Year: 2020
Type: article
Abstract: The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup T = ( T t ) t ∈ [ 0 , ∞ ) on ℓ 1 has bounded generator if the operator T 1 is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite-dimensional matrix semigroups of growing dimension — that is, it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite-dimensional systems. In this article, we show that such attempts must necessarily fail, that is, we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators Q of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite-dimensional Perron–Frobenius theory and Gelfand's T = id theorem.
DOI: 10.1112/blms.12416
Author: Jochen Glück
Institution: University of Passau
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,000
OpenAlex ID: W3015551100