🧐Functional Analysis Unit 6 – Bounded Linear Operators in Hilbert Spaces

Key Concepts and Definitions

• Hilbert spaces generalize Euclidean spaces to infinite dimensions while preserving the notion of an inner product
• Bounded linear operators map elements from one Hilbert space to another while maintaining linearity and boundedness
• Adjoint operators $T^*$ satisfy $\langle Tx, y \rangle = \langle x, T^*y \rangle$ for all $x$ and $y$ in the Hilbert space
• Spectrum of an operator $\sigma(T)$ consists of all scalars $\lambda$ for which $T - \lambda I$ is not invertible
  • Point spectrum $\sigma_p(T)$ contains eigenvalues of $T$
  • Continuous spectrum $\sigma_c(T)$ and residual spectrum $\sigma_r(T)$ are subsets of $\sigma(T)$
• Compact operators can be approximated by finite-rank operators and have useful spectral properties
• Self-adjoint operators $T$ satisfy $T = T^*$ and have real eigenvalues and orthogonal eigenvectors

Hilbert Space Fundamentals

• Hilbert spaces are complete inner product spaces over the field of real or complex numbers
• Inner product $\langle \cdot, \cdot \rangle$ induces a norm $\|\cdot\|$ and a metric $d(x, y) = \|x - y\|$
• Orthogonality in Hilbert spaces generalizes perpendicularity in Euclidean spaces
  • Two vectors $x$ and $y$ are orthogonal if $\langle x, y \rangle = 0$
• Orthonormal bases $\{e_n\}_{n=1}^\infty$ satisfy $\langle e_i, e_j \rangle = \delta_{ij}$ and span the Hilbert space
• Parseval's identity states that $\sum_{n=1}^\infty |\langle x, e_n \rangle|^2 = \|x\|^2$ for any $x$ in the Hilbert space
• Examples of Hilbert spaces include $L^2([a, b])$, $\ell^2(\mathbb{N})$, and $L^2(\mathbb{R}^n)$ with appropriate inner products

Types of Bounded Linear Operators

• Identity operator $I$ maps each element to itself and satisfies $IT = TI = T$ for any operator $T$
• Inverse operator $T^{-1}$ satisfies $TT^{-1} = T^{-1}T = I$ and exists only if $T$ is bijective
• Unitary operators $U$ preserve inner products and satisfy $U^*U = UU^* = I$
• Normal operators $N$ commute with their adjoints, i.e., $NN^* = N^*N$
  • Self-adjoint, unitary, and projection operators are special cases of normal operators
• Positive operators $P$ satisfy $\langle Px, x \rangle \geq 0$ for all $x$ in the Hilbert space
• Compact operators $K$ map bounded sets to relatively compact sets and have useful spectral properties

Properties of Bounded Linear Operators

• Linearity: $T(\alpha x + \beta y) = \alpha Tx + \beta Ty$ for scalars $\alpha, \beta$ and vectors $x, y$
• Boundedness: $\exists M \geq 0$ such that $\|Tx\| \leq M\|x\|$ for all $x$ in the Hilbert space
  • Operator norm $\|T\| = \sup\{\|Tx\| : \|x\| = 1\}$ quantifies the boundedness of $T$
• Continuity: Bounded linear operators are continuous, i.e., $x_n \to x \implies Tx_n \to Tx$
• Adjoints: Every bounded linear operator $T$ has a unique adjoint $T^*$ satisfying $\langle Tx, y \rangle = \langle x, T^*y \rangle$
  • Properties of adjoints: $(S + T)^* = S^* + T^*$, $(\alpha T)^* = \overline{\alpha}T^*$, and $(ST)^* = T^*S^*$
• Closed range theorem: For a bounded linear operator $T$, $\operatorname{ran}(T)$ is closed iff $\operatorname{ran}(T^*)$ is closed

Operator Norms and Continuity

• Operator norm $\|T\| = \sup\{\|Tx\| : \|x\| = 1\}$ measures the "size" of a bounded linear operator $T$
  • Equivalent definition: $\|T\| = \sup\{\|Tx\| : \|x\| \leq 1\} = \sup\{\|Tx\| / \|x\| : x \neq 0\}$
• Operator norm satisfies the triangle inequality $\|S + T\| \leq \|S\| + \|T\|$ and submultiplicativity $\|ST\| \leq \|S\|\|T\|$
• Bounded linear operators are continuous, and the operator norm quantifies the continuity
  • $T$ is Lipschitz continuous with Lipschitz constant $\|T\|$, i.e., $\|Tx - Ty\| \leq \|T\|\|x - y\|$
• Adjoint operator satisfies $\|T^*\| = \|T\|$, and $\|T^*T\| = \|T\|^2$
• Convergence in operator norm: A sequence of operators $\{T_n\}$ converges to $T$ if $\lim_{n\to\infty}\|T_n - T\| = 0$

Spectral Theory Basics

• Spectrum $\sigma(T)$ of an operator $T$ is the set of scalars $\lambda$ for which $T - \lambda I$ is not invertible
  • Eigenvalues $\lambda$ satisfy $Tx = \lambda x$ for some nonzero $x$ and form the point spectrum $\sigma_p(T)$
  • Continuous spectrum $\sigma_c(T)$ and residual spectrum $\sigma_r(T)$ are subsets of $\sigma(T) \setminus \sigma_p(T)$
• Spectral radius $r(T) = \sup\{|\lambda| : \lambda \in \sigma(T)\}$ satisfies $r(T) \leq \|T\|$
• Spectral mapping theorem: For any polynomial $p$, $\sigma(p(T)) = p(\sigma(T))$
• Compact operators have discrete spectra, with eigenvalues converging to 0 and eigenvectors forming an orthonormal basis
• Self-adjoint operators have real spectra and orthogonal eigenvectors, allowing for spectral decomposition

Applications in Functional Analysis

• Sturm-Liouville theory studies eigenvalue problems for self-adjoint differential operators
  • Applications in quantum mechanics (Schrödinger equation) and heat conduction (heat equation)
• Integral equations can be formulated as operator equations in Hilbert spaces
  • Fredholm and Volterra integral equations have applications in physics and engineering
• Fourier analysis relies on the Hilbert space structure of $L^2$ spaces and the properties of the Fourier transform
  • Applications in signal processing, image compression, and partial differential equations
• Wavelets, a generalization of Fourier analysis, provide localized basis functions for Hilbert spaces
  • Applications in signal denoising, image compression, and numerical analysis
• Quantum mechanics heavily relies on the Hilbert space formulation and the spectral theory of operators
  • Observables are represented by self-adjoint operators, and their spectra correspond to possible measurement outcomes

Common Examples and Problem-Solving Techniques

• Proving boundedness: Use the definition of the operator norm or the boundedness of the inner product
  • Example: Show that the integral operator $(Kf)(x) = \int_0^1 k(x, y)f(y)dy$ is bounded on $L^2([0, 1])$
• Finding adjoints: Use the defining property $\langle Tx, y \rangle = \langle x, T^*y \rangle$ and solve for $T^*$
  • Example: Find the adjoint of the left-shift operator $(Tx)_n = x_{n+1}$ on $\ell^2(\mathbb{N})$
• Spectrum and eigenvalues: Solve the characteristic equation $Tx = \lambda x$ or show that $T - \lambda I$ is not invertible
  • Example: Find the spectrum and eigenvalues of the multiplication operator $(Mf)(x) = xf(x)$ on $L^2([0, 1])$
• Compact operators: Show that the operator maps bounded sets to relatively compact sets or has finite-dimensional range
  • Example: Prove that the integral operator $(Kf)(x) = \int_0^1 k(x, y)f(y)dy$ with continuous $k$ is compact on $L^2([0, 1])$
• Spectral decomposition: For self-adjoint or compact operators, find eigenvalues and eigenvectors to construct the decomposition
  • Example: Find the spectral decomposition of the compact operator $(Kf)(x) = \int_0^1 \min(x, y)f(y)dy$ on $L^2([0, 1])$