Adjoint invariance of approximation numbers

For a bounded operator T : E →L[𝕜] F between Hilbert spaces over ℝ or ℂ, the operator and its adjoint have the same approximation numbers: aₙ(T†) = aₙ(T). Taking adjoints is an isometry for the operator norm, and it preserves the maps of rank at most n (ContinuousLinearMap.rank_adjoint_le_iff), so it carries the infimum defining aₙ(T) onto the one defining aₙ(T†). Summing over the first k indices, the Ky Fan gauges agree as well.

On finite-dimensional spaces this is the adjoint invariance of singular values; in general it is the reason the operator ideals built from the approximation numbers are closed under adjoints.

Main declarations

• ContinuousLinearMap.singularValue_adjoint: aₙ(T†) = aₙ(T).
• ContinuousLinearMap.kyFanGauge_adjoint: Kₖ(T†) = Kₖ(T).

References

• A. Pietsch, Eigenvalues and s-Numbers, Cambridge Studies in Advanced Mathematics 13, Cambridge University Press (1987), Chapter 2.
• I. C. Gohberg, M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs 18, AMS (1969), Chapter II.

@[simp]

theorem ContinuousLinearMap.singularValue_adjoint {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (T : E →L[𝕜] F) (n : ℕ) : (adjoint T).singularValue n = T.singularValue n

Adjoint invariance. A bounded operator between Hilbert spaces and its adjoint have the same approximation numbers: aₙ(T†) = aₙ(T).

@[simp]

theorem ContinuousLinearMap.kyFanGauge_adjoint {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] (T : E →L[𝕜] F) (k : ℕ) : (adjoint T).kyFanGauge k = T.kyFanGauge k

Ky Fan adjoint invariance. Kₖ(T†) = Kₖ(T).