4.4 Polar decomposition

Polar decomposition is a powerful tool in operator theory, allowing us to break down any bounded linear operator into simpler parts. It's like factoring complex numbers, but for infinite-dimensional spaces. This concept helps us understand operators better.

The theorem states that we can write any operator as T = UP, where U is a partial isometry and P is positive. This split gives us insight into the operator's "direction" and "magnitude", making it easier to analyze and work with.

Polar decomposition theorem

Definition and components

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• Polar decomposition theorem states any bounded linear operator T on a Hilbert space H can be uniquely written as T = UP
  • U represents a partial isometry
  • P denotes a positive operator
• Operator P in polar decomposition given by P = $(T^*T)^{1/2}$
  • $T^*$ represents the adjoint of T
• Partial isometry U has initial space $ker(T)^\perp$ and final space $[range](https://www.fiveableKeyTerm:Range)(T)$
• Generalizes polar form of complex numbers to infinite-dimensional spaces
• For finite-dimensional spaces, U can extend to a unitary operator
  • Makes decomposition analogous to matrix polar decomposition
• Closely related to singular value decomposition
• Provides method to factor an operator into "direction" (U) and "magnitude" (P)

Mathematical formulation and properties

• Expressed mathematically as T = UP
• U satisfies $U^*U = P_{ker(T)^\perp}$ and $UU^* = P_{range(T)}$
  • $P_{ker(T)^\perp}$ and $P_{range(T)}$ are orthogonal projections
• P is unique positive square root of $T^*T$
• For self-adjoint operators, U becomes sign operator
  • U = sgn(T) where sgn(T) = $P_{ker(T)^+} - P_{ker(T)^-}$
• Norm preservation property: $\|T\| = \|P\|$
• Relation to adjoint: $(UP)^* = PU^*$

Existence and uniqueness of polar decomposition

Existence proof

• Define P = $(T^*T)^{1/2}$ using functional calculus for positive operators
• Construct U initially on range of P by U(Px) = Tx for x in H
• Extend U to all of H
  • Set U = 0 on $ker(P) = ker(T)$
• Demonstrate U is well-defined
  • Show U preserves inner product on range of P
• Verify U satisfies properties of partial isometry
  • $U^*U = P_{ker(T)^\perp}$ and $UU^* = P_{range(T)}$
• Confirm T = UP by construction

Uniqueness proof

• Assume two polar decompositions T = $U_1P_1 = U_2P_2$
• Show $P_1 = P_2$ using properties of positive operators
  • $P_1^2 = P_2^2 = T^*T$
  • Uniqueness of positive square root implies $P_1 = P_2$
• Prove $U_1 = U_2$ on range of P
  • Use partial isometry properties and uniqueness of P
• Extend equality of $U_1$ and $U_2$ to all of H
  • Both are zero on $ker(T)$

Polar decomposition of specific operators

Shift operator decomposition

• Unilateral shift operator S on $l^2$ has polar decomposition S = S$\cdot I$
• Positive part P = $(S^*S)^{1/2} = I$ (identity operator)
  • $S^*S = I$ for unilateral shift
• Partial isometry U coincides with S itself
  • S already satisfies partial isometry properties
• Bilateral shift has similar decomposition
  • U is unitary in this case

Normal operator decomposition

• For normal operators (TT* = T*T), polar decomposition simplifies
• U becomes unitary and commutes with P
• Example: rotation operator R on $\mathbb{R}^2$
  • R = UI where U is rotation matrix and I is identity

Compact operator decomposition

• Compact operators have polar decompositions expressible via singular value decomposition
• T = $\sum_{n=1}^{\infty} s_n (·,e_n)f_n$
  • $s_n$ are singular values, $e_n$ and $f_n$ are orthonormal bases
• P = $\sum_{n=1}^{\infty} s_n (·,e_n)e_n$
• U = $\sum_{n=1}^{\infty} (·,e_n)f_n$ + arbitrary partial isometry on $ker(T)$

Applications of polar decomposition

Connection to singular value decomposition

• Polar decomposition T = UP closely related to singular value decomposition T = VSW*
  • V and W are unitary, S is diagonal
• Positive operator P in polar decomposition corresponds to W*SW in SVD
• U in polar decomposition relates to VW* in SVD
• Example: matrix A = $\begin{pmatrix} 3 & -4 \\ 0 & 5 \end{pmatrix}$
  • SVD: A = $U\Sigma V^*$
  • Polar: A = ($UV^*)(V\Sigma V^*$)

Relation to spectral theorem

• For normal operators, polar decomposition derives spectral theorem
• P provides magnitude information
• U provides phase information
• Spectral measure $E_\lambda$ related to polar decomposition
  • T = $\int \lambda dE_\lambda = U \int |\lambda| dE_\lambda$

Computational and physical applications

• Used to compute operator norm: $\|T\| = \|P\|$
• Calculates other operator functions: f(T) = Uf(P) for suitable f
• Quantum mechanics applications
  • Separates magnitude and phase of wavefunctions
  • Analyzes unitary evolution operators
• Provides geometric interpretation of linear transformations
  • U represents rotation/reflection
  • P represents scaling/shearing
• Image processing: used in image registration and morphing algorithms