application of functional calculus
• For compact operators, use the Fredholm alternative to determine the solvability of equations involving the operator
  • The spectrum of a compact operator consists only of the point spectrum and {0}, which simplifies the spectral analysis
• To apply the spectral mapping theorem, first determine the spectrum of the original operator $T$
  • Then, apply the function $f$ to the spectrum to obtain the spectrum of $f(T)$, keeping in mind the limitations of the theorem for different classes of functions
• When working with spectral projections or Riesz projections, use the Cauchy integral formula to express them in terms of the resolvent operator
  • This allows for the decomposition of the space into invariant subspaces corresponding to different parts of the spectrum
• For numerical approximation of eigenvalues and eigenfunctions, use iterative methods such as the power method or the inverse iteration method
  • These methods rely on the properties of the spectrum and the resolvent operator to efficiently compute the desired spectral information