Operator-norm stability of the numerical range

Using the same unit-vector witness for two operators shows that their numerical ranges mutually approximate one another to within the operator-norm distance.

theorem exists_mem_numericalRange_norm_sub_le_norm_sub {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℂ E] (A B : E →L[ℂ] E) {z : ℂ} (hz : z ∈ numericalRange A) : ∃ w ∈ numericalRange B, ‖z - w‖ ≤ ‖A - B‖

Every point of W(A) is within ‖A-B‖ of a point of W(B).

theorem numericalRange_mutually_approximates {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℂ E] (A B : E →L[ℂ] E) : (∀ z ∈ numericalRange A, ∃ w ∈ numericalRange B, ‖z - w‖ ≤ ‖A - B‖) ∧ ∀ w ∈ numericalRange B, ∃ z ∈ numericalRange A, ‖w - z‖ ≤ ‖A - B‖

Symmetric form: the numerical ranges of A and B mutually approximate one another within their operator-norm distance.