Stability Estimates of Markov Semigroups on Abstract States Spaces

Abstract

In order to successfully explore quantum systems which are perturbations of simple models, it is essential to understand the complexity of perturbation bounds. We must ask ourselves: How quantum many-body systems can be artificially engineered to produce the needed behavior. Therefore, it is convenient to make use of abstract framework to better understand classical and quantum systems. Thus, our investigation's purpose is to explore stability and perturbation bounds of positive C0-semigroups on abstract state spaces using the Dobrushin's ergodicity coefficient. Consequently, we obtain a linear relation between the stability of the semigroup and the sensitivity of its fixed point with respect to perturbations of C0-Markov semigroups. Our investigation leads to the discovery of perturbation bounds for the time averages of uniform asymptotically stable semigroups. A noteworthy mention is that we also prove the equivalence of uniform and weak ergodicities of the time averages Markov operators in terms of the ergodicity coefficient, which shines a new light onto this specific topic. Lastly, in terms of weighted averages, unique ergodicity of semigroups is also studied. Emphasis is put on the newly obtained results which is a new discovery in classical and non-commutative settings.

Generalized Dobrushin Coefficients on Banach Spaces

“…We merely present two results indicating possibilities in this direction. A recent paper [14] is another example supporting possible developments of this topic. We begin with recalling (cf.…”

“…It is well known that classical Markov operators, which are norm-mixing, form a dense and open subset of S 1 (see [3][4][5]19,20,23]). It has been recently extended to ordered Banach spaces satisfying some extra properties (see [10,14,17,24,27]). The category of abstract "norm mixing" elements in Banach algebras was studied in [31], but the obtained there results bring residuality (here, we deliver open denseness).…”

“…Clearly, C is a closed subset of L(X). The last theorem of the paper corresponds to [14,Theorem 3.8], where the considered semigroups belong to the C 0 class and their generators are generically unbounded [18, p. 15]. In particular, the topological structure of considered here sets C is not comparable with the one considered in [14].…”

“…The last theorem of the paper corresponds to [14,Theorem 3.8], where the considered semigroups belong to the C 0 class and their generators are generically unbounded [18, p. 15]. In particular, the topological structure of considered here sets C is not comparable with the one considered in [14]. We do realize that the presented below result is not surprising, but it is included for completeness of the topological structure of ergodic families of norm-continuous contraction semigroups, obtained with the use of the Dobrushin methods.…”

“…[1,2,5,6,10,11,29,30,35]). The topic of positive linear operators on ordered Banach spaces, which are not necessarily Banach lattices, has attracted significant attention (e.g., [8,[14][15][16][17][24][25][26][27][28]32])…”