4.1 Adjoint of a bounded linear operator

The adjoint of a bounded linear operator is a fundamental concept in operator theory. It extends the idea of matrix transposition to infinite-dimensional spaces, preserving inner product relationships. This powerful tool allows us to analyze operator properties and symmetries.

Adjoints play a crucial role in studying self-adjoint, normal, and unitary operators. They're essential in quantum mechanics for describing observables and enable the extension of finite-dimensional matrix concepts to infinite-dimensional spaces. Understanding adjoints is key to grasping operator behavior.

Adjoint of a Bounded Linear Operator

Definition and Properties

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• Adjoint of a bounded linear operator T on Hilbert space H represents unique operator T* satisfying $⟨Tx, y⟩ = ⟨x, T*y⟩$ for all x, y ∈ H
• T* maintains bounded linear operator status on H
• Preserves inner product structure of Hilbert space
• Analogous to conjugate transpose of matrix in finite-dimensional linear algebra (complex Hilbert spaces)
• Plays crucial role in studying self-adjoint, normal, and unitary operators
• Existence stems from Riesz representation theorem for Hilbert spaces
  • Theorem guarantees unique vector v ∈ H such that $f(x) = ⟨x, v⟩$ for any bounded linear functional f on H

Applications and Importance

• Enables analysis of operator properties through inner product relationships
• Facilitates study of operator symmetry and self-adjointness
• Crucial in quantum mechanics for describing observables and their measurements
• Used in functional analysis to characterize various operator classes (normal, unitary)
• Allows extension of finite-dimensional matrix concepts to infinite-dimensional spaces
• Provides tool for solving operator equations and analyzing spectral properties

Existence and Uniqueness of the Adjoint

Proof of Existence

• Utilizes Riesz representation theorem for existence proof
• For fixed y ∈ H, define linear functional $f_y(x) = ⟨Tx, y⟩$
• T's boundedness ensures fy boundedness as linear functional
• Apply Riesz representation theorem to fy, obtaining unique v ∈ H where $⟨Tx, y⟩ = ⟨x, v⟩$ for all x ∈ H
• Define Ty = v, establishing existence of adjoint operator T
• Process demonstrates well-defined nature of adjoint operator for all bounded linear operators on Hilbert space

Proof of Uniqueness

• Uniqueness follows from property: if two operators S and R satisfy $⟨Tx, y⟩ = ⟨x, Sy⟩ = ⟨x, Ry⟩$ for all x, y ∈ H, then S = R
• Proof by contradiction:
  • Assume S ≠ R
  • Then there exists y0 ∈ H such that Sy0 ≠ Ry0
  • By Hahn-Banach theorem, there exists x0 ∈ H such that $⟨x0, Sy0⟩ ≠ ⟨x0, Ry0⟩$
  • This contradicts the initial assumption
• Uniqueness ensures adjoint operator is well-defined and consistent

Computing Adjoints of Common Operators

Basic Operators and Their Adjoints

• Identity operator I on Hilbert space H: I* = I, as $⟨Ix, y⟩ = [⟨x, y⟩](https://www.fiveableKeyTerm:⟨x,_y⟩) = ⟨x, Iy⟩$ for all x, y ∈ H
• Left shift operator L on l²(ℤ): L* = R (right shift operator)
  • Example: L(x1, x2, x3, ...) = (x2, x3, x4, ...), L*(x1, x2, x3, ...) = (0, x1, x2, ...)
• Multiplication operator Mf on L²(X, μ): M_f* = M_f̄, where f̄ denotes complex conjugate of f
  • Example: For f(x) = x² on L²([0,1]), (Mfg)(x) = x²g(x), (M_f*g)(x) = x²g(x)

Composition and Special Cases

• Adjoint of composition: (ST)* = TS
• Finite-dimensional Hilbert spaces: computing adjoint equivalent to finding conjugate transpose of matrix representation
  • Example: For matrix A, A* is the conjugate transpose of A
• Diagonal operator adjoint obtained by taking complex conjugate of diagonal entries
  • Example: If D = diag(λ1, λ2, ...), then D* = diag(λ̄1, λ̄2, ...)
• Projection operator P onto closed subspace M: P* = P
  • Example: Orthogonal projection onto x-axis in ℝ², P(x,y) = (x,0), P* = P

Operator vs Adjoint: Norms and Spectra

Norm Relationships

• Norm equality: ||T|| = ||T*|| for any bounded linear operator T
• For normal operators (TT* = TT): ||T||² = ||T||² = ||TT|| = spectral radius of TT
• Triangle inequality: ||T + T*|| ≤ 2||T||
• Submultiplicativity: ||TT|| ≤ ||T|| · ||T|| = ||T||²

Spectral Properties

• Spectrum relationship: σ(T*) = {λ̄ : λ ∈ σ(T)}, where λ̄ denotes complex conjugate of λ
• Point spectrum (eigenvalues) of T*: complex conjugates of T's eigenvalues
  • Example: If λ is an eigenvalue of T, then λ̄ is an eigenvalue of T*
• Resolvent set of T*: complex conjugate of T's resolvent set
• Spectral radius equality: ρ(T) = ρ(T*), where ρ denotes spectral radius
• For self-adjoint operators (T = T*): spectrum is real
  • Example: Multiplication operator by real-valued function has real spectrum

Operator Classifications

• Self-adjoint operators: T = T*
  • Example: Hermitian matrices in finite-dimensional case
• Normal operators: TT* = T*T
  • Example: Unitary operators, where UU* = U*U = I
• Relationship between T and T* fundamental in classifying operators
  • Helps identify important operator classes with specific properties
  • Provides insights into operator behavior and spectral characteristics