Factoring a bounded operator through another one #

Let A : E →L[𝕜] G and T : E →L[𝕜] F be bounded operators with a common source, where G is an inner product space and F is a Banach space. Suppose ‖T x‖ ≤ C * ‖A x‖ for every x and some nonnegative C, and the closure of the range of A admits an orthogonal projection, as it does when G is complete. Then the rule A x ↦ T x is well defined and bounded on the range of A. It extends by continuity to the closure K of that range, and by zero on Kᗮ. The canonical factor T.factorThru A : G →L[𝕜] F satisfies T.factorThru A ∘L A = T and ‖T.factorThru A‖ ≤ C. It is the unique bounded operator V with V ∘L A = T that vanishes on (range A)ᗮ. This is the elementary direction of Douglas's factorization lemma.

When ‖T x‖ = ‖A x‖ for every x, the factor is isometric on K and is a contraction. When F and G are complete inner product spaces, the adjoint carries T back to A: (T.factorThru A)† ∘L T = A. The norm equality follows from equal Gram operators A† A = T† T when E is also a complete inner product space. Applied to a self-adjoint square root A of T† T, this is the Gram-contraction factorization of T. Applied to the modulus A = |T|, the factor is the polar partial isometry of T.

For complete inner product spaces in this isometric case, the factor is a partial isometry with initial space the closure of the range of A and final space the closure of the range of T; its initial and final projections are the orthogonal projections onto these spaces. When A is a normal endomorphism, for instance self-adjoint, the initial space is (ker T)ᗮ, and the factor is the unique V with V ∘L A = T vanishing on ker T. When A has dense range the factor is an isometry. Independently of the isometric case, the factor is T A⁻¹ when A is invertible, and for an endomorphism A with ‖T x‖ ≤ ‖A x‖ the factor U satisfies ‖T - U‖ ≤ ‖A - 1‖.

Main declarations #

ContinuousLinearMap.factorThru: the canonical factor of T through A.
ContinuousLinearMap.factorThru_comp: T.factorThru A ∘L A = T under a norm bound.
ContinuousLinearMap.factorThru_compLeftInverse_apply: the factor of f.compLeftInverse g through the inclusion of the range of g sends g x to f x.
ContinuousLinearMap.eq_factorThru: uniqueness of the factor vanishing on (range A)ᗮ.
ContinuousLinearMap.adjoint_factorThru_comp: (T.factorThru A)† ∘L T = A when ‖T x‖ = ‖A x‖ for every x.
ContinuousLinearMap.isPartialIsometry_factorThru: the factor is a partial isometry when ‖T x‖ = ‖A x‖ for every x.
ContinuousLinearMap.adjoint_factorThru_comp_factorThru, ContinuousLinearMap.factorThru_comp_adjoint_factorThru: its initial and final projections.
ContinuousLinearMap.ker_factorThru, ContinuousLinearMap.range_factorThru: its kernel is (range A)ᗮ and its range is the closure of the range of T.
ContinuousLinearMap.ker_factorThru_of_isStarNormal: for normal A, its kernel is ker T.
ContinuousLinearMap.factorThru_eq_comp_inverse: for invertible A, the factor is T A⁻¹.
ContinuousLinearMap.norm_sub_factorThru_le: ‖T - T.factorThru A‖ ≤ ‖A - 1‖.
ContinuousLinearMap.adjoint_comp_self_eq_adjoint_comp_self_iff: equal Gram operators are equivalent to equal pointwise norms.
ContinuousLinearMap.exists_contraction_of_gram_eq: equal Gram operators A† A = T† T give a contraction W with contractive adjoint, W ∘L A = T, and W† ∘L T = A.

References #

J. B. Conway, A Course in Functional Analysis, 2nd ed., Springer (1990), the treatment of partial isometries and the polar decomposition.
R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.