2006/11/23 by Bernhard H. Haak, Haak, Bernhard H., Jan van Neerven +1
Mathematics · #35R15 #47B10 #47D06 #60H15 #93B28 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:35R15 #msc:47B10 #msc:47D06 #msc:60H15 #msc:93B28
paper · pdf · doi:10.48550/arxiv.math/0611724
Version accepted for publication in Operators and Matrices
arxiv created 2011/10/19 · arxiv updated 2011/10/20
We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem dU(t) = AU(t) dt + B dW(t); U(0) = 0 Here, A is the generator of a strongly continuous semigroup of operators on a Banach space E, B is a bounded linear operator from a separable Hilbert space H into E, and W is an H-cylindrical Brownian motion. When A and B are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family \√λ R(λ, A)B : λ> 0\ is uniformly gamma-radonifying. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.