Comparing commuting bounded-generator semigroups #
This file gives the perturbation estimate needed to compare the bounded semigroups in the
Yosida approximation. If bounded operators A and B commute and their exponentials are bounded
by M and N at nonnegative times, then
‖exp (t A) x - exp (t B) x‖ ≤ M * N * t * ‖(A - B) x‖.
The proof applies Duhamel's formula to the difference of the two exponentials. Commutativity
moves A - B through the second exponential, after which the two exponential factors contribute
their bounds. This pointwise estimate is sharper than the generic Banach-algebra bound involving
exp (t ‖A‖) and exp (t ‖B‖); that generic bound is useless for Yosida approximations because
their operator norms grow with the approximation parameter.
The two constants are kept separate rather than fixed to one: the Yosida approximations of a
dissipative operator have contractive exponentials, but under the Hille--Yosida hypotheses for a
general growth constant M they only satisfy ‖exp (t A_lambda)‖ ≤ M, and the resulting
comparison carries the factor M ^ 2.
Main result #
TauCeti.Semigroups.norm_exp_smul_sub_exp_smul_apply_le_of_commute: the pointwise comparison estimate for commuting exponentials with uniform bounds.
References #
Engel--Nagel, One-Parameter Semigroups for Linear Evolution Equations, Section II.3.5; Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Chapter 1.
If A and B commute and their bounded-generator semigroups obey the uniform bounds M and
N at nonnegative times, then their orbits differ by at most
‖exp (t A) x - exp (t B) x‖ ≤ M * N * t * ‖(A - B) x‖ for t ≥ 0.
This is the bounded Duhamel estimate in the commuting case; the contraction case is M = N = 1.