5.1 Self-adjoint and normal operators

Self-adjoint and normal operators are key players in spectral theory. They're special types of linear operators with unique properties that make them super useful in quantum mechanics and other areas of math.

These operators have cool features like real eigenvalues for self-adjoint ones and orthogonal eigenvectors. They're the building blocks for understanding more complex operators and help us solve tricky math problems in physics and engineering.

Self-adjoint and Normal Operators

Definitions and Basic Properties

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• Self-adjoint operators satisfy $⟨Tx, y⟩ = ⟨x, Ty⟩$ for all x, y ∈ H in a Hilbert space H
• Normal operators commute with their adjoint $TT* = T*T$, where T* represents the adjoint of T
• Adjoint T* defined by $⟨Tx, y⟩ = ⟨x, T*y⟩$ for all x, y ∈ H
• Self-adjoint operators form a subset of normal operators with T = T*
• Spectrum of self-adjoint operators contains only real values
• Spectrum of normal operators can include complex values
• Both operator types play crucial roles in quantum mechanics and functional analysis (Schrödinger equation, observables)

Key Characteristics

• Eigenvalues of self-adjoint operators are always real numbers
• Eigenvectors corresponding to distinct eigenvalues of self-adjoint operators are orthogonal
• Spectral theorem for self-adjoint operators establishes existence of an orthonormal basis of eigenvectors in finite-dimensional Hilbert spaces
• Normal operators have equal norms for T and T*: $||T|| = ||T*|| = \sqrt{||T*T||}$
• Spectrum of normal operators remains unchanged under complex conjugation
• Polar decomposition of normal operator T expressed as $T = UP$, where U unitary and P positive self-adjoint, with U and P commuting

Properties of Self-adjoint and Normal Operators

Spectral Properties

• Self-adjoint operators have real-valued spectra consisting of eigenvalues and continuous spectrum
• Normal operators can have complex-valued spectra
• Eigenvectors of self-adjoint operators form an orthonormal basis (finite-dimensional case)
• Spectral theorem generalizes to infinite-dimensional spaces using spectral measures
• Continuous functional calculus allows defining functions of self-adjoint and normal operators
• Spectrum of normal operators closed under complex conjugation

Algebraic Properties

• Self-adjoint operators closed under addition and scalar multiplication by real numbers
• Normal operators closed under addition, scalar multiplication, and multiplication (when commuting)
• Commutator of two self-adjoint operators is skew-adjoint: $[A, B]^* = -[A, B]$
• Product of two commuting normal operators is normal
• Sum of two commuting normal operators is normal
• Inverse of an invertible normal operator is normal

Analytic Properties

• Norm of normal operator T equals spectral radius: $||T|| = \sup\{|\lambda| : \lambda \in \sigma(T)\}$
• Self-adjoint operators have real-valued numerical range
• Normal operators have numerical range equal to convex hull of spectrum
• Exponential of self-adjoint operator is positive and unitary
• Polar decomposition of normal operator unique and commutative

Self-adjoint vs Normal Operators

Similarities and Differences

• Self-adjoint operators always normal, but not all normal operators self-adjoint (rotation operators)
• Both types diagonalizable in appropriate basis (finite-dimensional case)
• Self-adjoint operators have real eigenvalues, normal operators can have complex eigenvalues
• Both types have orthogonal eigenvectors for distinct eigenvalues
• Normal operators include self-adjoint, skew-adjoint, and unitary operators as special cases
• Self-adjoint operators model physical observables in quantum mechanics, normal operators more general

Spectral Analysis

• Spectral theorem applies to both types, but with different implications
• Self-adjoint operators decomposed into real-valued spectral projections
• Normal operators decomposed into complex-valued spectral projections
• Functional calculus more straightforward for self-adjoint operators (real-valued functions)
• Normal operators require holomorphic functional calculus for general functions
• Simultaneous diagonalization possible for commuting normal operators, crucial in quantum mechanics (compatible observables)

Applications of Self-adjoint and Normal Operators

Quantum Mechanics

• Self-adjoint operators represent physical observables (position, momentum, energy)
• Eigenvalues of self-adjoint operators correspond to possible measurement outcomes
• Eigenvectors of self-adjoint operators represent quantum states
• Normal operators describe more general quantum transformations (unitary time evolution)
• Uncertainty principle derived from non-commuting self-adjoint operators
• Density matrices represented by positive semi-definite self-adjoint operators

Functional Analysis and Differential Equations

• Sturm-Liouville theory uses self-adjoint operators to solve boundary value problems
• Green's functions constructed using self-adjoint operators
• Spectral theory of self-adjoint operators applied to solve partial differential equations
• Fredholm alternative theorem formulated for self-adjoint operators
• Compact self-adjoint operators have discrete spectrum, important in integral equations
• Normal operators used in spectral theory of non-self-adjoint differential operators