Dissipativity under a bounded perturbation

Adding a bounded operator B to an unbounded operator A costs at most ‖B‖ of dissipativity: if A is dissipative then B + A - ‖B‖ I is again dissipative, because the triangle inequality absorbs B x into the extra ‖B‖ ‖x‖ gained by shifting the spectral parameter. Maximality is inherited too: the range condition for the perturbed operator comes from the Neumann perturbation of a resolvent point (ContinuousLinearMap.mem_resolventSet_vadd), applied at a spectral parameter large enough that ‖B‖ ‖R(lambda, A)‖ < 1.

Together these say that the Lumer--Phillips hypothesis set is stable under bounded perturbations, which is what makes the bounded perturbation theorem for generators (TauCeti.Semigroups.IsMDissipative.exists_stronglyContinuousSemigroup_generator_eq_vadd) a consequence of Lumer--Phillips.

Main results

• TauCeti.Semigroups.IsDissipative.subScalar_vadd: B + A - ‖B‖ I is dissipative.
• TauCeti.Semigroups.IsMDissipative.subScalar_vadd: B + A - ‖B‖ I is m-dissipative.

References

Engel--Nagel, One-Parameter Semigroups for Linear Evolution Equations, Section III.1; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 3, Theorem 1.1.

theorem TauCeti.Semigroups.IsDissipative.subScalar_vadd {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {A : X →ₗ.[ℝ] X} (hA : IsDissipative A) (B : X →L[ℝ] X) : IsDissipative (LinearPMap.subScalar (↑B +ᵥ A) ‖B‖)

Dissipativity survives a bounded perturbation, at the cost of shifting by ‖B‖: if A is dissipative and B is bounded, then B + A - ‖B‖ I is dissipative.

theorem TauCeti.Semigroups.IsMDissipative.subScalar_vadd {X : Type u_1} [NormedAddCommGroup X] [NormedSpace ℝ X] {A : X →ₗ.[ℝ] X} [CompleteSpace X] (hA : IsMDissipative A) (B : X →L[ℝ] X) : IsMDissipative (LinearPMap.subScalar (↑B +ᵥ A) ‖B‖)

Maximal dissipativity survives a bounded perturbation. If A is m-dissipative and B is bounded, then B + A - ‖B‖ I is m-dissipative.