4.3 Positive operators and square root of an operator

Positive operators and square roots are key concepts in operator theory. They build on self-adjoint operators, adding the condition that inner products with vectors are non-negative. This property links to important spectral characteristics and applications in quantum mechanics.

Square roots of positive operators exist and are unique. They're crucial in functional analysis and quantum theory. Computation methods vary for finite and infinite-dimensional cases, often using spectral decomposition or iterative techniques. These ideas extend our understanding of operator properties and relationships.

Positive Operators in Hilbert Spaces

Definition and Properties

• Positive operators defined as self-adjoint operators A on a Hilbert space H where $⟨Ax,x⟩ ≥ 0$ for all x in H
• Zero operator always qualifies as a positive operator satisfies the condition trivially
• Projection operators exemplify positive operators satisfying $⟨Px,x⟩ = ‖Px‖² ≥ 0$ for all x in H
• Multiplication operators Mf on L²(X,μ) defined by $(Mfg)(x) = f(x)g(x)$ are positive if and only if f(x) ≥ 0 almost everywhere
• Laplacian operator -Δ on L²(Ω) with Dirichlet boundary conditions represents a positive operator
• Covariance operator of a random vector in a Hilbert space always constitutes a positive operator

Examples in Various Contexts

• Identity operator I on any Hilbert space H serves as a positive operator ($⟨Ix,x⟩ = ‖x‖² ≥ 0$)
• Integral operators K on L²[a,b] with non-negative kernel k(x,y) ≥ 0 qualify as positive operators
• Gram matrix G of a set of vectors {v₁, ..., vₙ} in a Hilbert space forms a positive operator on ℂⁿ
• Density operators in quantum mechanics represent positive operators with trace 1
• Toeplitz operators with non-negative symbols on Hardy spaces constitute positive operators

Positive vs Self-Adjoint Operators

Spectral Characterization

• Spectral theorem for self-adjoint operators states every self-adjoint operator A has a spectral decomposition $A = ∫λdE(λ)$, where E represents the spectral measure of A
• Positive operators A satisfy $⟨Ax,x⟩ ≥ 0$ for all x in H implies $∫λd⟨E(λ)x,x⟩ ≥ 0$ for all x in H
• Non-negativity of the integral $∫λd⟨E(λ)x,x⟩$ for all x implies λ ≥ 0 almost everywhere with respect to the spectral measure
• Conversely, non-negative spectrum of A implies $∫λd⟨E(λ)x,x⟩ ≥ 0$ for all x in H, establishing A as positive
• Proof involves utilizing properties of spectral integrals and uniqueness of spectral decomposition
• Result establishes crucial connection between algebraic definition of positivity and spectral properties of the operator

Implications and Applications

• Positive operators always have non-negative eigenvalues (if they exist)
• Self-adjoint operators with non-negative eigenvalues not necessarily positive (consider finite-dimensional case)
• Positive operators have non-negative trace (if well-defined)
• Positive operators preserve order in the sense that if A and B are positive and A - B is positive, then A ≥ B
• Positive operators play crucial role in quantum mechanics (observables) and functional analysis (resolvent operators)

Square Root of Positive Operators

Definition and Existence

• Square root of positive operator A defined as positive operator B satisfying B² = A
• Existence guaranteed by functional calculus for self-adjoint operators applying function f(t) = √t to spectral decomposition of A
• Square root of A expressed as $√A = ∫√λdE(λ)$, where E represents spectral measure of A
• Square root itself constitutes positive operator and commutes with any operator commuting with A
• Existence of square root extends to bounded operators with non-negative real part (not necessarily self-adjoint)

Uniqueness and Properties

• Uniqueness of square root proved by showing if B and C are positive operators satisfying B² = A and C² = A, then B = C
• Proof of uniqueness involves using properties of positive operators and functional calculus
• Square root of positive operator always commutes with original operator ($A√A = √AA$)
• Norm of square root relates to norm of original operator: $‖√A‖ = √‖A‖$
• Square root preserves order: if A ≥ B ≥ 0, then √A ≥ √B

Computing Square Roots of Operators

Methods for Finite-Dimensional Operators

• Square root of diagonal operator D on ℓ²(N) with diagonal entries {dn} computed as diagonal operator with entries {√dn}
• Square root of positive definite matrix A computed using various methods:
  • Diagonalization: If A = PDP⁻¹, then √A = P√DP⁻¹, where √D represents diagonal matrix of square roots of eigenvalues
  • Cholesky decomposition: If A = LL*, then L represents square root of A
• Iterative methods (Newton's method, Denman-Beavers iteration) used for numerical computation of matrix square roots

Techniques for Infinite-Dimensional Operators

• Compact positive operators square root computed using spectral theorem for compact self-adjoint operators
• Infinite-dimensional spaces computation of square roots often involves approximation techniques or spectral methods
• Specific operators (Laplacian) may have explicit formulas for square root derived using Fourier analysis or other spectral techniques
• Functional calculus provides general method for computing square roots of unbounded positive operators
• Perturbation theory used to approximate square roots of operators close to known operators with computable square roots