The Kato–Rellich theorem

Let A be a self-adjoint partial linear map on a Hilbert space and V a symmetric partial linear map that is relatively bounded with respect to A: dom A ⊆ dom V and ‖V x‖ ≤ a * ‖x‖ + b * ‖A x‖ on dom A. If b < 1, then A + V is self-adjoint on dom A (IsSelfAdjoint.add_of_isRelativelyBounded). The scalar field is any RCLike field, so the real case is included.

Over ℂ the classical proof perturbs the deficiency operators A ± i μ. To treat real and complex scalars uniformly we use their real form, the pair shift B.pairShift μ (x, y) = (μ • x - B y, B x + μ • y) of TauCeti.Analysis.InnerProductSpace.LinearPMap.PairShift: a symmetric B is self-adjoint exactly when its pair shift is surjective for some μ ≠ 0. The perturbation V changes the pair shift of A by a map whose norm is at most (a / μ + b) times that of the pair shift, and a / μ + b < 1 once μ is large, so surjectivity survives (LinearMap.surjective_add_of_norm_le).

Main declarations

• IsSelfAdjoint.surjective_pairShift_add: the pair shift of A + V is surjective when a / μ + b < 1.
• IsSelfAdjoint.add_of_isRelativelyBounded: the Kato–Rellich theorem.

References

• T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer (1976), Theorem V.4.3.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness, Academic Press (1975), Theorem X.12.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer (2012), Theorem 8.5.

theorem IsSelfAdjoint.surjective_pairShift_add {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A V : E →ₗ.[𝕜] E} (hA : IsSelfAdjoint A) {a b : ℝ} (hAV : V.IsRelativelyBounded A a b) (ha : 0 ≤ a) (hb : 0 ≤ b) {μ : ℝ} (hμ : 0 < μ) (hθ : a / μ + b < 1) : Function.Surjective ⇑((A + V).pairShift μ)

The pair shift survives a relatively bounded perturbation. Let A be self-adjoint and let ‖V x‖ ≤ a * ‖x‖ + b * ‖A x‖ on dom A ⊆ dom V, with a, b ≥ 0. If μ > 0 and a / μ + b < 1, then the pair shift of A + V is surjective.

theorem IsSelfAdjoint.add_of_isRelativelyBounded {𝕜 : Type u_1} {E : Type u_2} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] {A V : E →ₗ.[𝕜] E} (hA : IsSelfAdjoint A) (hV : V.IsFormalAdjoint V) {a b : ℝ} (hAV : V.IsRelativelyBounded A a b) (hb : b < 1) : IsSelfAdjoint (A + V)

The Kato–Rellich theorem. Let A be a self-adjoint partial linear map on a Hilbert space and V a symmetric partial linear map that is relatively bounded with respect to A with constants a and b < 1, that is dom A ⊆ dom V and ‖V x‖ ≤ a * ‖x‖ + b * ‖A x‖ on dom A. Then A + V, whose domain is dom A, is self-adjoint.