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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Top images from around the web for Definitions and Basic Properties
Ergodicity probes: using time-fluctuations to measure the Hilbert space dimension – Quantum View original
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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∗∣∣=∣∣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{∣λ∣:λ∈σ(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)