Infinite Banach direct sums and diagonal $$C_0$$ C 0 -semigroups with applications to a stochastic particle system

Abstract

We show an alternative-axiomatic approach to the direct sums of countably many Banach spaces, called here B-direct sums (BDS). We develop "direct summing" of linear operators and the problems of generating the C 0 -semigroups in such "Banach direct sums of Banach spaces" by "the direct sums of the generators". We study special types of BDS, including M-BDS-the one closely related to Day's construction of a direct sum (M.M. Day, Normed linear spaces, 1973). The main abstract results of the paper concern C 0 -semigroup generation properties for diagonal operators in M-BDS type direct sums. The paper is motivated by stochastic particle systems and the problem of existing C 0 -semigroups defined by the corresponding hierarchies for marginal probability densities. We use our abstract M-BDS results to get a solution of the respective differential equation in the C 0 -semigroup sense.

Direct integrals of strongly continuous operator semigroups

“…As mentioned in the introduction, Theorem 4.2 says that decomposable generators generate decomposable C 0 -semigroups. We also have the following corollary which generalises [19,Theorem 4.4] in the Hilbert space case in a weaker way than Theorem 4.2.…”

“…r. Thus, Thus, [21, Theorem 3.2] with the additional assumption (7.2) follows by taking the counting measure. In fact, the only theory necessary in this specific discrete case is that of direct sums, covered in [19]. We also see from the above proof that if we only assume ess-sup demonstrating that the stronger condition (7.2) is indeed just as natural to assume as (7.1).…”

“…This corollary leads to another corollary, covering and generalising the counting measure case for Hilbert spaces in [19,Corollary 3.7], following its proof.…”

“…A natural question to ask is whether the sufficient conditions for a direct integral of operators to generate a C 0 -semigroup are also necessary conditions (see [19,Theorem 4.4]). A positive answer for this question is given in the following theorem.…”

“…In this article, we combine a recent approach to direct sums of C 0semigroups, done by Lachowicz and Moszyński [19] in 2016, with direct integral theory. Indeed, one can view our results as a generalisation and expansion of those in [19] when considering the setting of Hilbert spaces.…”