Research proof index

The continuous spectrum, σᶜ, of a partial linear map. A complex number z is in σᶜ T iff the range of T - z • 1 is not closed. -/ def continuousSpectrum (T : H →ₗ.[ℂ] H) : Set ℂ := {z : ℂ | ¬root.IsClosed ((T - z • 1).toFun.range : Set H)} @[inherit_doc continuousSpectrum] scoped notation "σᶜ" => continuousSpectrum lemma continuousSpectrum_eq (T...

If an operator is not closed then its resolvent set is empty.

The resolvent set is an open subset of ℂ.

For ‖ψ‖ = 1, zero standard deviation iff the eigenvector condition holds.

A centered commutator identity implies the Robertson–Schrödinger uncertainty bound.

Unitary conjugation preserves self-adjointness: if A is a self-adjoint operator on H and u : H ≃ₗᵢ[ℂ] H' is unitary, then u A u⁻¹ is self-adjoint on H'. Symmetry, dense domain, and the two deficiency surjectivities of A all transfer through u.